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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 4.2 · Optional

Parametric Functions Modeling Planar Motion

Interpret a pair of coordinate functions as an object's position over time, then locate its endpoints, coordinate extrema, and axis crossings.

Learning Goals

  • Interpret \(\mathbf p(t)=(x(t),y(t))\) as position at time \(t\).
  • Evaluate and organize horizontal and vertical positions in a table.
  • Determine leftmost, rightmost, lowest, and highest positions on a stated time interval.
  • Connect zeros of \(x(t)\) and \(y(t)\) with coordinate-axis crossings.
  • Distinguish a particle's location at one time from its complete path.

1. Position Is an Ordered-Pair Function

When the parameter \(t\) represents time,

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

gives the object's position in the plane at time \(t\). The function \(x(t)\) records horizontal position and \(y(t)\) records vertical position. If coordinates are measured in meters and \(t\) in seconds, then both component outputs have units of meters.

2. Separate Time, Components, and Location

QuantityMeaning
\(t\)Time input
\(x(t)\)Horizontal position at time \(t\)
\(y(t)\)Vertical position at time \(t\)
\(\mathbf p(t)\)Complete planar position at time \(t\)

3. Running Example

Suppose a small inspection robot follows

\[x(t)=t-2,\qquad y(t)=(t-1)^2-4,\qquad -1\le t\le4.\]

The time origin is a chosen reference; negative \(t\) means the model begins one second before that reference, not that elapsed time is negative.

4. Generate the Motion Table

\(t\)\(x(t)\)\(y(t)\)Position
\(-1\)\(-3\)\(0\)\((-3,0)\)
\(0\)\(-2\)\(-3\)\((-2,-3)\)
\(1\)\(-1\)\(-4\)\((-1,-4)\)
\(2\)\(0\)\(-3\)\((0,-3)\)
\(3\)\(1\)\(0\)\((1,0)\)
\(4\)\(2\)\(5\)\((2,5)\)

Each row is a snapshot. Reading downward keeps the positions in chronological order.

5. The Plane Graph Shows the Path

Parametric Functions Modeling Planar Motion example graphTime orders the points along the path even when the rectangular graph alone does not show direction.
Time orders the points along the path even when the rectangular graph alone does not show direction.

The plotted points form an upward-opening parabolic path. Labels or arrows associated with increasing \(t\) show the sequence of positions, while the unlabelled geometric curve alone does not show when the robot reaches a point.

6. Identify Initial and Final Positions

Use the endpoints of the model's time domain:

\[\mathbf p(-1)=(-3,0),\qquad \mathbf p(4)=(2,5).\]

The robot starts at \((-3,0)\) when \(t=-1\) and ends at \((2,5)\) when \(t=4\). These statements include both a position and its associated time.

7. Horizontal Extrema Come from \(x(t)\)

The leftmost and rightmost locations are determined by the minimum and maximum values of \(x(t)\) on the time domain. Since \(x(t)=t-2\) increases on \([-1,4]\),

\[\min x(t)=-3\text{ at }t=-1,\qquad \max x(t)=2\text{ at }t=4.\]

Thus the leftmost position is \((-3,0)\), and the rightmost position is \((2,5)\).

8. Vertical Extrema Come from \(y(t)\)

The lowest and highest locations are determined by the minimum and maximum values of \(y(t)\). The vertex form

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

shows a minimum of \(-4\) at \(t=1\). Checking the domain endpoints gives \(y(-1)=0\) and \(y(4)=5\), so the maximum on the restricted interval is 5 at \(t=4\).

The lowest position is \((-1,-4)\), and the highest is \((2,5)\).

9. Match Each Question to a Component

QuestionAnalyzeReport
Leftmost or rightmost?Minimum or maximum of \(x(t)\)Time and full point
Lowest or highest?Minimum or maximum of \(y(t)\)Time and full point
On the y-axis?Zeros of \(x(t)\)\((0,y(t))\)
On the x-axis?Zeros of \(y(t)\)\((x(t),0)\)

10. Zeros of \(x(t)\) Give y-Intercepts

A point lies on the y-axis when its horizontal coordinate is zero:

\[x(t)=t-2=0\quad\Longrightarrow\quad t=2.\]

Then \(y(2)=-3\), so the particle crosses the y-axis at

\[\mathbf p(2)=(0,-3).\]

11. Zeros of \(y(t)\) Give x-Intercepts

A point lies on the x-axis when its vertical coordinate is zero:

\[(t-1)^2-4=0\quad\Longrightarrow\quad t=-1\text{ or }t=3.\]

Evaluate the other component at both times:

\[\mathbf p(-1)=(-3,0),\qquad\mathbf p(3)=(1,0).\]

The path has two x-intercepts within the modeled interval.

12. Coordinate Signs Locate the Region

At \(t=0\), \(\mathbf p(0)=(-2,-3)\), so the robot is left of the y-axis and below the x-axis. At \(t=4\), \(\mathbf p(4)=(2,5)\), so it is right of the y-axis and above the x-axis.

Use both signs to describe a planar position; one component alone specifies only horizontal or vertical placement.

13. Position at a Time Is Not the Entire Path

RepresentationWhat it answers
\(\mathbf p(2)=(0,-3)\)Where is the object at one time?
Table of \(t,x(t),y(t)\)Where is it at several selected times?
Graph in the xy-planeWhich path is traced?
Graphs of x versus t and y versus tHow does each coordinate vary over time?

Keeping these views separate prevents a time input from being mistaken for an x-coordinate.

14. Use Technology Purposefully

  1. Enter both component functions in parametric mode.
  2. Set the exact time domain.
  3. Create a table containing \(t\), \(x(t)\), and \(y(t)\).
  4. Use TRACE to verify positions at important times.
  5. Inspect each component graph for extrema and zeros.
  6. Confirm every reported time by evaluating the full ordered pair.

15. AP Workflow and Common Errors

  1. State the time domain and coordinate units.
  2. Evaluate the requested position using both components.
  3. Use \(x(t)\) for horizontal extrema and \(y(t)\) for vertical extrema.
  4. Include endpoint values when finding absolute extrema on a closed interval.
  5. Solve \(x(t)=0\) for y-axis crossings and \(y(t)=0\) for x-axis crossings.
  6. Substitute each time into the other component to obtain a full point.
  7. Report the time, position, and contextual meaning requested.
  • Do not report \(t\) as one of the spatial coordinates.
  • Do not use a zero of \(x(t)\) as an x-intercept.
  • Do not state only the extreme coordinate value when a position is requested.
  • Do not ignore the time-domain endpoints.
  • Do not infer velocity or speed from position alone; those belong to a rate-of-change analysis.

Key Takeaways

  • Interpret parametric coordinates as position components and connect parameter values to location and direction.
  • Core relationship: \(\mathbf r(t)=\langle x(t),y(t)\rangle\)
  • Error check: Do not confuse the parameter value with an x-coordinate unless the formula says so.
Checkpoint · Topic 4.2
  1. For \(\mathbf p(t)=(2t-3,t^2-4)\), find the position at \(t=2\).
  2. On \(-1\le t\le3\), determine the leftmost and rightmost positions for \(x(t)=t+1,\ y(t)=2-t^2\).
  3. For the same motion, determine the lowest and highest positions on the interval.
  4. Explain why a zero of \(x(t)\) corresponds to a y-intercept rather than an x-intercept.
  5. Find all coordinate-axis crossings of \(x(t)=t-1,\ y(t)=t^2-1\) on \(-2\le t\le2\).