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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 4.3 · Optional

Parametric Functions and Rates of Change

Use the separate changes in horizontal and vertical position to describe planar direction and calculate the slope between two positions.

Learning Goals

  • Determine planar direction from the behavior of \(x(t)\) and \(y(t)\).
  • Calculate average horizontal and vertical rates of change over a time interval.
  • Interpret component rates with appropriate units.
  • Find the slope between two positions using the ratio of component rates.
  • Explain how different parametrizations can traverse the same curve differently.

1. Analyze Motion One Component at a Time

For \(\mathbf p(t)=(x(t),y(t))\), horizontal direction comes from the behavior of \(x(t)\), while vertical direction comes from \(y(t)\).

Component behavior as \(t\) increasesDirection
\(x(t)\) increasesRight
\(x(t)\) decreasesLeft
\(y(t)\) increasesUp
\(y(t)\) decreasesDown

Combine one horizontal statement and one vertical statement to describe the plane motion.

2. Four Common Direction Combinations

\(x(t)\)\(y(t)\)Planar direction
IncreasingIncreasingRight and up
IncreasingDecreasingRight and down
DecreasingIncreasingLeft and up
DecreasingDecreasingLeft and down

3. Compute Two Average Rates

On \(t_1\le t\le t_2\), calculate each component independently:

\[\operatorname{AROC}_x=\frac{x(t_2)-x(t_1)}{t_2-t_1},\qquad \operatorname{AROC}_y=\frac{y(t_2)-y(t_1)}{t_2-t_1}.\]

If position is measured in meters and time in seconds, both results have units of meters per second. They describe average component changes, not the total path length traveled.

4. Continue the Planar-Motion Model

Use the previous lesson's robot model:

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

The horizontal component increases throughout the domain. The vertical component decreases until \(t=1\), then increases after \(t=1\).

5. Direction Changes Along the Path

Parametric Functions and Rates of Change example graphHorizontal and vertical changes can be analyzed separately along the same path.
Horizontal and vertical changes can be analyzed separately along the same path.

The robot moves right and down before reaching \((-1,-4)\) at \(t=1\), then moves right and up. A single geometric point on the curve does not reveal this orientation without parameter information.

6. Example: Right and Down on \([0,1]\)

The endpoint positions are \(\mathbf p(0)=(-2,-3)\) and \(\mathbf p(1)=(-1,-4)\). Therefore,

\[\operatorname{AROC}_x=\frac{-1-(-2)}{1-0}=1,\qquad \operatorname{AROC}_y=\frac{-4-(-3)}{1-0}=-1.\]

On average, horizontal position increases 1 meter per second while vertical position decreases 1 meter per second, so the motion is right and down.

7. Example: Right and Up on \([1,2]\)

Now \(\mathbf p(1)=(-1,-4)\) and \(\mathbf p(2)=(0,-3)\), so

\[\operatorname{AROC}_x=1,\qquad \operatorname{AROC}_y=1.\]

The horizontal rate keeps the same sign, but the vertical average rate changes from negative to positive. This agrees with the turn from downward to upward motion at the path's lowest point.

8. Summarize Rates in an Interval Table

Interval\(\operatorname{AROC}_x\)\(\operatorname{AROC}_y\)Average direction
\([-1,0]\)\(1\)\(-3\)Right and down
\([0,1]\)\(1\)\(-1\)Right and down
\([1,2]\)\(1\)\(1\)Right and up
\([2,3]\)\(1\)\(3\)Right and up
\([3,4]\)\(1\)\(5\)Right and up

The growing vertical averages reflect the upward-opening shape, while the horizontal average remains constant.

9. The Same Point Can Have Different Directions

Both parametrizations below trace \(y=x^2\):

\[\mathbf f(t)=(t,t^2),\qquad \mathbf g(t)=(-t,t^2).\]

Function \(\mathbf f\) reaches \((1,1)\) at \(t=1\) while moving right and up. Function \(\mathbf g\) reaches \((1,1)\) at \(t=-1\) while moving left and down. The point and curve are the same, but the parameterized motions differ.

10. Different Parametrizations Change Timing Too

The functions

\[(t,t^2)\quad\text{and}\quad(2t,4t^2)\]

trace the same parabola \(y=x^2\), but equal changes in \(t\) produce different horizontal and vertical component changes. A curve specifies the path; a parametrization also specifies how inputs are assigned along that path.

11. Convert Component Rates into Secant Slope

If \(\operatorname{AROC}_x\ne0\), the slope between the two endpoint positions is

\[\frac{\operatorname{AROC}_y}{\operatorname{AROC}_x}=\frac{\dfrac{y(t_2)-y(t_1)}{t_2-t_1}}{\dfrac{x(t_2)-x(t_1)}{t_2-t_1}}=\frac{y(t_2)-y(t_1)}{x(t_2)-x(t_1)}=\frac{\Delta y}{\Delta x}.\]

The common time change cancels, leaving the ordinary slope of the secant line in the xy-plane.

12. Secant-Slope Example on \([2,4]\)

The positions are \(\mathbf p(2)=(0,-3)\) and \(\mathbf p(4)=(2,5)\). The component rates are

\[\operatorname{AROC}_x=\frac{2-0}{4-2}=1,\qquad \operatorname{AROC}_y=\frac{5-(-3)}{4-2}=4.\]

Thus the endpoint secant slope is

\[m_{\text{sec}}=\frac{4}{1}=4=\frac{5-(-3)}{2-0}.\]

13. A Secant Slope Summarizes Endpoints

On \([0,2]\), the robot moves from \((-2,-3)\) to \((0,-3)\). Therefore, \(\operatorname{AROC}_x=1\), \(\operatorname{AROC}_y=0\), and the secant slope is 0.

This does not mean the path is horizontal throughout the interval. The robot first moves down to \((-1,-4)\) and then back up. An average can hide changes occurring between the endpoints.

14. Handle Zero Horizontal Change

If \(\operatorname{AROC}_x=0\) but \(\operatorname{AROC}_y\ne0\), the endpoint secant is vertical and its slope is undefined. If both component averages are zero, the endpoint positions are the same; this does not prove the particle remained still between them.

Never divide by the horizontal component rate before checking that it is nonzero.

15. AP Workflow and Common Errors

  1. State the parameter interval and endpoint positions.
  2. Compute \(\operatorname{AROC}_x\) and \(\operatorname{AROC}_y\) separately in the same time order.
  3. Interpret each sign as right/left or up/down.
  4. Attach component-rate units.
  5. Divide vertical rate by horizontal rate only when the horizontal rate is nonzero.
  6. Interpret the ratio as an endpoint secant slope, not total speed.
  7. Check whether an average hides a direction change inside the interval.
  • Do not divide \(x\)-change by \(y\)-change when finding slope.
  • Do not mix different time intervals for the two component rates.
  • Do not interpret a negative vertical rate as leftward motion.
  • Do not claim equal endpoint positions mean no motion occurred.
  • Do not confuse average component rates with distance traveled per unit time.

Key Takeaways

  • Compute and interpret componentwise average rates of change for parametrically defined motion.
  • Core relationship: \(\frac{\Delta x}{\Delta t},\qquad\frac{\Delta y}{\Delta t}\)
  • Error check: Do not divide the y-change by the x-change when the question asks for change per unit time.
Checkpoint · Topic 4.3
  1. For \(x(t)=3-t,\ y(t)=t^2\), determine the average planar direction on \([1,2]\).
  2. Find both component average rates for \(\mathbf p(t)=(2t-1,t^2-4)\) on \([0,3]\) and include units.
  3. Use those rates to find the slope between the endpoint positions.
  4. Explain how \((t,t^2)\) and \((-t,t^2)\) traverse the same parabola differently.
  5. Describe what happens to the secant slope when the average horizontal rate is 0 but the average vertical rate is nonzero.