Chapter 7 — Front Matter
Purpose and Overview
Chapter 6 showed you how to model the relationship between two quantitative variables using a straight line. A linear model is a powerful tool, and it fits an enormous range of real situations surprisingly well. But the world does not always grow at a fixed rate. Populations double. Medications wear off. Projectiles rise and fall. Sound fades. Light dims. None of these patterns produces a straight line when graphed, and forcing a linear model onto curved data gives answers that can mislead everyone who reads them.
This chapter extends your statistical toolkit beyond linear regression. You will study four major families of nonlinear models — quadratic, exponential, logarithmic, and power — at a level that connects naturally to your work in Algebra I and Geometry. You will learn to look at a scatter plot and ask: What kind of pattern does this data seem to follow? What model fits the shape? How do I check whether that model actually works? And how do I interpret the model's numbers in plain language that describes the real situation?
You will also be introduced to data transformation — the technique of converting variable values using logarithms or other operations to reveal hidden patterns or linearize curved relationships. Transformation connects Chapter 7 to your future work in Algebra II, Precalculus, and college-level statistics and data science.
This chapter is about judgment, not just calculation. Statistics does not require you to find a single perfect model. It requires you to find a reasonable model, check its fit carefully using residual plots, and describe its meaning and limitations honestly.
Opening Question
A researcher records the number of bacteria in a laboratory dish every hour for eight hours. At hour zero, there are 100 bacteria. By hour eight, there are more than 25,000. When she plots these values on a coordinate plane, the pattern curves sharply upward — nothing like a straight line. Should she use a linear model anyway? What happens if she does? What model might describe her data better, and how would she know? By the end of this chapter, you will be able to answer these questions fully.
Why this chapter matters. Every scientific model, financial projection, public health forecast, and engineering calculation rests on a choice of model type. When you assume linear growth where the real relationship is exponential, your predictions can be drastically wrong — not slightly off, but off by orders of magnitude. Learning to distinguish among model types, and to evaluate fit using residual plots and context, is one of the most practical skills in applied statistics. It connects directly to biology, physics, chemistry, environmental science, economics, finance, medicine, engineering, and data science — fields that all depend on choosing the right model for the data.
14Sections
9Objectives
14Key Terms
12+Worked Examples
Learning Objectives
- 1Identify when a linear model is and is not appropriate for a given data set, using scatter plots and residual plots.
- 2Describe the shape, direction, and behavior of quadratic, exponential, logarithmic, and power models.
- 3Identify which model type is suggested by the shape of a scatter plot and the context of the data.
- 4Interpret the parameters — constants and coefficients — in nonlinear equations using the language of the real situation they describe.
- 5Use residual plots to evaluate and compare model fit.
- 6Transform data using logarithms to linearize certain nonlinear relationships.
- 7Compare competing models and choose the most reasonable one using scatter plot shape, residual plots, and context.
- 8Recognize real-world applications of nonlinear models in science, finance, and technology.
- 9Identify and avoid common nonlinear modeling mistakes, especially extrapolation and model confusion.
Sections 7.1 – 7.2
When a Line Is Not Enough
Recognizing Curvature · Reviewing Linear Models · Setting the Baseline
Section 7.1
Why Not All Data Are Linear
A linear model assumes that the relationship between two variables changes at a roughly constant rate. In Algebra I, you learned that slope measures this rate. A line with slope 3 means that for every one-unit increase in x, the value of y increases by exactly 3 units — the same fixed amount every time, whether x is small or large.
This assumption works well in many situations. If a car travels at a constant speed, the relationship between time and distance is linear. If a store charges a fixed price per item, cost scales linearly with quantity. But many real situations do not work this way at all.
Consider a few concrete examples. A savings account earning compound interest grows faster and faster over time — not by the same dollar amount each year, but by a fixed percentage of whatever is currently in the account. The more money is in the account, the larger each year's gain becomes. A radioactive substance loses mass by a fixed percentage of whatever remains, so the rate of loss slows as the quantity decreases. A ball thrown upward rises, reaches a peak, and then falls — a pattern that traces a symmetric curve, not a line. The loudness of sound fades quickly as you first move away from its source and then more slowly as you continue moving — again, not a constant rate.
In each of these cases, forcing a straight line onto the data would produce a model that misleads more than it informs. The line might pass near the middle of the data while badly missing both ends. The slope would have no meaningful interpretation because the actual rate of change is not constant. And predictions based on the linear model would grow worse and worse as x moves away from the center of the data.
Recognizing When a Line Is Not Enough
A linear model is probably not the right choice when you observe any of the following signals:
| Signal | What It Suggests |
|---|
| The scatter plot curves upward or downward rather than following a straight path. | A nonlinear model is likely needed. |
| The residual plot from a linear model shows a curved pattern rather than random scatter. | The linear model is systematically missing the true shape of the data. |
| The data changes by a roughly constant percentage rather than a constant amount per step. | An exponential model may be appropriate. |
| The data rises and then falls (or falls and then rises) in a symmetric way. | A quadratic model may be appropriate. |
| Growth is rapid at first and then levels off without ever turning back down. | A logarithmic model may be appropriate. |
| The subject matter involves acceleration, decay, scaling laws, or geometric patterns. | A nonlinear model matches the physics or biology of the situation. |
Student Warning
A common mistake is to fit a linear model and report its equation without ever examining the scatter plot or residual plot. Always look at the graph first. A curved scatter plot is a clear signal that a different model type may be needed — and the scatter plot is the first check you should always make before any modeling decision.
- Explain in your own words why a linear model is not appropriate for every data set. Give two examples of real situations where you would expect the data to follow a nonlinear pattern.
- A scatter plot shows data values that increase by roughly the same amount each step. Is a linear model likely to be appropriate? Explain.
- A scatter plot shows data values that appear to double every step. Which signal from the table above does this match? What model type does it suggest?
- What does it mean for a scatter plot to be "curved upward"? Describe what this would look like on a coordinate plane.
Section 7.2
Reviewing Linear Models
Before moving to nonlinear models, it is important to revisit what a linear model looks like and what its parts mean. This review gives you a clear baseline for comparison.
A linear model takes the form:
The Linear Model
ŷ = a + bx
In this equation, x is the explanatory variable (the input), ŷ is the predicted value of the response variable (the output), b is the slope (the predicted change in y for each one-unit increase in x), and a is the intercept (the predicted value of y when x = 0).
The graph of a linear model is a straight line. Residuals from a good linear fit should be small and scattered randomly above and below zero, with no visible pattern when plotted against x. If the residual plot shows a curved pattern — one where the residuals tend to be positive at the extremes and negative in the middle, or vice versa — this is a signal that the linear model is not capturing the true shape of the data.
Graph Instruction — What a Residual Plot Should Look Like
Draw a horizontal axis labeled with the explanatory variable x. Draw a vertical axis labeled "Residual (y − ŷ)." Mark a dashed horizontal line at residual = 0. Plot each residual as a point directly above or below its corresponding x-value. A good-fitting model produces points that scatter randomly above and below the zero line with no obvious pattern — some positive, some negative, arranged without structure. A curved or wave-shaped cluster of points is a warning sign that the model type is wrong.
Example 7.2.1
Reviewing a linear model and its residuals
- ProblemA student measures how far a toy car travels (in centimeters) when released from ramps of different heights (in centimeters). She fits the linear model ŷ = 5 + 12x. Identify the slope and intercept, and give each a meaning in context. Then explain what a curved residual plot would mean here.
- ThinkIdentify the model form, then connect each coefficient to the context. Consider what curved residuals would indicate about the model.
- RepresentLinear model: ŷ = 5 + 12x, where x = ramp height (cm) and ŷ = predicted distance traveled (cm).
- Interpret
Slope = 12: For each additional centimeter of ramp height, the car is predicted to travel 12 more centimeters.
Intercept = 5: When the ramp height is 0 cm, the model predicts the car travels 5 cm. This may not be physically meaningful — a ramp of zero height cannot launch a car.
- CheckIf x = 10 cm: ŷ = 5 + 12(10) = 125 cm. Does this seem reasonable? That depends on the actual data. The intercept should be interpreted with caution whenever x = 0 falls outside the range of the data or has no physical meaning.
- ResidualsIf the residual plot from this model showed a U-shape — negative residuals for small and large ramp heights, positive in the middle — this would mean the linear model is under-predicting at the extremes and over-predicting in the middle. The true relationship may curve, and a different model should be considered.
Answer: Slope = 12 cm per cm of height; intercept = 5 cm (interpret cautiously). A curved residual plot signals that the linear model misses the true pattern.
Key Idea — Residuals as Diagnostics
Residuals are not just leftovers from a calculation — they are diagnostic tools. A random residual plot confirms good model fit. A curved or patterned residual plot reveals what the model is missing. Every time you fit a model, examine the residual plot before drawing conclusions.
Sections 7.3 – 7.7
Four Families of Nonlinear Models
Quadratic · Exponential Growth · Exponential Decay · Logarithmic · Power
Section 7.3
Quadratic Models
A quadratic model takes the form:
Quadratic Model
ŷ = ax² + bx + c
The graph is a parabola — a symmetric curve that opens upward when a > 0 and downward when a < 0. The vertex is the highest point (maximum) if the parabola opens downward, or the lowest point (minimum) if it opens upward. The x-coordinate of the vertex is found at x = −b / (2a).
You encountered parabolas in Algebra I and Geometry. In statistics, the same shape describes certain data patterns. The height of a projectile over time follows a downward-opening parabola — it rises to a peak and then falls. The braking distance of a car as speed increases follows an upward-opening curve, because doubling speed more than doubles stopping distance. The profit of a business as a function of price might peak at an optimal price point and fall on either side, tracing a downward-opening parabola.
The key difference between a quadratic model and a linear model is that a quadratic has a rate of change that is not constant — it increases or decreases continuously as x changes. This is why the parabola curves rather than running straight.
Parameters in a Quadratic Model
| Parameter | Role | Interpretation |
|---|
a | Leading coefficient | Controls direction (up if a > 0, down if a < 0) and width. Larger |a| means a narrower, steeper curve. |
b | Linear term coefficient | Shifts the vertex horizontally. Together with a, determines the location of the vertex. |
c | Constant term | The value of ŷ when x = 0. Often the starting or initial value in context. |
| Vertex | Maximum or minimum | Located at x = −b/(2a). Represents the peak height, optimal price, maximum range, or minimum cost depending on context. |
Example 7.3.1
Interpreting a quadratic model for a falling object
- ProblemA ball is dropped from a building. Its height above the ground (in meters) after t seconds is modeled by ŷ = 80 − 5t². Identify the model type, interpret the parameters, and find when the ball hits the ground.
- ThinkThe equation has the form ŷ = at² + c (no linear term, so b = 0). The leading coefficient is negative, so the parabola opens downward — height decreases as time increases.
- RepresentQuadratic model: ŷ = −5t² + 80, where t = seconds elapsed and ŷ = height in meters.
- Calculate
At t = 0: ŷ = 80 − 5(0) = 80 m (starting height)
At t = 2: ŷ = 80 − 5(4) = 80 − 20 = 60 m
At t = 4: ŷ = 80 − 5(16) = 80 − 80 = 0 m (reaches the ground)
- CheckThe ball starts at 80 m and reaches 0 m at t = 4 s. Height decreases — consistent with a falling object. The decrease accelerates: in the first 2 s the ball falls 20 m; in the next 2 s it falls 60 m. This matches the physics of free fall, where speed increases over time.
- InterpretThe initial height is 80 meters. The coefficient −5 on t² reflects the acceleration due to gravity. The ball does not fall at a constant rate; it falls faster and faster. It hits the ground after exactly 4 seconds.
Answer: ŷ = 80 − 5t². The ball starts at 80 m and hits the ground at t = 4 seconds. The negative leading coefficient confirms the parabola opens downward (height decreases).
Tip — Direction of Opening Matters
Before interpreting a quadratic model, always identify the sign of the leading coefficient a. If a > 0, the parabola opens upward and the vertex is a minimum — the model reaches its lowest predicted value at the vertex. If a < 0, the parabola opens downward and the vertex is a maximum — the model reaches its highest predicted value at the vertex. The context will tell you which makes sense.
- A ball is launched upward and its height (in feet) after t seconds is given by ŷ = −16t² + 64t + 6. (a) What is the initial height? (b) Does the parabola open upward or downward? (c) Find the time at which the ball reaches its maximum height using x = −b/(2a). (d) What is the maximum height?
- A business models its daily profit (in dollars) based on price p (in dollars) using ŷ = −2p² + 80p − 400. (a) At what price is profit maximized? (b) What is the maximum profit? (c) What does it mean that the leading coefficient is negative?
- A student claims that a quadratic model always passes through the origin. Is this correct? Explain using the parameter c.
- Explain in plain language the difference between a quadratic model with a > 0 and one with a < 0. Give a real-world context for each.
Section 7.4
Exponential Growth Models
An exponential growth model takes the form:
Exponential Growth Model
ŷ = a · bx where b > 1 and a > 0
The key feature: the quantity is multiplied by a constant factor b for each one-unit increase in x. This is completely different from a linear model, where a constant amount is added. If b = 2, the quantity doubles with each step. If b = 1.05, it grows by 5% per step. Growth accelerates — the more there is, the more it increases each period.
In a linear model, the difference between successive y-values is constant. In an exponential growth model, the ratio between successive y-values is constant. This distinction is the heart of what makes exponential growth so different from linear growth — and why it accelerates.
Connecting to Percentage Growth
If a quantity grows by a percentage r per period, then the growth factor is b = 1 + r. For example, 3% annual growth means b = 1.03. A quantity doubling each period means b = 2 (which corresponds to 100% growth per period). The initial value a is the quantity when x = 0.
Example 7.4.1
Bacteria population — exponential growth
- ProblemA laboratory dish starts with 200 bacteria. The count doubles every hour. Write an exponential growth model and predict the population after 3 hours and after 6 hours.
- ThinkDoubling means the growth factor is b = 2. The initial value is a = 200. The explanatory variable x represents hours elapsed.
- RepresentExponential growth model: ŷ = 200 · 2x, where x = hours elapsed.
- Calculate
At x = 0: ŷ = 200 · 1 = 200 bacteria
At x = 3: ŷ = 200 · 2³ = 200 · 8 = 1,600 bacteria
At x = 6: ŷ = 200 · 2⁶ = 200 · 64 = 12,800 bacteria
- CheckEach hour the count doubles: 200 → 400 → 800 → 1,600 → 3,200 → 6,400 → 12,800. The ratio between consecutive values is always 2. This confirms the exponential pattern.
- InterpretThe bacteria population starts at 200 and doubles each hour. After 6 hours, the model predicts 12,800 — 64 times the starting value. This dramatic growth illustrates why exponential models cannot be extrapolated far without caution; real populations eventually hit resource limits that slow growth.
Answer: ŷ = 200 · 2x. After 3 hours: 1,600 bacteria. After 6 hours: 12,800 bacteria.
Data Ethics Note
Exponential growth models are commonly used in public health to describe the early stages of disease spread. However, unchecked exponential growth always eventually slows because of resource limits, acquired immunity, behavior changes, or medical interventions. Using an exponential model to make long-range predictions during an epidemic — without checking whether the growth rate has already changed — can dramatically overstate the scale of future harm. Responsible modeling always states the time period for which a model is expected to apply.
- A town's population was 5,000 in 2010 and has grown by 4% per year since then. (a) Identify the initial value a and the growth factor b. (b) Write an exponential model where x = years since 2010. (c) Predict the population in 2020.
- Explain why the ratio of consecutive values in an exponential model is constant, while the difference of consecutive values in a linear model is constant. Use a small numerical example to illustrate both.
- A savings account begins with $1,000 and earns 6% annual interest, compounded annually. (a) Write an exponential model for the balance. (b) What is the balance after 10 years? (c) By what factor does the balance grow each year?
- A researcher claims that because the first 5 data points fit an exponential model well, the model can be used to predict values 50 years in the future. What statistical concern should you raise?
Section 7.5
Exponential Decay Models
An exponential decay model has the same form as an exponential growth model — ŷ = a · bx — but here the growth factor satisfies 0 < b < 1. Because b is less than 1, each step multiplies the quantity by a fraction, so the quantity decreases. The decrease is fast at first and slows over time, approaching but never quite reaching zero.
The shape of an exponential decay curve is the mirror image of a growth curve: it starts high on the left, drops steeply, and then gradually levels off to the right. This leveling-off shape is a distinctive visual signature that distinguishes exponential decay from a straight line declining at a constant rate.
Connecting to Percentage Decrease
If a quantity decreases by a percentage r per period, then b = 1 − r.
A drug losing 20% of its concentration per hour: b = 0.80
A radioactive substance losing 10% of its mass per year: b = 0.90
A car losing 18% of its value per year: b = 0.82
Half-Life
A special case of exponential decay is half-life — the time it takes for a quantity to reduce to half its value. Radioactive materials and some medications are routinely described by their half-lives in physics, environmental science, and medicine. If the half-life is h time units, then after each period of length h, the remaining amount is multiplied by 1/2.
Example 7.5.1
Medication concentration — exponential decay
- ProblemA patient takes a medication. The drug's concentration in the bloodstream begins at 400 mg/L and decreases by 25% each hour. Write a model and find the concentration after 4 hours.
- ThinkA 25% decrease each hour means 75% of the concentration remains each hour. So b = 1 − 0.25 = 0.75. The initial value is a = 400 mg/L.
- RepresentExponential decay model: ŷ = 400 · (0.75)x, where x = hours elapsed.
- Calculate
At x = 1: ŷ = 400 · 0.75 = 300 mg/L
At x = 2: ŷ = 400 · (0.75)² = 400 · 0.5625 = 225 mg/L
At x = 4: ŷ = 400 · (0.75)⁴ = 400 · 0.3164 ≈ 126.6 mg/L
- CheckEach hour retains 75% of the previous value: 400 → 300 → 225 → 168.75 → 126.6. The ratio of consecutive values is consistently 0.75. The values decrease and slow — consistent with exponential decay.
- InterpretAfter 4 hours, roughly 31.6% of the original dose remains — less than a third. Medical professionals use this kind of model to schedule dosing intervals: if the drug is only effective above a certain concentration threshold, they need to know how quickly it drops below that level.
Answer: ŷ = 400 · (0.75)x. After 4 hours, the concentration is approximately 126.6 mg/L.
Tip — Growth Factor vs. Growth Rate
The growth factor b and the growth rate r are related but different. The growth factor b is the number you multiply by each step. The growth rate r is the percentage change expressed as a decimal. For growth: b = 1 + r. For decay: b = 1 − r. If b = 0.82, then r = 0.18, meaning the quantity decreases by 18% per period. Always distinguish between the two when writing interpretations.
- A car is purchased for $28,000 and loses 18% of its value each year. (a) Write an exponential decay model. (b) What is the car's value after 3 years? After 5 years? (c) Interpret the decay factor 0.82 in plain language.
- A radioactive substance has a half-life of 10 years. A sample starts at 640 grams. Write a decay model using b = 0.5 and x = number of 10-year periods. Find the amount after 30 years.
- Explain why an exponential decay curve approaches but never reaches zero. What does this mean for interpreting models of medication concentration or radioactive decay?
- Two medications: Drug A loses 30% of its concentration per hour; Drug B loses 15% per hour. Both start at 200 mg/L. Which drug has a higher concentration after 5 hours? Show your calculations.
Section 7.6
Logarithmic Models as an Introduction
A logarithmic model takes the form:
Logarithmic Model
ŷ = a + b · ln(x) where x > 0
The function ln(x) is the natural logarithm of x — the inverse of the natural exponential function. The logarithmic function grows quickly at first but slows as x increases. The graph curves upward steeply at small values of x and then flattens out, never reaching a fixed ceiling but adding smaller and smaller amounts with each step.
If you have studied logarithms in Algebra II or Precalculus, you already know that the natural logarithm is the inverse of the exponential function ex. If you have not yet studied logarithms formally, the key idea for this chapter is the shape: logarithmic growth is rapid early and slow late, which is the opposite of exponential growth.
Logarithmic models appear in situations where something improves or grows quickly at first but gains slow down with continued input. A new student practicing a skill improves rapidly in the first few sessions but continues to gain at a diminishing rate. A biological population grows quickly when resources are abundant and food is easy to find, but slows as competition increases. The decibel scale for sound and the Richter scale for earthquake magnitude are both logarithmic — each step on the scale represents a tenfold increase in physical intensity, but our perception of loudness or shaking grows more slowly.
Example 7.6.1
A typing practice learning curve
- ProblemA student practices typing. Her speed (in words per minute) after x weeks of daily practice is approximately modeled by ŷ = 10 + 18 · ln(x). Find her predicted speed after 1, 4, 20, and 100 weeks. Describe the rate of improvement.
- ThinkThis is a logarithmic model. Improvements are large early and smaller later. Use a calculator or the fact that ln(1) = 0, ln(4) ≈ 1.386, ln(20) ≈ 2.996, ln(100) ≈ 4.605.
- RepresentLogarithmic model: ŷ = 10 + 18 · ln(x), where x = weeks of practice.
- Calculate
x = 1 week: ŷ = 10 + 18(0) = 10 wpm
x = 4 weeks: ŷ = 10 + 18(1.386) ≈ 10 + 25.0 = 35 wpm
x = 20 weeks: ŷ = 10 + 18(2.996) ≈ 10 + 53.9 = 64 wpm
x = 100 weeks:ŷ = 10 + 18(4.605) ≈ 10 + 82.9 = 93 wpm
- CheckFrom week 1 to week 4 (3 more weeks), speed jumps 25 wpm. From week 4 to week 20 (16 more weeks), it gains 29 wpm. From week 20 to week 100 (80 more weeks), it gains only 29 more wpm. Gains are large early, small late — consistent with the logarithmic shape.
- InterpretThe student improves rapidly in the first few weeks and continues to improve, but each additional week of practice produces a smaller gain than the week before. After 100 weeks, the model predicts about 93 wpm. This pattern — fast early improvement slowing over time — matches the experience of many learners.
Answer: Predicted speeds: 10, 35, 64, and 93 wpm. Each block of time produces smaller gains — the signature of logarithmic growth.
Student Warning — No True Maximum
Unlike an exponential decay curve that approaches zero, or a bounded model that levels off at a ceiling, the logarithmic model never stops growing — it always increases, just more and more slowly. If a real-world situation has a true ceiling (a maximum possible skill level, a saturation concentration, or a fixed carrying capacity), the logarithmic model eventually becomes unrealistic. Always state the range of x for which your model is meant to apply.
- A park's visitor count (in thousands) after x years of a marketing campaign is modeled by ŷ = 5 + 3 · ln(x). (a) Find predicted visitor counts at x = 1, 5, 10, and 50 years. (b) Do the gains increase, stay the same, or decrease over time? (c) What does this say about the effectiveness of the marketing campaign over time?
- Explain the difference between logarithmic growth and exponential growth in terms of how the rate of change behaves over time.
- The Richter scale measures earthquake magnitude using a logarithmic model. An earthquake of magnitude 5 releases roughly 32 times more energy than a magnitude 4 earthquake, not twice as much. How does the logarithmic relationship explain why the Richter scale numbers feel "smaller" than the actual energy differences?
Section 7.7
Power Models as an Introduction
A power model takes the form:
Power Model
ŷ = a · xb where a > 0 and b is any real number
The behavior of the power model depends entirely on the value of b. When b > 1, the curve rises faster than a line (accelerating growth). When 0 < b < 1, the curve rises but decelerates (like a square-root function). When b < 0, the curve decreases and approaches zero as x grows.
Power models appear frequently in physics, biology, and engineering, often arising from geometric or physical relationships. When a quantity depends on area, it scales with the square. When it depends on volume, it scales with the cube. These geometric relationships produce power models naturally.
| Value of b | Shape | Example Context |
|---|
b = 2 | Parabolic — rises quickly | Area of a circle as radius increases; braking distance as speed increases |
b = 3 | Cubic — rises very steeply | Volume of a sphere; weight scaling with length for fish or other animals |
b = 0.5 | Square root — rises then levels | Speed of a falling object as a function of height; wave speed and depth |
b = 1.5 | Between linear and quadratic | Kepler's Third Law: orbital period vs. distance from Sun |
b < 0 | Decreasing and approaching zero | Gravity weakening with distance; light intensity diminishing with distance |
Example 7.7.1
Weight of a fish — power model
- ProblemFor a species of freshwater fish, the weight w (in grams) is modeled by ŷ = 0.012 · L3.1, where L is the fish's length in centimeters. Find the predicted weights for fish of lengths 20 cm and 40 cm. Does doubling the length double the weight?
- ThinkThis is a power model with b = 3.1. Weight grows much faster than length because the exponent is greater than 1.
- RepresentPower model: ŷ = 0.012 · L3.1
- Calculate
At L = 20 cm: ŷ = 0.012 · (20)³·¹ ≈ 0.012 · 9,189 ≈ 110 g
At L = 40 cm: ŷ = 0.012 · (40)³·¹ ≈ 0.012 · 78,088 ≈ 937 g
- CheckThe 40 cm fish is twice the length of the 20 cm fish. Is it twice the weight? 937 / 110 ≈ 8.5 — the 40 cm fish is about 8.5 times heavier, not 2 times heavier. This makes intuitive sense: a fish twice as long is roughly twice as long, twice as wide, and twice as deep, so it occupies roughly 2³ = 8 times the volume.
- InterpretDoubling the length of a fish increases its predicted weight by a factor of about 8.5 — not 2. This is the signature of a power model with exponent near 3: the relationship is geometric, not linear. Biologists and fisheries managers use this kind of model to estimate fish populations from length measurements alone.
Answer: 20 cm fish: ≈ 110 g. 40 cm fish: ≈ 937 g. Doubling length multiplies weight by ≈ 8.5, not 2.
- The power generated by a wind turbine is approximately proportional to the cube of wind speed: ŷ = 0.5 · v³, where v is wind speed in m/s and ŷ is power in watts. (a) Find the power at v = 4 m/s, 6 m/s, and 8 m/s. (b) If wind speed doubles, by what factor does power increase?
- According to Kepler's Third Law, the orbital period T (in years) of a planet is related to its average distance from the Sun d (in astronomical units, AU) by T = d1.5. Earth is at d = 1 AU. Find the orbital periods for planets at d = 4 AU and d = 9 AU.
- A power model has b = 0.5 (a square root model). Describe how the rate of increase changes as x grows. Compare this to the linear case where b = 1.
Sections 7.10 – 7.13
Evaluating and Interpreting Models
Parameters in Context · Residual Plots · Choosing a Model · Real-World Applications
Section 7.10
Interpreting Parameters in Context
Every model has parameters — the constants and coefficients that define its specific shape and scale. Two exponential models can have the same algebraic form but behave very differently because their parameters are different. Interpreting parameters in context means connecting the numbers in the equation to the real-world situation they describe. A parameter without a context interpretation is statistically incomplete.
Exponential Parameters in Context
In the model ŷ = a · bx, the parameter a is the initial value — the quantity when x = 0. The parameter b is the growth or decay factor — the multiplier applied per unit of x. To find the growth rate in percentage terms, compute r = b − 1 for growth (so if b = 1.08, the rate is 8% per period) or r = 1 − b for decay (so if b = 0.90, the rate is 10% decrease per period).
Quadratic Parameters in Context
In the model ŷ = ax² + bx + c, the parameter c is the initial value (ŷ when x = 0). The vertex represents the maximum or minimum of the model, which often has a specific physical meaning: the peak height of a projectile, the time at which a disease reaches peak incidence, or the price that maximizes revenue. The sign of a determines whether the vertex is a maximum (a < 0) or minimum (a > 0).
Example 7.10.1
Car depreciation — interpreting exponential parameters
- ProblemA car's resale value is modeled by V̂ = 28,000 · (0.82)t, where V is value in dollars and t is years since purchase. Identify and interpret each parameter. Then find the value after 5 years.
- ThinkThis is an exponential decay model. Identify a and b, then compute the decay rate r and the value at t = 5.
- RepresentExponential decay: V̂ = 28,000 · (0.82)t
- Calculate
a = 28,000: The car's purchase price (value at t = 0) was $28,000.
b = 0.82: Each year the car retains 82% of its value from the prior year.
Decay rate r = 1 − 0.82 = 0.18: The car loses 18% of its value each year.
At t = 5: V̂ = 28,000 · (0.82)⁵ = 28,000 · 0.3707 ≈ $10,380
- CheckAfter 5 years the car is worth about $10,380, which is roughly 37% of its original price. Losing 18% per year for five years compounded is consistent with retaining about 37% — more than 5 × 18% = 90% lost, because percentages compound on a shrinking base.
- InterpretThe car was purchased for $28,000. It loses 18% of its value each year. After five years, its predicted value is approximately $10,380 — a loss of more than $17,000. Car buyers, insurance companies, and accountants use exponential depreciation models like this to estimate asset values over time.
Answer: a = $28,000 (purchase price); b = 0.82 (retains 82% per year); decay rate = 18%/year. After 5 years: ≈ $10,380.
Example 7.10.2
Projectile — interpreting the vertex of a quadratic model
- ProblemA ball is launched upward. Its height (in meters) after t seconds is ŷ = −4.9t² + 20t + 1.5. Find the maximum height and the time at which it occurs. Interpret each parameter.
- ThinkThis is a downward-opening parabola (a < 0). The vertex is the maximum. Use x = −b/(2a) with a = −4.9 and b = 20.
- Calculate
Time at vertex: t = −20 / (2 · (−4.9)) = −20 / (−9.8) ≈ 2.04 seconds
Maximum height: ŷ = −4.9(2.04)² + 20(2.04) + 1.5
= −4.9(4.162) + 40.8 + 1.5
= −20.4 + 40.8 + 1.5 ≈ 21.9 meters
- Checkc = 1.5 means the ball starts at 1.5 meters above the ground — consistent with being launched from about waist height. The maximum of 21.9 m at about 2 seconds is physically reasonable for a ball thrown upward.
- InterpretThe ball starts at 1.5 m above the ground. It reaches a maximum height of approximately 21.9 m at t ≈ 2.04 seconds, then falls back down. The coefficient −4.9 reflects the effect of gravity (half the gravitational acceleration on Earth).
Answer: Maximum height ≈ 21.9 m occurs at t ≈ 2.04 seconds. Initial height = 1.5 m (the constant c).
- A savings account is modeled by B̂ = 2,500 · (1.06)t. (a) What was the initial deposit? (b) What is the annual growth factor? (c) What is the annual growth rate as a percentage? (d) Find the balance after 8 years.
- A population model is given by P̂ = 12,000 · (0.95)t, where t is years. (a) What is the initial population? (b) What is the decay rate? (c) Is the population growing or shrinking? (d) After how many years does the model predict the population will fall below 8,000? (Use trial and error.)
- For the quadratic model ŷ = −3x² + 24x + 5: (a) Identify the initial value. (b) Does the parabola open upward or downward? (c) Find the x-coordinate of the vertex. (d) Find the maximum value. (e) Give a real-world context this model could describe.
- Why is it incomplete to say "the growth factor is 1.12" without specifying the units of x? Give an example where the same growth factor could mean very different things depending on whether x is measured in days, months, or years.
Section 7.11
Residual Plots
From Chapter 6, recall that a residual is the difference between an observed data value and the value predicted by a model:
Residual Definition
residual = y − ŷ
A positive residual means the model under-predicted the actual value. A negative residual means the model over-predicted. A good model produces residuals that are small and show no systematic pattern.
A residual plot graphs the residuals on the vertical axis against the explanatory variable x on the horizontal axis. For a well-fitting model, residuals should be approximately centered at zero with no obvious pattern — just random scatter above and below the zero line. If the residual plot shows a curved pattern, a fan shape (widening spread), or a wave, the model is not capturing the true structure of the data, and a different model should be considered.
Graph Instruction — Constructing and Reading a Residual Plot
Draw a horizontal axis labeled with the explanatory variable x. Draw a vertical axis labeled "Residual (y − ŷ)." Draw a dashed horizontal line at residual = 0. Plot each point directly above or below its corresponding x-value at the height of its residual. Then examine the pattern:
Random scatter above and below zero: Good model fit. The model captures the main structure of the data.
Curved pattern (U-shape or inverted-U): The model type is probably wrong. A linear model fitted to curved data shows this pattern. Try a nonlinear model.
Fan shape (increasing spread): The variability of the data increases as x grows — a condition called heteroscedasticity. The model may still be right in form, but its precision varies across the range of x.
Cluster or wave pattern: The model is missing a systematic structure — possibly a seasonal cycle, group effect, or trend within a trend.
Example 7.11.1
Using a residual plot to detect the wrong model type
- ProblemA student fits a linear model to population data recorded over 10 years. The residual plot shows a clear pattern: residuals are positive for years 1–3, negative for years 4–7, and positive again for years 8–10. What does this pattern mean? What should the student do?
- ThinkA pattern where residuals start positive, dip negative, and return to positive traces a U-shape — the classic signature of a linear model applied to curved data.
- AnalyzeThe U-shape means the linear model over-predicts in the early and late years and under-predicts in the middle years. Equivalently, the data values in the middle are lower than the line but the data at both ends are higher — the data curve upward at both ends and sag in the middle. Wait — or is it the reverse? Positive residuals at the edges mean the actual values are above the line at the edges, and negative residuals in the middle mean actual values are below the line in the middle. This suggests the data curve like a U — not the linear model's straight path.
- InterpretThe U-shaped residual plot strongly suggests the data follow a curved pattern — possibly exponential or quadratic — rather than a straight line. The linear model is systematically wrong, not just noisy.
- ConclusionThe student should look at the original scatter plot again and consider exponential, quadratic, or logarithmic models depending on whether the pattern shows consistent multiplication, a peak, or rapid-early-slow-late growth.
Answer: The U-shaped residual pattern indicates the linear model is not appropriate. The student should consider a nonlinear model — the residual plot identifies the problem but does not specify which nonlinear model to use.
Student Warning — Residual Plots Do Not Choose the Model
A residual plot tells you whether your current model is adequate. It does not tell you which model to use instead. To choose the right nonlinear model, you must also look at the original scatter plot, consider the context and subject matter, and possibly try several candidates. The residual plot is the final check, not the first step.
- A researcher fits a linear model to data and finds the following residuals in order of increasing x: +4.2, +3.8, +1.1, −2.4, −4.9, −3.1, +0.8, +3.5, +4.7. Describe the pattern in these residuals. Does the linear model fit well? What pattern does the residual sequence suggest?
- A student fits an exponential model to data. Her residual plot shows points scattered randomly above and below the zero line with no obvious pattern. (a) Does this suggest a good fit? (b) Does a random residual plot guarantee the exponential model is "correct"? Explain the difference between a good statistical fit and the model being right for the context.
- Explain why examining the scatter plot and the residual plot together gives better model evaluation than examining either one alone.
- A residual plot for a nonlinear model shows a fan shape — residuals are small for small x-values and larger for large x-values. What does this mean for predictions at large values of x?
Section 7.12
Choosing a Reasonable Model
Choosing a model is not guesswork. It is a structured process that combines visual analysis, contextual knowledge, and statistical checking. The following six-step process will guide you through model selection in a systematic way.
Six-Step Model Selection Process
Step 1 — Look at the scatter plot. What does the overall shape suggest? Linear? Curved upward? Leveling off? A peak?
Step 2 — Consider the context. What does the subject matter tell you? Does this kind of quantity typically grow exponentially, decay, peak, or level off?
Step 3 — Fit candidate models. Using a graphing calculator or statistical software, fit the models suggested by Steps 1 and 2. Compare how well each fits visually.
Step 4 — Examine residual plots. For each candidate model, examine the residual plot. A good-fitting model produces random residual scatter. A curved or patterned residual plot indicates a poor fit.
Step 5 — Consider the limitations. Does the model make sense throughout the range of x? Does it predict reasonable values outside the observed data? Are there conditions under which the model would break down?
Step 6 — State the model and its limitations. Report the model equation, interpret the parameters in context, and describe known limitations — especially about extrapolation.
Example 7.12.1
Compound interest — choosing between linear and exponential
- SituationA savings account starts with $1,000 and earns 6% interest per year, compounded annually. A student has recorded the balance each year for 10 years and must choose between a linear model and an exponential model.
- ThinkCompound interest grows by a fixed percentage each year, not a fixed dollar amount. This is the definition of exponential growth, not linear growth. This context alone strongly suggests the exponential model before looking at any plot.
- Scatter PlotThe data curve upward. A straight line would pass through the middle of the data but under-predict in the later years and over-predict in the earlier years — consistent with a curve, not a line.
- Residual Check (Linear)The residuals from the linear model form an upward U-shape — negative in the middle years, positive at both ends. This confirms the linear model is wrong.
- Exponential FitThe exponential model is ŷ = 1,000 · (1.06)t. Its residual plot shows random scatter near zero — a good fit.
- LimitationsThe model assumes a constant 6% annual rate. If the interest rate changes, the model becomes inaccurate. The model should not be extrapolated far beyond 10 years without checking whether the rate assumption still holds.
- ConclusionThe exponential model ŷ = 1,000 · (1.06)t is the correct choice. Forcing a linear model on compound interest data would underestimate future balances and mislead financial planning.
Answer: Exponential model ŷ = 1,000 · (1.06)t. The context (compound interest), scatter plot shape (curved upward), and residual plots (linear model: U-shape; exponential: random) all confirm the exponential model.
- For each situation, describe which step in the six-step process is most decisive, and explain why. (a) The scatter plot curves upward with accelerating growth. (b) The residual plot from an exponential model shows a systematic U-shape. (c) The context involves compound interest. (d) The model predicts a negative population in 50 years.
- A linear model for a data set has a residual plot with a clear wave pattern. An exponential model for the same data has a residual plot that is approximately random. Which model fits better? What additional information would help you decide which model is truly appropriate?
- A student fits a power model ŷ = 3.2 · x2.1 to a data set. The residual plot looks approximately random. The student wants to use this model to predict y at x = 500, but the original data only go up to x = 40. What concern should you raise?
Section 7.13
Nonlinear Models in Science, Finance, and Technology
Nonlinear models are not abstract mathematics. They are the foundation of practical work in many fields that students will encounter in college courses and careers. This section connects the models studied in this chapter to a broad range of applications.
Biology and Medicine
Bacterial population growth in early stages follows exponential growth. Drug concentration in the bloodstream decays exponentially, making dosing schedules a direct application of the exponential decay model. Many ecological populations grow logistically — a pattern where exponential growth is modified by resource limits and eventually levels off. Tumor growth can also be modeled as exponential in early stages. Epidemiologists use exponential and logistic models to predict the spread of infectious disease and evaluate the effect of interventions.
Physics and Environmental Science
The distance a free-falling object travels is quadratic in time: d = ½gt². The intensity of light decreases with the square of the distance from the source — an inverse power model. Orbital periods of planets follow Kepler's power law. Radioactive decay and the half-lives of pollutants in ecosystems follow exponential decay, making these models essential for environmental monitoring, nuclear waste management, and carbon dating in archaeology.
Finance and Economics
Compound interest, investment growth, and inflation are all exponential. Loan balances decrease exponentially (with compounding) as regular payments are made. Some financial risks — including insurance premiums for catastrophic events — are modeled with power functions. Demand curves in economics are often modeled as power functions of price.
Technology
The growth of computer processing power was described for several decades by Moore's Law, an approximately exponential doubling of transistor count every two years. Network effects — where a platform becomes more valuable as more users join — can grow super-linearly and sometimes approximately exponentially. Signal attenuation in wireless communications follows logarithmic or power models depending on the medium and frequency.
Example 7.13.1
Radioactive decay — environmental science
- ProblemA radioactive isotope has a half-life of 30 years. A sample begins with 500 grams. Write a model for the amount remaining after t years. Find the amount after 90 years.
- ThinkHalf-life means the quantity halves every 30 years. The decay factor per 30-year period is b = 0.5. If we count x in 30-year periods, we can write the model directly.
- RepresentLet x = number of 30-year half-life periods elapsed. Then ŷ = 500 · (0.5)x. Equivalently, in terms of years t: ŷ = 500 · (0.5)t/30.
- Calculate
At t = 90 years (x = 3 periods): ŷ = 500 · (0.5)³ = 500 · 0.125 = 62.5 grams
- CheckAfter 30 years: 250 g. After 60 years: 125 g. After 90 years: 62.5 g. Each period halves the amount. Three halvings from 500: 500 → 250 → 125 → 62.5. ✓
- InterpretAfter 90 years — three full half-lives — only 62.5 grams remain, one-eighth of the original 500. Nuclear engineers, environmental scientists, and archaeologists all use half-life calculations like this to predict when a radioactive material will reach a safe level or to date ancient organic materials.
Answer: ŷ = 500 · (0.5)t/30. After 90 years: 62.5 grams remain.
- A radioactive material has a half-life of 20 years. A sample starts at 800 grams. (a) Write an exponential decay model using t in years. (b) How much remains after 60 years? (c) After how many half-lives will less than 10% of the original amount remain?
- A town's population was 10,000 in 2010. It has grown at approximately 5% per year since then. (a) Write an exponential growth model. (b) Predict the population in 2030. (c) In what year does the model predict the population first exceeds 30,000? (Use trial and error or estimation.)
- A student invests $500 at 7% annual interest, compounded annually. (a) Write the model. (b) Find the balance after 15 years. (c) The rule of 72 states that doubling time ≈ 72 ÷ interest rate. Use this rule to estimate when the investment doubles. Check by computing directly.
- The velocity v (in m/s) of an object that has fallen a height h meters under gravity is given by v = √(2 · 9.8 · h) ≈ √(19.6h). This is a power model with b = 0.5. Compute v for h = 5, 20, 80, and 320 meters. Does doubling the fall height double the velocity? Explain why or why not.
Section 7.14
Common Nonlinear Modeling Mistakes
Six Major Errors and How to Avoid Them
Section 7.14
Common Nonlinear Modeling Mistakes
Nonlinear modeling introduces new opportunities for error that go beyond simple arithmetic mistakes. These are conceptual errors — misunderstandings about model types, model fit, or the responsible use of models for prediction. Learning to recognize and avoid them is as important as learning to build the models themselves.
Mistake 1 — Assuming All Curved Data Are Exponential
"The scatter plot curves upward, so I'll use an exponential model."
Not every upward curve is exponential. A quadratic also curves upward. A power model can curve upward. A logarithmic model curves upward and then levels off. Before labeling any curved data as exponential, examine the residual plot of the exponential model and compare it to other candidates. Check whether consecutive y-values have a constant ratio (exponential) or whether the curve has a peak (quadratic) or leveling (logarithmic).
Mistake 2 — Reporting a Model Without Checking Residuals
"The model looks close to the data on the scatter plot, so it fits."
A model can appear visually close to the data while still producing a systematic residual pattern that reveals important misfit. Always examine the residual plot after fitting a model. A model that looks "close enough" on a scatter plot may still fail the residual check — and a failed residual check means the model type is wrong, not just imprecise.
Mistake 3 — Extrapolating Nonlinear Models Too Far
"The exponential model fits well for 10 years, so I'll use it to predict 50 years from now."
Nonlinear models — especially exponential ones — can produce extreme values very quickly when extrapolated beyond the range of the observed data. Real populations cannot grow without limit. Real drug concentrations cannot fall below zero. Exponential growth always eventually slows because of resource limits, saturation, or physical constraints. Always state the range of x for which the model is reasonable, and treat predictions far outside that range with great skepticism.
Mistake 4 — Ignoring Units and Context When Interpreting Parameters
"The growth factor is 1.08."
This statement is incomplete. A growth factor of 1.08 means very different things depending on whether x is measured in days, months, years, or decades. Always state the units of x when interpreting a growth or decay factor. "The quantity grows by 8% per year" is a complete interpretation; "the growth factor is 1.08" is not.
Mistake 5 — Using an Exponential Model When Growth Eventually Slows
"The early data fit an exponential model, so the exponential model describes the full pattern."
Early-stage growth in biology, social media adoption, and technology often appears exponential, but real systems eventually run into limits — resource constraints, market saturation, or physical boundaries. Using an exponential model to predict long-term behavior in such systems overstates growth. Recognizing when to switch from an exponential to a bounded or logistic model is an important judgment skill in applied statistics.
Mistake 6 — Confusing Correlation r with Model Type Choice
"The correlation r = 0.91, so the linear model fits well."
A high correlation coefficient r does not guarantee that a linear model is appropriate. If the data are curved, a linear model might still show a moderately high r while missing the true shape. Correlation measures linear association only — it does not evaluate model type. Always base model selection on the scatter plot shape, the residual plot, and the context of the data — not on r alone.
Chapter 7 — Back Matter
Chapter Summary
- Sec 7.1Linear models assume a constant rate of change. When data curve, have a constant ratio between consecutive values, peak and fall, or level off, a linear model is not appropriate. Look at the scatter plot first, and examine residual plots to confirm.
- Sec 7.2A linear model ŷ = a + bx serves as the baseline. Residuals from a good linear fit are randomly scattered around zero. A curved residual pattern signals that a nonlinear model is needed.
- Sec 7.3Quadratic models ŷ = ax² + bx + c produce parabolas. When a < 0 the parabola opens downward with a maximum vertex; when a > 0 it opens upward with a minimum vertex. The vertex occurs at x = −b/(2a).
- Sec 7.4Exponential growth models ŷ = a · bx with b > 1 describe quantities that multiply by a constant factor per step. The initial value is a; the growth factor is b; the growth rate is r = b − 1. Consecutive ratios are constant.
- Sec 7.5Exponential decay models have 0 < b < 1. The decay rate is r = 1 − b. The curve decreases and approaches but never reaches zero. Half-life is a special case: after each half-life period, the quantity is multiplied by 0.5.
- Sec 7.6Logarithmic models ŷ = a + b · ln(x) describe rapid early growth that slows over time. The model has no true maximum — it always grows, but more and more slowly. Useful for learning curves, diminishing returns, and sensory perception scales.
- Sec 7.7Power models ŷ = a · xb produce a variety of curve shapes depending on b. Common in physics, biology, and engineering where geometric or physical scaling laws produce natural power relationships.
- Sec 7.8Data transformation uses logarithms to linearize curved relationships. Taking ln(y) linearizes exponential data; taking ln(x) and ln(y) linearizes power data. The slope of the transformed line identifies the growth factor or exponent.
- Sec 7.9Comparing model shapes requires asking whether the data go up or down, whether the rate accelerates or decelerates, whether there is a peak, and whether the pattern levels off. Context guides model choice alongside visual inspection.
- Sec 7.10Parameters must be interpreted in context. In exponential models: a is the initial value, b is the factor, and r = b − 1 or 1 − b is the rate. In quadratic models: c is the initial value, and the vertex gives the maximum or minimum and when it occurs.
- Sec 7.11Residual plots (y − ŷ vs. x) diagnose model fit. Random scatter near zero indicates good fit. A curved pattern indicates the model type is wrong. Residual plots do not specify which model to use — they only identify failures.
- Sec 7.12Six-step model selection: (1) examine scatter plot; (2) consider context; (3) fit candidates; (4) check residual plots; (5) consider limitations; (6) state the model and its limitations. Choosing a model requires judgment, not just computation.
- Sec 7.13Nonlinear models appear throughout biology, medicine, physics, environmental science, finance, economics, and technology. Recognizing which model applies in which field connects statistical modeling to real-world scientific and professional practice.
- Sec 7.14Six major mistakes: assuming all curves are exponential; skipping residual checks; extrapolating too far; omitting units from parameter interpretations; applying exponential models beyond their valid range; confusing correlation r with model type suitability.
Key Terms Review
nonlinear model
A statistical model whose equation produces a curved graph rather than a straight line. Includes quadratic, exponential, logarithmic, and power models.
quadratic model
A model of the form ŷ = ax² + bx + c. Its graph is a parabola with a maximum (a < 0) or minimum (a > 0) at the vertex.
exponential growth
A model ŷ = a · bx with b > 1. The quantity is multiplied by a constant factor b per unit of x. Consecutive ratios are constant.
exponential decay
A model ŷ = a · bx with 0 < b < 1. The quantity decreases by a constant percentage per unit of x and approaches but never reaches zero.
logarithmic model
A model ŷ = a + b · ln(x). Describes quantities that grow quickly at first and then more slowly, without a fixed maximum.
power model
A model ŷ = a · xb. The shape depends on b. Common in physics and biology where geometric scaling produces natural power relationships.
growth factor
The base b in an exponential model. The multiplier applied to the quantity per unit of x. Growth rate r = b − 1 for growth; r = 1 − b for decay.
half-life
The time required for an exponentially decaying quantity to reduce to half its current value. After each half-life, the quantity is multiplied by 0.5.
residual
The difference y − ŷ between an observed data value and the value predicted by a model. Positive residuals indicate under-prediction; negative indicate over-prediction.
residual plot
A graph of residuals on the vertical axis versus x on the horizontal axis. Random scatter near zero indicates good model fit. Curved patterns indicate the wrong model type.
data transformation
Converting variable values using a mathematical operation — such as taking logarithms — to reveal patterns or linearize a nonlinear relationship.
linearization
Transforming variables so that a nonlinear relationship becomes approximately linear. Taking ln(y) linearizes exponential data; taking ln(x) and ln(y) linearizes power data.
parameters
The constants and coefficients in a model equation — such as a, b, and c — that determine its specific shape, scale, and behavior.
extrapolation
Using a model to predict values far outside the range of the observed data. Carries much greater risk of error than interpolation and must be described with appropriate limitations.
Concept Review Questions
- Explain in your own words why a linear model is not appropriate for every data set. Give two examples of real situations where you would expect the data to follow a nonlinear pattern.
- What is the key feature that distinguishes exponential growth from linear growth? Answer in terms of what remains constant in each model.
- A scatter plot shows values that increase steeply and then level off — never turning downward. Which model types could describe this pattern? What is the key difference between them?
- A residual plot from a linear model shows a clear U-shape. What does this tell you about the linear model? What should the analyst do next?
- Explain the difference between the growth factor b and the growth rate r in an exponential model. If b = 1.12, what is r?
- What is meant by the half-life of a radioactive substance? Write a general exponential decay model using half-life h and initial value a.
- A student plots x vs. ln(y) and finds the points follow a straight line with slope 0.45. What can the student conclude about the original relationship between x and y? What is the value of the growth factor b?
- Why does a high correlation coefficient r not guarantee that a linear model is appropriate? What should be examined instead?
Skill Practice
For each situation, identify the most likely model type (linear, quadratic, exponential growth, exponential decay, logarithmic, or power) and justify your choice.
- A ball thrown upward: height vs. time.
- Compound interest: account balance vs. years.
- Drug clearance: concentration vs. hours.
- Learning speed: skill level vs. practice days.
- Constant-speed car: distance vs. time.
- Fish weight vs. fish length.
- Earthquake energy vs. Richter scale magnitude.
- Cooling coffee: temperature vs. minutes.
- A population model is P̂ = 5,000 · (1.04)t. (a) Initial population? (b) Annual growth rate? (c) Population after 10 years?
- A car's value is V̂ = 35,000 · (0.85)t. (a) Purchase price? (b) Annual percentage loss? (c) Value after 6 years?
- For ŷ = −4.9t² + 20t + 2: (a) Initial value? (b) Time at maximum? (c) Maximum value?
- For ŷ = 12 + 9 · ln(x): (a) Value at x = 1? (b) Value at x = e ≈ 2.718? (c) How does improvement change over time?
- Residuals from a linear model in order of increasing x are: −5, −3, 0, +4, +8, +13, +19. Describe the pattern. Does the linear model fit well? What type of model might fit better?
- Residuals from an exponential model scatter randomly above and below zero with no pattern. Does this confirm the exponential model is correct? Explain what else you would need to check.
- Describe what a fan-shaped residual plot looks like and what concern it raises for making predictions.
- Data: x = 0, 1, 2, 3, 4 and y = 50, 110, 242, 533, 1173. Compute ln(y) for each point. Compute the differences in ln(y). Does the pattern suggest exponential growth? Estimate the growth factor b.
- A student plots ln(x) vs. ln(y) for a data set and finds a straight line with slope 2.3 and intercept 0.69. (a) What kind of model does this suggest? (b) What is the exponent b? (c) Since ln(a) = 0.69, find a. (d) Write the original power model.
Applications
- A radioactive material has a half-life of 20 years and starts at 800 grams. (a) Write a model using t in years. (b) How much remains after 60 years? (c) After how many half-lives will less than 10% remain?
- A town's population was 10,000 in 2010 and grows at 5% per year. (a) Write the model. (b) Predict the 2030 population. (c) In what year does the model first predict a population above 30,000?
- An investor puts $500 into an account at 7% annual compound interest. (a) Write the model. (b) Balance after 15 years? (c) Use the Rule of 72 to estimate doubling time, then verify directly.
- A student records typing speed: Week 1: 12 wpm; Week 4: 28 wpm; Week 16: 45 wpm; Week 64: 62 wpm. Does the pattern look linear or logarithmic? Explain using the differences in wpm gained per period of practice.
- Wind turbine power: ŷ = 0.5v³. (a) Find power at v = 3, 6, 9 m/s. (b) If wind speed doubles, by what factor does power increase? (c) What type of model is this?
- A medication is administered at 320 mg/L and loses 30% per hour. (a) Write the decay model. (b) Find concentration at hours 2, 4, and 6. (c) After how many hours does concentration fall below 50 mg/L?
Error Analysis
Each problem shows a student's reasoning or work. Identify the error and write a correct response.
Problem 1
"The scatter plot curves upward, so I'll use an exponential model. I don't need to check the residual plot."
What is wrong with this reasoning? What should the student do before committing to the exponential model?
Problem 2
A student interprets b = 1.15 in an exponential model as "the quantity grows by 15%." They do not mention any units.
What is missing from this interpretation? Write a complete version.
Problem 3
A student fits a linear model to data and reports r = 0.93. They conclude: "The model fits very well because r is close to 1."
Is this a complete or correct conclusion? What else should the student examine?
Problem 4
A student uses an exponential growth model fitted to 5 years of data to predict population 100 years from now. They report the prediction without any caveats.
What concern should a careful analyst raise? What should the student add to make this prediction responsibly reported?
Problem 5
A student says: "In the model ŷ = −5t² + 20t + 3, the value c = 3 is the maximum height."
What is the error? What does c actually represent, and how do you find the maximum height?
Problem 6
A student plots x vs. ln(y) and finds a straight line. They conclude: "The data definitely follow an exponential model."
Is this conclusion too strong? What does the transformed linear pattern confirm, and what additional checking is still appropriate?
Chapter 7 Checkpoint
Chapter 7 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 7
Section 1 — Vocabulary and Concepts
- Define growth factor and growth rate. If b = 1.09 in an exponential model, what is the growth rate?
- Explain what a curved pattern in a residual plot indicates about the current model.
- What is linearization, and why is it useful in statistical modeling?
Section 2 — Model Identification
- A hospital tracks flu admissions for 6 weeks. The counts are approximately: 40, 87, 189, 411, 893, 1940. Do consecutive counts suggest linear or exponential growth? How can you tell?
- A student practices guitar. Her accuracy (%) improves quickly at first but slows: 30%, 52%, 61%, 67%, 71%, 74%, 76%. Which model type — exponential growth, quadratic, or logarithmic — best describes this pattern? Explain.
Section 3 — Interpretation
- A model for car depreciation is V̂ = 24,000 · (0.88)t. (a) What was the car's purchase price? (b) What is the annual decay rate as a percentage? (c) Find the value after 4 years.
- A quadratic model for a launched ball is ŷ = −5t² + 30t + 2. (a) What is the initial height? (b) At what time is the maximum height reached? (c) What is the maximum height?
Section 4 — Residuals and Model Choice
- A student fits a linear model to 10 data points. The residuals in order of increasing x are: +6, +4, +1, −2, −5, −7, −5, −2, +2, +5. (a) Describe the pattern. (b) Does the linear model fit well? (c) What type of model might fit better?
- A student fits an exponential model to data. The residual plot shows random scatter near zero. Does this confirm the exponential model is perfect? What else would you check before reporting conclusions?
Section 5 — Applications and Error Analysis
- A savings account starts at $2,000 and earns 5% annual interest compounded annually. (a) Write the model. (b) Find the balance after 12 years. (c) Approximately how many years does it take to double? (Use the Rule of 72.)
- A student reports: "My exponential model has r = 0.97, so it fits better than any other model." Identify at least two problems with this statement and explain how to evaluate model fit more carefully.
Bridge Note — Looking Ahead
Chapter 8: Probability Foundations
Chapter 7 completed Part III of this book — the section on relationships, models, and algebraic statistics. You now have a complete modeling toolkit: linear models from Chapter 6, and quadratic, exponential, logarithmic, and power models from Chapter 7. These tools will appear throughout the rest of your mathematical education. In Algebra II and Precalculus, you will develop a deeper algebraic understanding of exponential and logarithmic functions — including the laws of logarithms and inverse functions. In Calculus, you will find exact maxima and minima analytically, and you will study rates of change of exponential and logarithmic functions with precision. In Data Science, you will fit these models to much larger and messier data using software, and you will evaluate fit with additional criteria beyond residual plots. In Biology, Chemistry, Physics, and Environmental Science, these models will appear in context after context — because the real world does not always change at a constant rate.
Chapter 8 begins Part IV: Probability Foundations. Probability is the language of uncertainty — and statistics, at its core, is the discipline of reasoning under uncertainty. You will learn how to assign probabilities to outcomes, calculate probabilities of events, represent sample spaces with tables and diagrams, and use probability rules to reason about risk, games, science, and decision-making.
MGU Library Connection
- Appendix G: Correlation and Regression Reference — summary of linear and nonlinear model types, forms, and graph shapes
- Appendix B: Geometry and Graphing Review — parabolas, coordinate graphs, and transformation basics
- Appendix F: Summary Statistics Formula Sheet — residual definitions and model comparison criteria
- Algebra I, Chapter 6 and 7: Linear equations and function models — prerequisites for regression and model building
- Algebra II: Exponential and logarithmic functions — deeper algebraic treatment of the models introduced here
- Precalculus: Transformations, inverse functions, and function composition — extensions of linearization
- Data Science Foundations: Fitting nonlinear models with software, cross-validation, and model comparison metrics
- MGU Dictionary entries: exponential function, logarithm, power function, residual, model fit, half-life, data transformation