The Central Idea

A graph is a set of points together with whatever structure tells us how those points are generated or interpreted. Cartesian function notation is one important method, but it is not the only one.

Three especially useful alternatives are:

Representation Changes, Geometry Does Not

The same geometric curve may admit several representations. For example, the unit circle can be written in Cartesian form as

\[ x^2+y^2=1, \]

in parametric form as

\[ x=\cos t,\qquad y=\sin t, \]

and in polar form as

\[ r=1. \]

These formulas look different, but they describe the same set of points when their variables are given suitable domains.

Why Alternative Graphing Systems Matter

A representation is useful when it makes important structure easier to see or calculate. Alternative graphing systems are especially helpful when:

Part I: Parametric Graphs

In a parametric representation, the coordinates are given as functions of a third variable, usually called a parameter.

\[ \boxed{x=f(t),\qquad y=g(t)}. \]

The parameter \(t\) may represent time, angle, distance traveled, or simply an auxiliary variable used to generate the curve.

The Parameter Generates Points

Suppose

\[ x=t+1,\qquad y=t^2. \]

Each allowed value of \(t\) produces an ordered pair. For example,

\[ t=-1\Rightarrow(0,1),\qquad t=0\Rightarrow(1,0),\qquad t=1\Rightarrow(2,1). \]

Plotting all generated points traces the parametric curve.

A Parameter Is Not Necessarily an Axis

In an ordinary two-dimensional parametric graph, the visible axes are still the \(x\)- and \(y\)-axes. The parameter \(t\) usually does not appear as a third spatial axis.

Instead, \(t\) determines which point \((x,y)\) is produced.

Building a Parametric Table

A table is often the simplest way to begin a parametric graph.

For

\[ x=2t, \qquad y=t+3, \]

we may use:

\(t\)\(x=2t\)\(y=t+3\)Point
\(-1\)\(-2\)\(2\)\((-2,2)\)
\(0\)\(0\)\(3\)\((0,3)\)
\(1\)\(2\)\(4\)\((2,4)\)

Eliminating the Parameter

Sometimes a parametric system can be converted to a Cartesian relation by solving one equation for the parameter and substituting into the other.

From

\[ x=2t, \]

we get

\[ t=\frac{x}{2}. \]

Substitute into \(y=t+3\):

\[ \boxed{y=\frac{x}{2}+3}. \]

Eliminating the Parameter Can Reveal the Curve

Consider

\[ x=t+1, \qquad y=t^2. \]

Since \(t=x-1\),

\[ y=(x-1)^2. \]

The parametric equations therefore trace a parabola. The Cartesian equation reveals the familiar family immediately.

Do Not Lose Parameter Restrictions

Eliminating the parameter can accidentally hide the original parameter domain. If

\[ x=t, \qquad y=t^2, \qquad 0\le t\le2, \]

then eliminating \(t\) gives \(y=x^2\), but only for

\[ \boxed{0\le x\le2}. \]

The entire parabola is not part of the parametric graph.

Orientation on a Parametric Curve

A parametric curve can contain more information than the geometric set of points. As the parameter increases, the curve may be traced in a particular orientation or direction.

Arrows placed on a parametric graph commonly show the direction of increasing parameter values.

Example: Orientation on a Line

For

\[ x=t, \qquad y=2t+1, \]

increasing \(t\) increases \(x\), so the line is traced from lower-left toward upper-right.

Replacing \(t\) with \(-t\) can preserve the same geometric line while reversing the tracing direction.

The Same Curve Can Have Different Parametrizations

The equations

\[ x=t,\qquad y=t^2 \]

and

\[ x=2s,\qquad y=4s^2 \]

both describe the parabola \(y=x^2\) when the parameters range over all real numbers. The rate at which the points are generated differs, but the geometric locus is the same.

A Parametric Curve Need Not Be a Function of \(x\)

Parametric equations can describe curves that fail the vertical-line test. The standard example is the circle

\[ x=\cos t, \qquad y=\sin t. \]

No single function \(y=f(x)\) describes the entire circle, but the parametric form does so naturally.

Parametric Circle

Let

\[ x=r\cos t, \qquad y=r\sin t. \]

Then

\[ x^2+y^2 =r^2\cos^2t+r^2\sin^2t =r^2. \]

Thus the curve is

\[ \boxed{x^2+y^2=r^2}. \]

Parameter Interval for One Trip Around a Circle

The choice

\[ 0\le t\le2\pi \]

traces the circle once. Allowing \(t\) to continue beyond \(2\pi\) retraces points already visited.

Parametric domains therefore control not only which points appear but also how many times a curve may be traversed.

Parametric Ellipse

The equations

\[ x=a\cos t, \qquad y=b\sin t \]

give

\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \]

The parameters \(a\) and \(b\) control the horizontal and vertical semiaxes.

Parametric Motion

Parametric equations are particularly natural when a point moves. If

\[ x=x(t),\qquad y=y(t), \]

then the pair \((x(t),y(t))\) gives the position at parameter value \(t\). When \(t\) is time, the equations describe a trajectory.

Position and Path Are Different Ideas

A parametric system contains two related layers of information:

Two objects can travel along the same path with different parameterizations.

A Point May Be Reached More Than Once

Different parameter values can generate the same coordinate pair. On a circle,

\[ (\cos 0,\sin0)=(1,0) \]

and

\[ (\cos2\pi,\sin2\pi)=(1,0). \]

The repeated point does not create a new geometric location, but it may matter when interpreting motion or traversal.

Finding Parametric Intercepts

To find an \(x\)-intercept of a parametric curve, solve

\[ y(t)=0 \]

for allowed parameter values and then calculate the corresponding \(x(t)\). For a \(y\)-intercept, solve \(x(t)=0\) and calculate \(y(t)\).

Common Error: Eliminating the Parameter Too Early

Converting to Cartesian form can be useful, but it may hide orientation, parameter restrictions, repeated traversal, or the meaning of the parameter.

Use elimination when it clarifies the geometry, not as an automatic first step.

Part II: Polar Graphs

Cartesian coordinates locate a point by horizontal and vertical displacement. Polar coordinates locate a point using:

A polar point is written

\[ \boxed{(r,\theta)}. \]

The Pole and Polar Axis

In the polar system, the origin is called the pole. The ray corresponding to the positive Cartesian \(x\)-axis is the polar axis.

Angles are usually measured counterclockwise when positive and clockwise when negative.

Plotting a Polar Point

To plot

\[ \left(3,\frac{\pi}{4}\right), \]

begin at the pole, turn through an angle of \(\pi/4\), and move \(3\) units in that direction.

Polar to Cartesian Conversion

The right-triangle relationships give

\[ \boxed{x=r\cos\theta}, \] \[ \boxed{y=r\sin\theta}. \]

These formulas convert a polar coordinate into Cartesian coordinates.

Example: Convert a Polar Point

Convert

\[ \left(2,\frac{\pi}{3}\right) \]

to Cartesian form.

\[ x=2\cos\frac{\pi}{3}=1, \] \[ y=2\sin\frac{\pi}{3}=\sqrt3. \]

Therefore,

\[ \boxed{(x,y)=(1,\sqrt3)}. \]

Cartesian to Polar Distance

Squaring and adding the conversion formulas gives

\[ x^2+y^2=r^2. \]

For the conventional choice \(r\ge0\),

\[ \boxed{r=\sqrt{x^2+y^2}}. \]

Cartesian to Polar Angle

When \(x\ne0\), division gives

\[ \tan\theta=\frac{y}{x}. \]

However, inverse tangent alone does not always identify the correct quadrant. The signs of \(x\) and \(y\), or a quadrant-aware angle function, must be used to choose the correct direction.

Polar Coordinates Are Not Unique

The same point can have infinitely many polar descriptions because adding a full revolution leaves the direction unchanged:

\[ (r,\theta)=(r,\theta+2\pi k), \qquad k\in\mathbb Z. \]

Negative Radius

Polar coordinates also permit a negative radius. The point

\[ (-r,\theta) \]

is located \(r\) units in the direction opposite \(\theta\). Equivalently,

\[ \boxed{(-r,\theta)=(r,\theta+\pi)}. \]

Example: Equivalent Polar Coordinates

The point

\[ \left(-2,\frac{\pi}{6}\right) \]

can also be written

\[ \boxed{\left(2,\frac{7\pi}{6}\right)}. \]

Both descriptions identify the same Cartesian point.

What Is a Polar Equation?

A polar equation relates \(r\) and \(\theta\). A common form is

\[ r=f(\theta). \]

For each allowed angle, the equation specifies one or more radial positions.

The Simplest Polar Circle

The equation

\[ \boxed{r=a} \]

describes all points a fixed distance \(|a|\) from the pole. For positive \(a\), this is a circle centered at the origin with radius \(a\).

A Constant Angle Gives a Line

The equation

\[ \boxed{\theta=\alpha} \]

describes the line through the pole oriented at angle \(\alpha\). Allowing both positive and negative \(r\) reaches both directions along that line.

Converting a Polar Equation to Cartesian Form

Suppose

\[ r=4\cos\theta. \]

Multiply both sides by \(r\):

\[ r^2=4r\cos\theta. \]

Using \(r^2=x^2+y^2\) and \(r\cos\theta=x\),

\[ x^2+y^2=4x. \]

Complete the square:

\[ \boxed{(x-2)^2+y^2=4}. \]

The polar equation therefore describes a circle centered at \((2,0)\) with radius \(2\).

Polar Form Can Make Symmetry Natural

Curves organized around the origin often have simple polar descriptions. Equations involving \(\sin\theta\), \(\cos\theta\), or multiples of \(\theta\) can encode rotational or reflective structure that would be cumbersome in Cartesian form.

Rose Curves

Equations such as

\[ r=a\cos(n\theta) \]

or

\[ r=a\sin(n\theta) \]

can produce petal-like rose curves. The number and orientation of petals depend on \(n\) and on whether sine or cosine is used.

At this stage, the main goal is recognizing that polar form can encode repeated angular structure compactly.

Cardioid-Type Curves

Equations of the form

\[ r=a(1\pm\cos\theta) \]

or

\[ r=a(1\pm\sin\theta) \]

generate cardioid-shaped curves. Their orientation follows the angular term and sign.

Polar Spirals

A rule such as

\[ r=a\theta \]

increases radial distance as the angle increases. The result is a spiral rather than a closed curve.

A Polar Table

Just as a Cartesian function can be graphed from a table of \(x\)- and \(y\)-values, a polar graph can be sampled using values of \(\theta\) and the resulting radii.

For \(r=2\cos\theta\), one might examine

\(\theta\)\(r\)
\(0\)\(2\)
\(\pi/2\)\(0\)
\(\pi\)\(-2\)
\(3\pi/2\)\(0\)

Negative Polar Values Can Fold the Trace

When \(r\) changes sign, the plotted point moves to the opposite direction from the stated angle. This can cause a polar curve to loop, repeat, or form petals in ways that are not obvious from a simple table of positive radii.

A Polar Graph Is Still a Set of Cartesian Points

Polar notation changes how a point is specified, but the resulting graph still occupies the ordinary plane. Every polar point can be translated into Cartesian coordinates with

\[ x=r\cos\theta, \qquad y=r\sin\theta. \]

Common Error: Treating \((r, heta)\) Like \((x,y)\)

In a polar ordered pair, the second coordinate is an angle, not a vertical displacement. The point

\[ \left(3,\frac{\pi}{2}\right) \]

is not the Cartesian point \((3,\pi/2)\). It lies three units from the pole in the direction \(\pi/2\), so its Cartesian coordinate is \((0,3)\).

Common Error: Ignoring the Quadrant

The equation

\[ \tan\theta=\frac{y}{x} \]

does not by itself distinguish angles that differ by \(\pi\). When converting from Cartesian coordinates, always use the signs of \(x\) and \(y\) to identify the correct quadrant.

Part III: Discrete Graphs

A discrete graph contains separated points rather than every point along a continuous curve or interval.

The separation may come from the domain, from the nature of the objects being counted, or from the fact that observations occur only at selected inputs.

Discrete Domains

Consider the rule

\[ y=2x+1 \]

with domain

\[ x\in\{0,1,2,3\}. \]

The formula is linear, but the graph contains only four points. The line between those points is not part of the relation unless the domain is expanded.

Same Formula, Different Graph

The rule \(y=2x+1\) can produce very different graphs depending on the domain.

The formula alone does not determine the complete graph; the domain matters.

Sequences Are Naturally Discrete

A sequence is commonly indexed by integers such as

\[ n=1,2,3,\ldots \]

so the graph of a sequence consists of isolated points \((n,a_n)\). Connecting those points creates an interpolation, not the sequence itself.

Example: Arithmetic Sequence as a Graph

Let

\[ a_n=3n+2, \qquad n\in\{1,2,3,4\}. \]

The graph contains

\[ (1,5),\quad(2,8),\quad(3,11),\quad(4,14). \]

These points align on a line, but the sequence contains only the listed integer inputs.

Discrete Does Not Mean Random

A discrete graph may follow an exact deterministic rule, such as a sequence, or it may represent measured observations. The word discrete refers to separated allowed points, not to unpredictability.

Discrete Data and Discrete Functions

A table of measurements can produce a discrete relation even when the underlying phenomenon is continuous. For example, temperature recorded once per hour gives isolated observations, even though temperature changes between measurements.

Distinguish the data that were recorded from the model that may later be fitted between them.

When Should Discrete Points Be Connected?

Connecting discrete points is appropriate only when the connecting curve or segment has a defined meaning.

Domain and Range of a Discrete Graph

The domain is the set of displayed or permitted \(x\)-coordinates, and the range is the set of corresponding \(y\)-coordinates.

For the points

\[ \{(1,4),(2,7),(5,7),(8,10)\}, \]

the domain is

\[ \boxed{\{1,2,5,8\}}, \]

and the range is

\[ \boxed{\{4,7,10\}}. \]

Repeated output values are listed only once when range is written as a set.

Discrete Graphs Can Still Be Functions

A graph does not need to be continuous to represent a function. A discrete relation is a function whenever each allowed input has exactly one output.

The vertical-line test still applies.

Discrete Relations Need Not Be Functions

If a discrete relation contains both

\[ (2,5) \]

and

\[ (2,9), \]

then the input \(2\) has two outputs, so the relation is not a function.

Open and Closed Points in Discrete Settings

A filled point normally indicates that a coordinate pair belongs to the graph. An open point can indicate an excluded value or a limiting location, especially when a discrete graph is combined with piecewise or continuous pieces.

Always interpret the marker together with the stated domain or rule.

Integer Restrictions Matter

If a formula is valid only for integer inputs, solving the corresponding real-valued equation may produce candidates that are not actually in the domain.

A graphical solution must satisfy both the formula and the domain restriction.

Parametric and Discrete Ideas Can Combine

A parametric system need not use every real parameter value. If

\[ x=n, \qquad y=n^2, \qquad n\in\mathbb Z, \]

then the parameter generates a discrete set of points on the parabola \(y=x^2\).

Polar and Discrete Ideas Can Combine

A polar relation may also be sampled at selected angles only. For example,

\[ r=2, \qquad \theta\in\left\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\} \]

gives four isolated points on a circle rather than the entire circle.

Representation as a Choice of Coordinates and Domain

These three graphing systems emphasize two design decisions:

  1. How are coordinates generated?
  2. Which values are allowed to generate them?

Parametric and polar forms alter the coordinate-generation rule. Discrete graphs emphasize restrictions on the allowed generating values.

Comparing Cartesian and Parametric Forms

Cartesian form often makes vertical dependence clear:

\[ y=f(x). \]

Parametric form instead separates the coordinates:

\[ x=f(t),\qquad y=g(t). \]

This separation is useful when neither coordinate naturally serves as the independent variable for the entire curve.

Comparing Cartesian and Polar Forms

Cartesian coordinates emphasize horizontal and vertical displacement. Polar coordinates emphasize radial distance and angle.

Curves centered on rotation or radial symmetry are often simpler in polar form, while curves organized around vertical dependence may be simpler in Cartesian form.

Graph Shape Depends on the Full Specification

A complete graphing description may require more than a formula. Relevant information can include:

Graphing Parametric Curves by Hand

  1. Identify the permitted parameter interval.
  2. Choose strategically useful parameter values.
  3. Compute the corresponding \(x\)- and \(y\)-coordinates.
  4. Plot the ordered pairs.
  5. Determine whether the points form a continuous path.
  6. Add orientation arrows when direction matters.
  7. Check whether eliminating the parameter clarifies the curve.

Graphing Polar Curves by Hand

  1. Identify the allowed angle interval.
  2. Look for obvious symmetry or repetition.
  3. Evaluate \(r\) at strategic angles.
  4. Plot each radial position carefully, including negative radii.
  5. Trace the curve in increasing \(\theta\) when orientation helps.
  6. Use Cartesian conversion when it reveals a familiar equation.

Graphing Discrete Relations

  1. Identify the allowed inputs exactly.
  2. Compute or read the output for each allowed input.
  3. Plot only the included ordered pairs.
  4. Do not connect points unless the model explicitly includes intermediate values.
  5. Read domain and range as sets when appropriate.

Choosing Strategic Parameter Values

Good sample values are not always equally spaced arbitrary decimals. Choose values that expose structure, such as:

Scale Still Matters

Alternative coordinate systems do not eliminate the need for careful graph scale. A poor Cartesian viewing window can hide a parametric loop, compress a polar petal, or make separated discrete points appear almost continuous.

Scale is part of interpretation, not merely decoration.

Sampling Can Hide Structure

Graphing technology often approximates a parametric or polar curve by sampling many parameter or angle values. If the sampling is too coarse, the display can miss loops, sharp changes, narrow petals, or repeated behavior.

A graphing display should therefore be checked against the formula and known structure.

A Graphing Window Can Hide an Entire Branch

Just as with Cartesian graphs, a restricted viewing window may show only part of a parametric or polar curve. The visible image is evidence, not automatically the complete mathematical object.

Exact and Approximate Graph Information

Tables and graphing technology often provide approximate coordinates. Algebraic conversion can sometimes recover exact information.

For example, a polar point might appear numerically near \((1.414,1.414)\), while exact conversion reveals

\[ \left(\sqrt2,\sqrt2\right). \]

Checking a Parametric Point

To verify that a point belongs to a parametric curve, find an allowed parameter value that produces both coordinates simultaneously.

It is not enough for one parameter value to produce the correct \(x\)-coordinate and a different parameter value to produce the correct \(y\)-coordinate.

Checking a Polar Point

To verify a polar point, substitute a valid angular description into the polar equation and check the resulting radius. Because polar coordinates are not unique, an equivalent representation of the same Cartesian point may also work.

Checking a Discrete Point

A candidate point on a discrete graph must satisfy both conditions:

  1. its input belongs to the discrete domain;
  2. its output matches the rule or recorded relation.

Common Error: Connecting Every Set of Points

A collection of plotted points does not automatically imply that every point between them belongs to the graph. Connecting points without justification can change the domain, range, solution set, and meaning of the relation.

Common Error: Ignoring Orientation

Two parametric systems can trace the same geometric curve in opposite directions. If the parameter represents time or another ordered process, orientation is part of the information and should not be discarded.

Common Error: Assuming One Parameter Value per Point

Closed or self-intersecting parametric curves can revisit a point. A geometric point is not required to correspond to a unique parameter value.

Common Error: Assuming Polar Coordinates Are Unique

Because angles can differ by full revolutions and radius can change sign with a half-turn adjustment, many polar ordered pairs can identify the same point.

Common Error: Treating a Discrete Pattern as Continuous

If a sequence is defined only for integer indices, a point such as \((2.5,a_{2.5})\) is not automatically meaningful. Extending the formula to real inputs creates a new continuous model, not part of the original sequence unless that extension is explicitly defined.

Example: Analyze a Parametric Segment

Consider

\[ x=2t-1, \qquad y=t+4, \qquad -1\le t\le2. \]

At the parameter endpoints:

\[ t=-1\Rightarrow(-3,3), \] \[ t=2\Rightarrow(3,6). \]

Eliminating \(t\):

\[ t=\frac{x+1}{2}, \] \[ y=\frac{x+1}{2}+4 =\frac{x}{2}+\frac92. \]

The graph is the segment of this line from \((-3,3)\) to \((3,6)\), traced in that direction as \(t\) increases.

Example: Analyze a Parametric Parabola

Consider

\[ x=t+2, \qquad y=t^2-1. \]

Since \(t=x-2\),

\[ \boxed{y=(x-2)^2-1}. \]

The Cartesian form reveals a parabola with vertex \((2,-1)\). If \(t\) is unrestricted, every real \(x\)-value occurs exactly once.

Example: Analyze a Polar Circle

Consider

\[ r=6\cos\theta. \]

Multiply by \(r\):

\[ r^2=6r\cos\theta. \]

Convert:

\[ x^2+y^2=6x. \]

Complete the square:

\[ \boxed{(x-3)^2+y^2=9}. \]

The graph is a circle centered at \((3,0)\) with radius \(3\).

Example: Analyze a Discrete Relation

Suppose

\[ y=x^2-1, \qquad x\in\{-2,-1,0,1,2\}. \]

The points are

\[ (-2,3),\ (-1,0),\ (0,-1),\ (1,0),\ (2,3). \]

The domain is

\[ \{-2,-1,0,1,2\}, \]

and the range is

\[ \{-1,0,3\}. \]

The points lie on the parabola \(y=x^2-1\), but the continuous arcs between them are not part of this discrete relation.

Intersections in Parametric Form

Intersections involving a parametric curve can be found by matching its coordinates to another relation. For example, to find where

\[ x=f(t),\qquad y=g(t) \]

meets the line \(y=3\), solve

\[ g(t)=3 \]

and then compute the associated \(x\)-coordinates from \(f(t)\).

Intersections in Polar Form

Polar intersections require care because the same point can have different polar coordinates. Solving two equations for the same \(r\) at the same \(\theta\) finds many intersections, but special points such as the pole may be represented by different angular values.

Graphical and Cartesian checks are useful when completeness matters.

Zeros in Alternative Representations

In Cartesian function notation, a zero is an input where \(y=0\). For a parametric curve, an \(x\)-axis crossing occurs when \(y(t)=0\). For a polar curve, the pole occurs whenever \(r=0\).

Similar geometry can therefore correspond to different algebraic conditions.

Symmetry Across Representations

Symmetry remains a geometric property even when notation changes. A Cartesian circle, a parametrized circle, and the polar equation \(r=a\) all have the same geometric symmetries.

The most convenient algebraic symmetry test, however, depends on the chosen representation.

Transformations Across Representations

Transformations can be expressed in parametric and polar systems as well. For example, adding constants to parametric coordinates,

\[ x=f(t)+h, \qquad y=g(t)+k, \]

translates the entire parametric curve by \((h,k)\).

Polar transformations can be more representation-specific because changing \(r\) or \(\theta\) alters radial distance or angular position directly.

Modeling with Parametric Graphs

Parametric representations are useful when two quantities depend on a shared variable. Examples include horizontal and vertical position depending on time, or two measured quantities changing together as a process advances.

The graph shows how the pair evolves, while the parameter records progression through the process.

Modeling with Polar Graphs

Polar models are useful when distance from a central point depends on direction. Examples include radial boundaries, rotating sensors, directional intensity, orbital-style geometry, and periodic angular patterns.

Modeling with Discrete Graphs

Discrete graphs are natural for counts, scheduled observations, sequence terms, inventory levels at reporting times, and any quantity defined only at separated inputs.

The discreteness should reflect the problem, not merely the way the graph was drawn.

A Reliable Representation-Choice Procedure

  1. Identify what the variables mean.
  2. Determine whether one coordinate naturally depends on the other.
  3. Look for a shared parameter such as time or angle.
  4. Ask whether radial distance and direction are more natural than \(x\) and \(y\).
  5. Determine whether the allowed inputs are continuous or discrete.
  6. Choose the representation that exposes the structure with the least unnecessary complexity.
  7. Convert to another representation when doing so provides a useful check.

Practice 1: Generate Parametric Points

For

\[ x=t-1, \qquad y=2t+3, \]

find the points corresponding to \(t=-1,0,2\).

Practice 2: Eliminate the Parameter

Convert

\[ x=3t+2, \qquad y=t-4 \]

to a Cartesian equation.

Practice 3: Preserve a Parameter Restriction

For

\[ x=t, \qquad y=t^2+1, \qquad -2\le t\le1, \]

write the Cartesian equation together with the correct \(x\)-restriction.

Practice 4: Convert Polar to Cartesian Coordinates

Convert

\[ \left(4,\frac{\pi}{6}\right) \]

to Cartesian coordinates.

Practice 5: Equivalent Polar Coordinate

Rewrite

\[ \left(-3,\frac{\pi}{4}\right) \]

using a positive radius.

Practice 6: Convert a Polar Equation

Convert

\[ r=8\cos\theta \]

to Cartesian form and identify the curve.

Practice 7: Discrete Domain and Range

For the relation

\[ \{(-2,5),(0,1),(3,5),(7,9)\}, \]

state the domain and range.

Practice 8: Function or Not?

Determine whether

\[ \{(1,4),(2,6),(2,8),(3,10)\} \]

represents a function.

Practice 9: Discrete Sequence Graph

For

\[ a_n=2^n, \qquad n\in\{0,1,2,3\}, \]

list the plotted points and explain whether they should automatically be connected.

Practice 10: Compare Representations

The curve is given parametrically by

\[ x=5\cos t, \qquad y=5\sin t. \]

Identify the Cartesian equation and a simple polar equation describing the same geometric circle.

Practice Answers

Practice 1

\[ t=-1\Rightarrow(-2,1), \] \[ t=0\Rightarrow(-1,3), \] \[ t=2\Rightarrow(1,7). \]

Practice 2

From \(y=t-4\),

\[ t=y+4. \]

Substitute into \(x=3t+2\):

\[ x=3(y+4)+2=3y+14. \]

Thus

\[ \boxed{y=\frac{x-14}{3}}. \]

Practice 3

Since \(x=t\),

\[ \boxed{y=x^2+1,\qquad -2\le x\le1}. \]

Practice 4

\[ x=4\cos\frac{\pi}{6}=2\sqrt3, \] \[ y=4\sin\frac{\pi}{6}=2. \]

Therefore,

\[ \boxed{(2\sqrt3,2)}. \]

Practice 5

Add \(\pi\) to the angle and reverse the sign of the radius:

\[ \boxed{\left(3,\frac{5\pi}{4}\right)}. \]

Practice 6

Multiply by \(r\):

\[ r^2=8r\cos\theta. \]

Convert:

\[ x^2+y^2=8x. \]

Complete the square:

\[ \boxed{(x-4)^2+y^2=16}. \]

The curve is a circle centered at \((4,0)\) with radius \(4\).

Practice 7

The domain is

\[ \boxed{\{-2,0,3,7\}}, \]

and the range is

\[ \boxed{\{1,5,9\}}. \]

Practice 8

No. The input \(2\) is paired with both \(6\) and \(8\), so the relation fails the function requirement.

Practice 9

The points are

\[ \boxed{(0,1),(1,2),(2,4),(3,8)}. \]

They should not automatically be connected because the sequence is defined only at the listed integer indices. Connecting them would introduce intermediate points that are not terms of the sequence.

Practice 10

Squaring and adding gives

\[ x^2+y^2=25. \]

Thus the Cartesian equation is

\[ \boxed{x^2+y^2=25}, \]

and a simple polar equation is

\[ \boxed{r=5}. \]

What to Remember

The most useful habit is to separate the geometric object from the language used to generate it. A line, circle, trajectory, sequence, or collection of observations can often be described in several ways, and each representation highlights different information.

Where We Go Next

Once a graph has been represented appropriately, we can use it to estimate quantities that may be difficult to obtain exactly. Intersections, zeros, extrema, solutions of equations, and solutions of inequalities can often be approximated by inspecting or numerically refining graphical information.

The next lesson, Graphical Estimation and Solution Methods, develops systematic ways to use graphs as computational tools while keeping the difference between visual approximation and exact mathematical proof clear.