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AP Precalculus: Functions Involving Parameters, Vectors, and Matrices — Practice Questions & Answers
Unit 4 widens the idea of a function beyond the familiar y as a function of x. You learn to describe motion and curves with parametric equations, handle implicitly defined relations and conic sections, represent quantities that have both size and direction with vectors, and organize and transform data with matrices. Although the College Board does not test this unit on the AP exam, it builds the foundation for calculus, linear algebra, and physics, so the concepts are well worth mastering.
74 practice questions available for this topic — here are 10 with full answers and explanations.
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Practice questions with answers
1
Multiple choice · Easy
Point A is at $(1,2)$ and point B is at $(4,6)$. What is the component form of the vector $\vec{AB}$?
$\langle 3,4\rangle$
$\langle 5,8\rangle$
$\langle -3,-4\rangle$
$\langle 3,8\rangle$
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Why To find $\vec{AB}$ subtract the initial point from the terminal point: $\langle 4-1, 6-2\rangle = \langle 3,4\rangle$.
2
Multiple choice · Easy
What is the magnitude of the vector $\langle 3,4\rangle$?
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Why The magnitude is $\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=\sqrt{25}=5$.
3
Multiple choice · Easy
Compute $\langle 2,3\rangle + \langle 1,-5\rangle$.
$\langle 3,-2\rangle$
$\langle 3,8\rangle$
$\langle 1,8\rangle$
$\langle 3,2\rangle$
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Why Add corresponding components: $\langle 2+1, 3+(-5)\rangle = \langle 3,-2\rangle$.
4
Multiple choice · Easy
Compute the scalar multiple $3\langle 2,-1\rangle$.
$\langle 6,-3\rangle$
$\langle 5,2\rangle$
$\langle 6,-1\rangle$
$\langle 2,-3\rangle$
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Why Multiply each component by 3: $\langle 3\cdot 2, 3\cdot(-1)\rangle = \langle 6,-3\rangle$.
5
Multiple choice · Easy
What is the determinant of $\begin{pmatrix}2 & 3\\1 & 4\end{pmatrix}$?
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Why The determinant of a $2\times 2$ matrix is $ad-bc = (2)(4)-(3)(1) = 8-3 = 5$.
6
Multiple choice · Easy
Compute $\begin{pmatrix}1 & 2\\3 & 4\end{pmatrix} + \begin{pmatrix}0 & 1\\2 & 1\end{pmatrix}$.
$\begin{pmatrix}1 & 3\\5 & 5\end{pmatrix}$
$\begin{pmatrix}1 & 2\\5 & 5\end{pmatrix}$
$\begin{pmatrix}1 & 3\\6 & 5\end{pmatrix}$
$\begin{pmatrix}0 & 2\\6 & 4\end{pmatrix}$
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Why Add corresponding entries: $\begin{pmatrix}1+0 & 2+1\\3+2 & 4+1\end{pmatrix} = \begin{pmatrix}1 & 3\\5 & 5\end{pmatrix}$.
7
Fill in the blank · Easy
The magnitude of the vector $\langle 6,8\rangle$ is
Check answer
Answer:
10
Why The magnitude is $\sqrt{6^{2}+8^{2}}=\sqrt{36+64}=\sqrt{100}=10$.
8
Fill in the blank · Easy
In the $2\times 2$ identity matrix, the entries on the main diagonal equal 1 and all other entries equal
Check answer
Answer:
0
Why The identity matrix $\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}$ has 1's on the main diagonal and 0's elsewhere.
10
Multiple choice · Medium
Compute the dot product $\langle 3,-2\rangle \cdot \langle 4,5\rangle$.
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Why The dot product is $(3)(4)+(-2)(5) = 12-10 = 2$.
Key terms in Functions Involving Parameters, Vectors, and Matrices
Parameter: An independent variable, often $t$, used to express coordinates of points along a curve.
Parametric Function: A function in which $x$ and $y$ are each defined in terms of a parameter.
Parametric Equations: A pair of equations $x = f(t)$ and $y = g(t)$ describing a curve using a parameter.
Parametrization: The process of expressing a curve using a parameter.
Plane Curve: A set of points traced in the plane by parametric equations.
Orientation: The direction in which a parametric curve is traced as the parameter increases.
Direction of Motion: The path direction indicated by increasing parameter values along a parametric curve.
Eliminating the Parameter: Combining parametric equations to obtain a single equation in $x$ and $y$.
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