Unit 4 check

Unit 4 check: Functions Involving Parameters, Vectors, and Matrices

14 questions from 14 lessons in Unit 4. Answer each one, and if you miss, follow the link back to where it is taught, then try again. Your score counts first tries.

Not assessed on the AP Exam. Unit 4 is part of the course, but nothing in it is on the exam.

Question 1 · from 4.1 · Fill the statement

The parametric function is $f (t) = (t^{2} - 1, 3 t)$ . Tap a blank, then tap an answer.

The single independent variable of $f$ is , called the parameter. In $f (t) = (x (t), y (t))$ , the letters $x$ and $y$ are names of two of it. The output $f (2)$ is the point .

Question 2 · from 4.2 · Calculate

A particle's position at time $t$ is $f (t) = (t^{2} - 6 t + 5, 2 t + 1)$ for $0 \leq t \leq 5$ . What is the least $x$ -coordinate the particle reaches, the farthest left its path goes?

Your answer

Question 3 · from 4.3 · Decide and justify

Compare $f (t) = (t - 1, (t - 1)^{2})$ and $g (t) = (1 - t, (1 - t)^{2})$ , each for $- 1 \leq t \leq 3$ . Which is true?

Which reason supports your answer?

Question 4 · from 4.4 · Fill the statement

A ball rolls in a straight line at a steady speed from $(- 2, 5)$ to $(6, - 1)$ , in meters, in 4 seconds, with $t$ in seconds. Tap a blank, then tap a number.

In the ball's parametric function, the initial position has $x$ -coordinate . Each second, $x$ changes by meters and $y$ changes by meters, and the domain is $0 \leq t \leq$ .

Setup for question 5

You need paper for this one. Take the equation $x = 3 y - y^{3}$ . It is already solved for $x$ , so pick values of $y$ and compute $x$ . Here are nine solutions.

Draw axes from $- 3$ to 3 both ways, with $x$ across and $y$ up. Plot the nine points and join them in order of $y$ , from the bottom of the table to the top. You should get an S shape lying on its side. Then answer the two questions.

Solutions of $x = 3 y - y^{3}$
$y$	$x$
$- 2$	$2$
$- 1.5$	$- 1.125$
$- 1$	$- 2$
$- 0.5$	$- 1.375$
$0$	$0$
$0.5$	$1.375$
$1$	$2$
$1.5$	$1.125$
$2$	$- 2$

Question 5 · from 4.5 · Calculate

How many points of the graph of $x = 3 y - y^{3}$ have $x = 0$ ?

Your answer

Setup for question 6

A stone arch over a canal is shaped like a parabola. An engineer puts it on a grid in meters, with the water surface along the $x$ -axis. The top of the arch is at $(10, 8)$ , and its feet touch the water at $(2, 0)$ and $(18, 0)$ . Work on paper. You need the arch's equation first, then you can ask what fits under it.

Question 6 · from 4.6 · Decide and justify

A canal arch is a parabola with its top at $(10, 8)$ and its feet at $(2, 0)$ and $(18, 0)$ , in meters, with the water along $y = 0$ . A barge carries a load 12 meters wide that stands 4 meters above the water, centered under the top of the arch at $x = 10$ . Does the load clear the arch?

Which reason supports your answer?

Setup for question 7

A level designer moves three objects along paths on a grid measured in tiles. Each path is a parametric function, given in the table.

Three paths in the level
Path	$x (t)$	$y (t)$	Interval of $t$
$A$	$2 + 3 \cos t$	$- 1 + 3 \sin t$	$0 \leq t \leq 2 π$
$B$	$1 + 2 \tan t$	$- 3 + 5 \sec t$	$0 \leq t \leq 2 π, \cos t \neq 0$
$C$	$t^{2} - 1$	$t$	$- \infty < t < \infty$

Question 7 · from 4.7 · Select all

Path C is $(x (t), y (t)) = (t^{2} - 1, t)$ for every real $t$ . Which two equations does Path C satisfy for every $t$ ? Pick two.

Select 2 answers, then check.

Setup for question 8

A survey drone flies straight legs over a field. Its log records each leg as a vector on a map in meters, with $x$ measured east and $y$ measured north.

On a windy morning the log also records two velocities: $\vec{a}$ , the drone's motion through the air, with magnitude 33 kilometers per hour, and $\vec{w}$ , the wind, with magnitude 7 kilometers per hour. Drawn tail to tail, $\vec{a}$ and $\vec{w}$ make an angle of $\frac{π}{3}$ , and the drone's speed over the ground, $‖ \vec{a} + \vec{w} ‖$ , is 37 kilometers per hour.

Three legs of the flight, in meters
Leg	Vector
$\vec{p}$	$⟨ - 4 \sqrt{3}, - 4 ⟩$
$\vec{q}$	$⟨ 12, - 5 ⟩$
$\vec{r}$	$⟨ - 3, 17 ⟩$

Question 8 · from 4.8 · Calculate

A drone flies the leg $\vec{q} = ⟨ 12, - 5 ⟩$ and then the leg $\vec{r} = ⟨ - 3, 17 ⟩$ , in meters. How far is it, in meters, from the point where the leg $\vec{q}$ began? That is, what is $‖ \vec{q} + \vec{r} ‖$ ?

Your answer

Setup for question 9

A tugboat T is tracked from a harbor tower at the origin. Its position $p (t)$ is in kilometers, with east as the positive $x$ direction and north as the positive $y$ direction, and its velocity $v (t)$ is in kilometers per hour. Time $t$ is in hours.

Position and velocity of the tugboat T
$t$ (hours)	$p (t)$ (kilometers)	$v (t)$ (kilometers per hour)
$0$	$⟨ 17, 14 ⟩$	$⟨ - 6, 1 ⟩$
$1$	$⟨ 12, 14 ⟩$	$⟨ - 4, - 1 ⟩$
$2$	$⟨ 9, 12 ⟩$	$⟨ - 2, - 3 ⟩$
$3$	$⟨ 8, 8 ⟩$	$⟨ 0, - 5 ⟩$
$4$	$⟨ 9, 2 ⟩$	$⟨ 2, - 7 ⟩$

Question 9 · from 4.9 · Multiple choice

At $t = 3$ hours the tugboat T's position is $⟨ 8, 8 ⟩$ kilometers from the tower and its velocity is $⟨ 0, - 5 ⟩$ kilometers per hour, with north as the positive $y$ direction. Which describes T's motion at $t = 3$ ?

Question 10 · from 4.10 · Decide and justify

Let $A = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]$ and $B = [\begin{matrix} 1 & 0 \\ 3 & 1 \end{matrix}]$ . Is $A B$ equal to $B A$ ?

Which reason supports your answer?

Question 11 · from 4.11 · Fill the statement

Let $A = [\begin{matrix} 2 & 1 \\ 7 & 4 \end{matrix}]$ and $B = [\begin{matrix} 4 & - 1 \\ - 7 & 2 \end{matrix}]$ . Multiply $A$ by $B$ , then complete the statement. Tap a blank, then tap an answer.

$A B$ = , so $B$ = .

Setup for question 12

A drawing app stores the four corners of a logo as the columns of a matrix $C$ , each corner as the vector from the origin to it. It transforms the logo with $L (\vec{v}) = M \vec{v}$ , and the image corners are the columns of $M C$ .

The corner matrix C and the transformation matrix M
Name	Matrix
$C$	$[\begin{matrix} 0 & 3 & 3 & 1 \\ 0 & 0 & 2 & 2 \end{matrix}]$
$M$	$[\begin{matrix} 1 & - 2 \\ 2 & 1 \end{matrix}]$

Question 12 · from 4.12 · Multiple choice

A drawing app transforms a logo with corners $(0, 0)$ , $(3, 0)$ , $(3, 2)$ and $(1, 2)$ by $L (\vec{v}) = M \vec{v}$ , where $M = [\begin{matrix} 1 & - 2 \\ 2 & 1 \end{matrix}]$ . Which corner is its own image?

Question 13 · from 4.13 · Spot the error

A student finds the inverse of the transformation $L$ that rotates every vector $\frac{π}{6}$ radians counterclockwise about the origin. Tap the first line that goes wrong.

Question 14 · from 4.14 · Decide and justify

A gym has 800 members, each on a monthly plan or an annual plan. Each year, 30% of monthly members switch to annual and 10% of annual members switch to monthly. With monthly first, $T = [\begin{matrix} 0.7 & 0.1 \\ 0.3 & 0.9 \end{matrix}]$ . Is 300 monthly and 500 annual members the steady state?

Which reason supports your answer?

0 of 14 answered.