The Central Idea
Consider three functions:
\[ f(x)=5, \] \[ g(x)=3x, \]and
\[ h(x)=3x+5. \]All three are simple, but they represent three different structures.
- \(f(x)=5\) is constant: the output never changes.
- \(g(x)=3x\) is linear: the output changes at a constant rate and the graph passes through the origin.
- \(h(x)=3x+5\) is affine: it has the same constant rate of change as \(g\), but it has been translated vertically.
These distinctions let us separate rate from starting value.
A Note About the Word “Linear”
In many school algebra courses, every function of the form
\[ y=mx+b \]is called a linear function.
That convention is common and useful in elementary algebra.
In more precise mathematical language, however:
\[ \boxed{f(x)=mx} \]is linear, while
\[ \boxed{f(x)=mx+b} \]with \(b\ne0\) is affine.
Malone Math will recognize the common school usage while also preserving this more structural distinction.
Why the Distinction Matters
The difference becomes important later because a true linear map preserves the origin:
\[ f(0)=0. \]An affine function generally does not:
\[ f(0)=b. \]Affine functions can be understood as linear functions followed by a translation.
Constant Functions
A constant function has the form
\[ \boxed{f(x)=c} \]where \(c\) is fixed.
No matter which allowed input is chosen, the output remains the same.
Constant Function Example
Let
\[ f(x)=7. \]Then
\[ f(-10)=7, \] \[ f(0)=7, \] \[ f(100)=7. \]The input changes, but the output does not.
Graph of a Constant Function
The graph of
\[ f(x)=c \]is a horizontal line at height \(c\).
Its slope is
\[ \boxed{0}. \]Rate of Change of a Constant Function
For two distinct inputs \(x_1\) and \(x_2\),
\[ \frac{f(x_2)-f(x_1)}{x_2-x_1} = \frac{c-c}{x_2-x_1} = 0. \]A constant function has zero average rate of change over every interval in its domain.
Constant Does Not Mean Zero
The function
\[ f(x)=8 \]is constant even though its output is not zero.
The special constant function
\[ f(x)=0 \]is called the zero function.
The Zero Function Is Special
The zero function
\[ f(x)=0 \]is both constant and linear.
It passes through the origin and satisfies the structural properties of a linear map.
A nonzero constant function is affine, but not linear in the stricter sense.
Linear Functions
In the structural sense used here, a real one-variable linear function has the form
\[ \boxed{f(x)=mx} \]where \(m\) is a constant.
The Parameter \(m\)
The parameter
\[ m \]controls the rate of change.
If
\[ m>0, \]the function is increasing.
If
\[ m<0, \]the function is decreasing.
If
\[ m=0, \]the function becomes the zero function.
Linear Functions Pass Through the Origin
For
\[ f(x)=mx, \]evaluate at zero:
\[ f(0)=m(0)=0. \]Therefore, every linear function of this form contains the point
\[ \boxed{(0,0)}. \]Linear Functions and Direct Proportion
The relationship
\[ y=mx \]is also a direct proportionality.
The ratio
\[ \frac{y}{x} \]is constant for nonzero \(x\):
\[ \frac{y}{x} = \frac{mx}{x} = m. \]So the slope, unit rate, and constant of proportionality are the same number.
Example: Constant Speed
Suppose a vehicle travels at a constant speed of \(60\) miles per hour and begins measuring distance from zero.
Then
\[ d(t)=60t. \]This is a linear function.
The slope
\[ 60 \]is both the rate of change and the constant of proportionality.
Affine Functions
An affine function has the form
\[ \boxed{f(x)=mx+b}. \]It combines:
- a linear scaling \(mx\);
- a translation by \(b\).
Affine as Linear Plus Translation
Write
\[ f(x)=mx+b. \]Define the linear part
\[ L(x)=mx. \]Then
\[ \boxed{f(x)=L(x)+b}. \]The parameter \(b\) translates the linear graph vertically.
The Meaning of \(b\)
For
\[ f(x)=mx+b, \]evaluate at zero:
\[ f(0)=b. \]So \(b\) is the \(y\)-intercept.
The graph contains the point
\[ \boxed{(0,b)}. \]Example: Taxi Fare
Suppose a taxi charges a \$4 starting fee plus \$2.50 per mile.
The total cost can be modeled by
\[ C(m)=2.50m+4. \]The rate of change is
\[ 2.50\text{ dollars per mile}, \]and the initial value is
\[ 4\text{ dollars}. \]This is affine rather than directly proportional because the cost is already positive at \(m=0\).
Constant Rate of Change
Both linear and affine functions have constant rate of change.
For
\[ f(x)=mx+b, \]take two inputs \(x_1\) and \(x_2\):
\[ \frac{f(x_2)-f(x_1)}{x_2-x_1} = \frac{(mx_2+b)-(mx_1+b)}{x_2-x_1}. \]The \(b\)-terms cancel:
\[ = \frac{m(x_2-x_1)}{x_2-x_1} = m. \]Thus,
\[ \boxed{\text{average rate of change}=m} \]on every interval.
The Intercept Does Not Change the Rate
Compare
\[ f(x)=3x \]and
\[ g(x)=3x+10. \]Both have slope
\[ 3. \]The second graph is simply shifted upward by \(10\).
The rate is unchanged.
Constant Functions Are Affine
Set
\[ m=0 \]in the affine form:
\[ f(x)=0x+b=b. \]So every constant function is an affine function.
But Most Constant Functions Are Not Linear
For a function to be linear in the strict sense, it must satisfy
\[ f(0)=0. \]A constant function
\[ f(x)=c \]has
\[ f(0)=c. \]Therefore, it is linear only when
\[ \boxed{c=0}. \]Hierarchy of These Functions
In one-variable real algebra:
- the zero function is both constant and linear;
- every linear function is affine;
- every constant function is affine;
- not every affine function is linear;
- not every affine function is constant.
Slope-Intercept Form
The familiar equation
\[ \boxed{y=mx+b} \]is slope-intercept form.
It exposes two important features immediately:
- \(m\): slope or constant rate of change;
- \(b\): \(y\)-intercept or initial value.
Slope as Rise Over Run
For two points
\[ (x_1,y_1) \]and
\[ (x_2,y_2), \]with \(x_1\ne x_2\), slope is
\[ \boxed{ m= \frac{y_2-y_1}{x_2-x_1} }. \]This compares output change to input change.
Slope Is a Rate
The units of slope are
\[ \frac{\text{output units}}{\text{input units}}. \]Examples include:
\[ \frac{\text{miles}}{\text{hour}}, \] \[ \frac{\text{dollars}}{\text{item}}, \]or
\[ \frac{^\circ\text{C}}{\text{minute}}. \]Positive Slope
If
\[ m>0, \]then larger inputs produce larger outputs.
The function is strictly increasing on its real domain.
Negative Slope
If
\[ m<0, \]then larger inputs produce smaller outputs.
The function is strictly decreasing.
Zero Slope
If
\[ m=0, \]then
\[ f(x)=b. \]The graph is horizontal and the function is constant.
Tables of Affine Functions
An affine function has equal output differences when the inputs increase by equal amounts.
For example:
| \(x\) | \(f(x)\) |
|---|---|
| 0 | 5 |
| 1 | 8 |
| 2 | 11 |
| 3 | 14 |
The first differences are constant:
\[ 8-5=3, \] \[ 11-8=3, \] \[ 14-11=3. \]This suggests
\[ f(x)=3x+5. \]Equal Differences Require Equal Input Steps
Constant output differences are meaningful only when the input differences are also equal.
If the input spacing is irregular, compare rates instead:
\[ \frac{\Delta y}{\Delta x}. \]Irregularly Spaced Table Example
| \(x\) | \(y\) |
|---|---|
| 1 | 5 |
| 3 | 9 |
| 8 | 19 |
From \(x=1\) to \(x=3\):
\[ \frac{9-5}{3-1}=2. \]From \(x=3\) to \(x=8\):
\[ \frac{19-9}{8-3}=2. \]The constant rate suggests an affine function with slope \(2\).
Finding the Intercept from a Point
Suppose the slope is
\[ m=2 \]and the graph contains
\[ (1,5). \]Use
\[ y=mx+b. \]Substitute:
\[ 5=2(1)+b. \]So
\[ b=3. \]Therefore,
\[ \boxed{f(x)=2x+3}. \]Point-Slope Form
If a line has slope \(m\) and passes through
\[ (x_1,y_1), \]then
\[ \boxed{ y-y_1=m(x-x_1) }. \]This is point-slope form.
Why Point-Slope Form Works
Slope gives
\[ m= \frac{y-y_1}{x-x_1}. \]Multiply by
\[ x-x_1: \] \[ y-y_1=m(x-x_1). \]Point-slope form is simply the slope definition rewritten as an equation.
Two-Point Example
Find the affine function through
\[ (2,7) \]and
\[ (5,16). \]First find slope:
\[ m= \frac{16-7}{5-2} = \frac93 = 3. \]Use point-slope form:
\[ y-7=3(x-2). \]Simplify:
\[ y=3x+1. \]So
\[ \boxed{f(x)=3x+1}. \]One Point Is Not Enough
Infinitely many different affine functions can pass through one point.
To determine one unique nonvertical line, we generally need:
- two distinct points;
- or one point and the slope;
- or equivalent information.
Two Points with the Same \(x\)-Coordinate
Suppose the points are
\[ (4,1) \]and
\[ (4,8). \]They determine the vertical line
\[ x=4. \]This relation is not a function \(y=f(x)\), because the input \(x=4\) corresponds to more than one \(y\)-value.
Not Every Straight Line Is a Function of \(x\)
Every nonvertical line can be written
\[ y=mx+b. \]A vertical line cannot.
So “straight line” and “function of \(x\)” are related but not identical ideas.
Standard Form of a Line
A line can also be written
\[ \boxed{Ax+By=C}. \]If
\[ B\ne0, \]solve for \(y\):
\[ y=-\frac{A}{B}x+\frac{C}{B}. \]This reveals the slope
\[ -\frac{A}{B}. \]Affine Relations in Two Variables
The equation
\[ Ax+By=C \]is an affine relation in the plane.
When \(B\ne0\), it defines \(y\) as an affine function of \(x\).
When \(B=0\), the equation describes a vertical line instead.
Linear Equations Versus Linear Functions
The phrase linear equation is often used for equations whose variables appear only to the first power.
That terminology is broader than the stricter function-theory meaning of a linear map.
Context matters.
Zero of an Affine Function
For
\[ f(x)=mx+b, \qquad m\ne0, \]a zero satisfies
\[ mx+b=0. \]So
\[ \boxed{x=-\frac{b}{m}}. \]The \(x\)-Intercept
The corresponding \(x\)-intercept is
\[ \boxed{ \left(-\frac{b}{m},0\right) }. \]If \(m=0\), a constant function either has no zeros or is the zero function.
Constant Function Zeros
If
\[ f(x)=c \]with
\[ c\ne0, \]then the function has no zeros.
If
\[ f(x)=0, \]then every input in the domain is a zero.
Affine Functions Are Usually One-to-One
If
\[ m\ne0, \]then
\[ f(x)=mx+b \]is strictly monotonic.
Therefore, it is injective on \(\mathbb{R}\).
Constant Functions Are Not One-to-One
If the domain contains more than one input,
\[ f(x)=c \]assigns the same output to every input.
So a nontrivial constant function is not injective.
Affine Functions and Inverses
If
\[ m\ne0, \]then
\[ f(x)=mx+b \]is bijective from
\[ \mathbb{R}\to\mathbb{R}. \]Its inverse is
\[ \boxed{ f^{-1}(x)=\frac{x-b}{m} }. \]Verify the Inverse
Let
\[ f(x)=mx+b. \]Then
\[ f^{-1}(f(x)) = \frac{(mx+b)-b}{m} = x. \]The inverse undoes the affine transformation.
Affine Functions as Transformations
The function
\[ f(x)=mx+b \]can be seen as two steps:
\[ x \longmapsto mx \longmapsto mx+b. \]First scale by \(m\), then translate by \(b\).
Composition of Affine Functions
Let
\[ f(x)=ax+b \]and
\[ g(x)=cx+d. \]Then
\[ (g\circ f)(x) = c(ax+b)+d. \]So
\[ (g\circ f)(x) = acx+cb+d. \]The composition is again affine.
Affine Functions Form a Reusable Family
Because affine functions remain affine under composition, they are especially useful for:
- unit conversions;
- coordinate changes;
- financial models;
- measurement calibration;
- computer graphics;
- linear approximation.
Temperature Conversion
The Fahrenheit-Celsius relationship is
\[ F(C)=\frac95C+32. \]This is affine.
The slope
\[ \frac95 \]rescales temperature intervals, while
\[ 32 \]shifts the zero point.
Why Temperature Conversion Is Not Direct Proportion
If Celsius temperature doubles, Fahrenheit temperature does not generally double.
That is because the nonzero intercept changes the relationship.
Affine does not automatically mean proportional.
Linear Versus Affine Scaling
A linear function
\[ f(x)=mx \]preserves the origin and scales all values relative to zero.
An affine function
\[ f(x)=mx+b \]scales and then shifts the reference point.
Structural Property of Linear Functions
A true linear function satisfies
\[ f(x+y)=f(x)+f(y). \]For
\[ f(x)=mx, \]we have
\[ f(x+y)=m(x+y)=mx+my. \]Thus,
\[ f(x+y)=f(x)+f(y). \]Scaling Property of Linear Functions
A true linear function also satisfies
\[ f(cx)=cf(x). \]Indeed,
\[ f(cx)=m(cx)=c(mx)=cf(x). \]Affine Functions Usually Do Not Preserve Addition
Let
\[ f(x)=mx+b. \]Then
\[ f(x+y)=m(x+y)+b. \]But
\[ f(x)+f(y)=mx+b+my+b = m(x+y)+2b. \]These are equal only in special cases.
The translation term breaks linearity.
Affine Difference Property
Although affine functions do not generally preserve addition, they do preserve differences up to the common scale \(m\):
\[ f(x_2)-f(x_1) = m(x_2-x_1). \]This is the algebra behind constant slope.
Midpoint Property
Affine functions preserve weighted averages.
For example, the midpoint input
\[ \frac{x_1+x_2}{2} \]maps to the midpoint of the outputs:
\[ f\left(\frac{x_1+x_2}{2}\right) = \frac{f(x_1)+f(x_2)}{2}. \]This geometric property becomes important later in affine geometry.
More General Affine Combinations Preview
If weights satisfy
\[ \lambda_1+\lambda_2=1, \]then an affine function satisfies
\[ f(\lambda_1x_1+\lambda_2x_2) = \lambda_1f(x_1)+\lambda_2f(x_2). \]This generalizes the midpoint property.
Constant, Linear, and Affine from Tables
When input steps are equal:
- constant outputs suggest a constant function;
- constant nonzero first differences suggest an affine function;
- if that affine pattern also passes through \((0,0)\), it is linear in the strict sense.
Classification Example
| \(x\) | \(f(x)\) |
|---|---|
| 0 | 0 |
| 1 | 4 |
| 2 | 8 |
| 3 | 12 |
The rate of change is constant:
\[ 4. \]The function also passes through the origin.
So the rule is
\[ \boxed{f(x)=4x}, \]which is linear and affine.
Affine but Not Linear Example
| \(x\) | \(f(x)\) |
|---|---|
| 0 | 3 |
| 1 | 7 |
| 2 | 11 |
| 3 | 15 |
The slope is still
\[ 4, \]but
\[ f(0)=3. \]So
\[ \boxed{f(x)=4x+3} \]is affine but not linear in the strict sense.
Constant Example
| \(x\) | \(f(x)\) |
|---|---|
| \(-2\) | 9 |
| 0 | 9 |
| 5 | 9 |
The output never changes.
So
\[ \boxed{f(x)=9}. \]Worked Example: Build an Affine Function from a Rate and Initial Value
A storage tank contains \(120\) liters initially and fills at \(15\) liters per minute.
Initial amount:
\[ 120. \]Rate:
\[ 15. \]After \(t\) minutes:
\[ \boxed{V(t)=15t+120}. \]This is affine.
Worked Example: Direct Proportion
A machine produces \(8\) parts per minute and starts from zero completed parts.
After \(t\) minutes:
\[ \boxed{P(t)=8t}. \]This is linear and directly proportional.
Worked Example: Constant Relationship
A flat monthly membership fee is \$30 regardless of usage.
If usage is \(u\),
\[ \boxed{C(u)=30}. \]The function is constant.
Worked Example: Find a Line Through Two Points
Find the affine function through
\[ (-1,4) \]and
\[ (3,12). \]Slope:
\[ m= \frac{12-4}{3-(-1)} = \frac84 = 2. \]Use
\[ y=2x+b. \]Substitute \((-1,4)\):
\[ 4=2(-1)+b. \]So
\[ b=6. \]Therefore,
\[ \boxed{f(x)=2x+6}. \]Worked Example: Determine Whether a Table Is Affine
| \(x\) | \(y\) |
|---|---|
| 0 | 2 |
| 2 | 8 |
| 5 | 17 |
Check rates:
\[ \frac{8-2}{2-0} = 3, \] \[ \frac{17-8}{5-2} = 3. \]The rate is constant, so the data are consistent with an affine function.
Since
\[ f(0)=2, \]the rule is
\[ \boxed{f(x)=3x+2}. \]Worked Example: Not Affine
| \(x\) | \(y\) |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 7 |
The first rate is
\[ \frac{3-1}{1-0}=2. \]The second is
\[ \frac{7-3}{2-1}=4. \]The rate changes.
Therefore, the table is not affine.
A Reliable Classification Workflow
- Check whether the output changes at all.
- If it does not, the function is constant.
- If it changes, check whether the rate of change is constant.
- If the rate is constant, the function is affine.
- If the affine function also satisfies \(f(0)=0\), it is linear in the strict sense.
A Reliable Formula Workflow
- Rewrite the formula in the form \(mx+b\) when possible.
- Identify \(m\) as the constant rate of change.
- Identify \(b=f(0)\) as the initial value or \(y\)-intercept.
- If \(m=0\), classify the function as constant.
- If \(b=0\), classify it as linear in the strict sense.
- Otherwise, classify it as affine but not linear.
A Reliable Modeling Workflow
- Identify the input and output quantities.
- Determine the constant rate of change, if one exists.
- Determine the output when the input is zero.
- Write \(f(x)=mx+b\).
- Attach units to \(m\) and interpret \(b\) in context.
- Restrict the domain if the real situation does not allow all real inputs.
Common Errors
Calling Every \(mx+b\) Function Linear Without Recognizing the Affine Distinction
The common school convention is acceptable in context, but mathematically \(b\ne0\) introduces a translation and breaks strict linearity.
Assuming Constant Means Zero
Any function \(f(x)=c\) is constant.
Assuming Constant Rate Means Direct Proportion
Direct proportion additionally requires
\[ f(0)=0. \]Ignoring Units of Slope
Slope is a rate and should carry output units per input unit in applied problems.
Using Equal Output Differences with Unequal Input Steps
Compare
\[ \frac{\Delta y}{\Delta x} \]when the input spacing changes.
Forgetting the Initial Value
The intercept \(b\) often has important contextual meaning.
Treating a Vertical Line as \(y=f(x)\)
A vertical line fails the vertical-line test and is not a function of \(x\).
Practice
1. Classify a Function
Classify
\[ f(x)=6. \]Check Answer
It is a constant function.
It is also affine with
\[ m=0,\qquad b=6. \]It is not linear in the strict sense because
\[ f(0)=6\ne0. \]2. Linear Function
Classify
\[ f(x)=-4x. \]Check Answer
It is linear and affine.
Its slope is
\[ \boxed{-4}. \]3. Affine but Not Linear
Classify
\[ f(x)=5x+2. \]Check Answer
It is affine but not linear in the strict sense because
\[ f(0)=2. \]4. Find the Function
A function has slope \(3\) and \(y\)-intercept \(-5\). Find the formula.
Check Answer
\[ \boxed{f(x)=3x-5}. \]5. Find the Slope
Find the slope through
\[ (2,5) \]and
\[ (6,13). \]Check Answer
\[ m= \frac{13-5}{6-2} = \frac84 = \boxed{2}. \]6. Build from Two Points
Find the affine function through
\[ (1,4) \]and
\[ (3,10). \]Check Answer
\[ m= \frac{10-4}{3-1} = 3. \]Use
\[ 4=3(1)+b, \]so
\[ b=1. \]Therefore,
\[ \boxed{f(x)=3x+1}. \]7. Direct Proportion or Affine?
A service costs \$12 initially plus \$4 per hour.
Is the relationship directly proportional?
Check Answer
No.
The model is
\[ C(t)=4t+12. \]It is affine, but not directly proportional because
\[ C(0)=12\ne0. \]8. Zero Function
Why is
\[ f(x)=0 \]both constant and linear?
Check Answer
Its output is constant, and it also has the form
\[ f(x)=mx \]with
\[ m=0. \]It passes through the origin and preserves linear structure.
Key Ideas
- A constant function has the form \(f(x)=c\) and slope \(0\).
- A linear function in the structural sense has the form \(f(x)=mx\).
- An affine function has the form \(f(x)=mx+b\).
- School algebra often uses “linear function” for all functions \(mx+b\); the stricter distinction separates linear scaling from translation.
- Every linear function is affine.
- Every constant function is affine.
- The zero function is both constant and linear.
- A true linear function passes through the origin.
- An affine function has constant rate of change \(m\) and initial value \(b\).
- Direct proportionality is the special affine case \(y=mx\).
- Equal rates of change identify affine behavior even when input spacing is irregular.
- Nonvertical lines can be represented as affine functions of \(x\).
- Affine functions with nonzero slope are one-to-one and invertible over \(\mathbb{R}\).
- Affine functions can be understood as linear transformations followed by translations.
- Affine structure becomes important in geometry, coordinate transformations, modeling, and higher-dimensional mathematics.
Where This Leads
Constant, linear, and affine functions establish the basic language of constant rate and straight-line dependence.
From here, we can study slope more deeply, equations of lines, parallel and perpendicular relationships, intercept forms, systems of linear and affine equations, linear interpolation, affine combinations, linear transformations, and applications where rate and starting value must be separated carefully.