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AP Course

AP Precalculus

Study nine focused AP Precalculus units through topic lessons, original graph examples, quizzes, and practice sets.

Follow the College Board topic sequence across polynomial, rational, exponential, logarithmic, trigonometric, polar, parametric, vector, and matrix functions.

Lessons
Change in TandemRates of ChangeRates of Change in Linear and Quadratic FunctionsPolynomial Functions and Rates of ChangePolynomial Functions and Complex ZerosPolynomial Functions and End BehaviorRational Functions and End BehaviorRational Functions and ZerosRational Functions and Vertical AsymptotesRational Functions and HolesEquivalent Representations of Polynomial and Rational ExpressionsTransformations of FunctionsFunction Model Selection and Assumption ArticulationFunction Model Construction and ApplicationChange in Arithmetic and Geometric SequencesChange in Linear and Exponential FunctionsExponential FunctionsExponential Function ManipulationExponential Function Context and Data ModelingCompeting Function Model ValidationComposition of FunctionsInverse FunctionsLogarithmic ExpressionsInverses of Exponential FunctionsLogarithmic FunctionsLogarithmic Function ManipulationExponential and Logarithmic Equations and InequalitiesLogarithmic Function Context and Data ModelingSemi-log PlotsPeriodic PhenomenaSine, Cosine, and TangentSine and Cosine Function ValuesSine and Cosine Function GraphsSinusoidal FunctionsSinusoidal Function TransformationsSinusoidal Function Context and Data ModelingThe Tangent FunctionInverse Trigonometric FunctionsTrigonometric Equations and InequalitiesThe Secant, Cosecant, and Cotangent FunctionsEquivalent Representations of Trigonometric FunctionsTrigonometry and Polar CoordinatesPolar Function GraphsRates of Change in Polar FunctionsParametric FunctionsParametric Functions Modeling Planar MotionParametric Functions and Rates of ChangeParametrically Defined Circles and LinesImplicitly Defined FunctionsConic SectionsParametrization of Implicitly Defined FunctionsVectorsVector-Valued FunctionsMatricesThe Inverse and Determinant of a MatrixLinear Transformations and MatricesMatrices as FunctionsMatrices Modeling Contexts
Quizzes
Practice Problems Open a unit to choose from 58 topic-aligned practice sets.

Unit 7 · Topic 4.1 · Optional

Parametric Functions

Use one independent parameter to generate two dependent coordinates, organize the resulting points in a table, and trace a plane curve in parameter order.

Learning Goals

  • Interpret \(x(t)\) and \(y(t)\) as coordinate functions of one parameter.
  • Evaluate a parametric function and build a numerical table.
  • Sketch a curve in increasing-\(t\) order with correct orientation.
  • Use a restricted parameter domain to identify start and end points.
  • Relate a parametric representation to a rectangular equation without losing its restrictions.

1. One Input Generates a Point

A parametric function in the coordinate plane has two component equations:

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

Together they define the ordered-pair output

\[\mathbf f(t)=\bigl(x(t),y(t)\bigr).\]

The independent variable \(t\) is the parameter. For each allowed \(t\), both component functions are evaluated at the same input to produce one point in \(\mathbb R^2\).

2. Identify the Roles of the Variables

SymbolRole
\(t\)Independent parameter
\(x(t)\)Horizontal coordinate generated by \(t\)
\(y(t)\)Vertical coordinate generated by \(t\)
\((x(t),y(t))\)Point on the plane curve

The parameter may represent time, but it can also be any variable used to order points along a curve.

3. Evaluate Both Components

Let \(x(t)=t-1\) and \(y(t)=t^2-4\). At \(t=2\),

\[x(2)=1,\qquad y(2)=0,\]

so \(\mathbf f(2)=(1,0)\). Do not pair component values produced by different parameter inputs.

4. Build a Table in Increasing-\(t\) Order

For \(-2\le t\le3\), evaluate both formulas at each selected input:

\(t\)\(x(t)=t-1\)\(y(t)=t^2-4\)Point
\(-2\)\(-3\)\(0\)\((-3,0)\)
\(-1\)\(-2\)\(-3\)\((-2,-3)\)
\(0\)\(-1\)\(-4\)\((-1,-4)\)
\(1\)\(0\)\(-3\)\((0,-3)\)
\(2\)\(1\)\(0\)\((1,0)\)
\(3\)\(2\)\(5\)\((2,5)\)

5. Plot and Connect in Parameter Order

Parametric Functions example graphThe parameter traces the parabola in a specific direction as t increases.
The parameter traces the parabola in a specific direction as t increases.

Plot the \((x,y)\) pairs, connect neighboring points smoothly, and add arrows in the direction of increasing \(t\). Here the curve moves from left to right through the vertex \((-1,-4)\).

6. The Parameter Domain Selects the Curve Segment

The restriction \(-2\le t\le3\) gives

\[\mathbf f(-2)=(-3,0),\qquad \mathbf f(3)=(2,5).\]

These are the start and end points. The parametric graph contains only the portion reached as \(t\) moves across the stated domain, including both endpoints because the inequalities are closed.

7. Orientation Is Part of the Representation

The rectangular image shows where the curve lies. The parameter additionally records the order in which its points are generated.

When \(t\) increases from \(-2\) to \(3\), \(x=t-1\) increases, so this example is traced from left to right. Reversing the parameter can trace the same geometric path in the opposite direction.

8. Eliminate the Parameter

Solving \(x=t-1\) gives \(t=x+1\). Substitute into the y-component:

\[y=t^2-4=(x+1)^2-4.\]

The rectangular equation identifies an upward-opening parabola with vertex \((-1,-4)\). Because \(-2\le t\le3\), the corresponding rectangular restriction is \(-3\le x\le2\).

9. What Elimination Can Hide

A rectangular equation may identify the geometric curve but omit information carried by the parameter.

  • The starting and ending parameter values
  • The direction in which the curve is traced
  • Whether a point is visited more than once
  • How parameter inputs are assigned to positions

Always carry the parameter-domain restriction into the final description.

10. One Curve Can Have Many Parametrizations

ParametrizationDomainResult
\(x=t,\ y=t^2\)\(-2\le t\le2\)\(y=x^2\), left to right
\(x=-t,\ y=t^2\)\(-2\le t\le2\)\(y=x^2\), right to left
\(x=2t,\ y=4t^2\)\(-1\le t\le1\)\(y=x^2\), left to right with a different parameter scale

Equivalent geometric images need not be identical parametric functions because the same \(t\)-value may produce different points or order them differently.

11. A Parametric Curve Need Not Be \(y\) as a Function of \(x\)

Let

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

Eliminating \(t\) gives \(x=y^2\), a sideways parabola. Most positive x-values correspond to two y-values, so the curve fails the vertical-line test, yet the parametric function is valid because each \(t\) still produces exactly one ordered pair.

12. Different Inputs Can Reach the Same Point

For \(x=t^2-1\) and \(y=t^3-t\),

\[\mathbf f(-1)=(0,0),\qquad \mathbf f(1)=(0,0).\]

The two parameter inputs reach the same location. A graphing table should retain the \(t\)-column so repeated visits and their order are not lost.

13. Create a Simple Parametrization

A rectangular function \(y=g(x)\) can be written parametrically by choosing

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

For example, \(y=2x+3\) on \(-1\le x\le4\) becomes

\[x=t,\qquad y=2t+3,\qquad -1\le t\le4.\]

This choice traces the segment from \((-1,1)\) to \((4,11)\) as \(t\) increases.

14. Use Technology Purposefully

  1. Select parametric graphing mode.
  2. Enter \(x(t)\) and \(y(t)\) as a matched pair.
  3. Set the exact \(t\)-minimum and \(t\)-maximum.
  4. Choose a sufficiently small \(t\)-step.
  5. Use TRACE to compare displayed points with a table.
  6. Confirm the start point, end point, and orientation.

15. AP Workflow and Common Errors

  1. Identify the parameter and its domain.
  2. Evaluate both components at the same selected inputs.
  3. Record the ordered pairs in increasing-\(t\) order.
  4. Plot the points and mark start and end points.
  5. Connect the points smoothly and add orientation arrows.
  6. Eliminate the parameter when a rectangular equation is useful.
  7. Preserve every restriction when describing the resulting curve.
  • Do not treat \(t\) as an additional coordinate axis.
  • Do not pair \(x(t_1)\) with \(y(t_2)\).
  • Do not connect points in left-to-right order unless that matches increasing \(t\).
  • Do not graph the entire rectangular curve when the parameter domain is restricted.
  • Do not assume every parametric curve must pass the vertical-line test.

Key Takeaways

  • Represent planar relations by defining x and y separately as functions of a parameter.
  • Core relationship: \(x=f(t),\quad y=g(t)\)
  • Error check: A parametric relation need not pass the vertical-line test to be valid.
Checkpoint · Topic 4.1
  1. For \(x=2t-1,\ y=t^2+2\), find the point generated by \(t=-2\).
  2. Make a table for \(x=t+1,\ y=3-t\) on \(-1\le t\le2\), then state its start and end points.
  3. Eliminate \(t\) from \(x=t-2,\ y=t^2\) and include the x-domain corresponding to \(0\le t\le3\).
  4. Compare the orientation of \(x=t,\ y=t^2\) with \(x=-t,\ y=t^2\) on \([-2,2]\).
  5. Explain what information can be lost when a parametric representation is replaced by a rectangular equation.