Q: The weight $W$ of an object varies inversely as the square of the distance $d$ from the center of the Earth. At sea level ($d = 3978$ miles), an astronaut weighs $220$ pounds. What is the astronaut's weight when he is $200$ miles above the surface?

$W = 220 \times \left(\frac{3978}{4178}\right)^2 \approx 199$ pounds

Q: Miguel deposits $8,000$ into two accounts: a CD that offers $4\%$ interest rate and a Mutual Bond with $5\%$ interest rate. If the total interest earned after a year is $380$, how much was deposited into each account?

$x = 3,000$ in CD, $y = 5,000$ in Bond

Q: Find the domain and all asymptotes of the function $f(x) = \frac{x^2 + 3x + 1}{4x^2 - 9}$.

Domain: $x \neq \frac{3}{2}, x \neq -\frac{3}{2}$; Vertical asymptotes: $x = \frac{3}{2}, x = -\frac{3}{2}$

Q: While on top of a $24$-ft building, an object is thrown into the air. The height of the object is given by $h(t) = -16t^2 + 24t + 24$, where $t$ denotes the time (in seconds) after the ball was thrown. What is the maximum height of the ball and when does it occur?

Maximum height: $42$ ft at $t = 0.75$ sec

Question 1

Find the $x$- and $y$-intercepts of the line $5x - 2y = 10$ and graph the line. What are the intercepts?

Accepted Answer

$x$-intercept: $(2, 0)$; $y$-intercept: $(0, -5)$

Question 2

Given the function $f(x) = 2x^2 - x + 5$, evaluate $f(-1)$.

Accepted Answer

$f(-1) = 8$

Question 3

Find the equation of the line containing the points $(4, -10)$ and $(-8, -12)$. What is the slope?

Accepted Answer

Slope $m = \frac{-2}{12} = -\frac{1}{6}$

Question 4

Find the equation of the line parallel to $2x + 3y = -6$ and passing through the point $(-1, 3)$.

Accepted Answer

$2x + 3y = 7$

Question 5

The weight $W$ of an object varies inversely as the square of the distance $d$ from the center of the Earth. At sea level ($d = 3978$ miles), an astronaut weighs $220$ pounds. What is the astronaut's weight when he is $200$ miles above the surface?

Accepted Answer

$W = 220 	imes \left(\frac{3978}{4178}ight)^2 \approx 199$ pounds

Question 6

Find the equation of the line perpendicular to $2x + 3y = -6$ and passing through the point $(-1, 3)$.

Accepted Answer

$3x - 2y = -9$

Question 7

Solve the inequality $2x - 1 < x - 3$ and express the solution in interval notation.

Accepted Answer

$x < -2$

Question 8

A boat moves at a rate of $11$ mi/h in still water. If the river current flows at a rate of $5$ mi/h, how long does it take for the boat to travel $33$ miles upstream?

Accepted Answer

$\frac{33}{6}$ hours

Question 9

Miguel deposits $8,000$ into two accounts: a CD that offers $4\%$ interest rate and a Mutual Bond with $5\%$ interest rate. If the total interest earned after a year is $380$, how much was deposited into each account?

Accepted Answer

$x = 3,000$ in CD, $y = 5,000$ in Bond

Question 10

Given $m(x) = x^3 - 5x$, find and simplify the difference quotient $\frac{m(x+h) - m(x)}{h}$.

Accepted Answer

$3x^2h + 3xh^2 + h^3 - 5h$

Question 11

Given the quadratic equation $y = 5x^2 - 10x + 7$, what is the axis of symmetry?

Accepted Answer

$x = 1$

Question 12

Find the domain and all asymptotes of the function $f(x) = \frac{x^2 + 3x + 1}{4x^2 - 9}$.

Accepted Answer

Domain: $x 
eq \frac{3}{2}, x 
eq -\frac{3}{2}$; Vertical asymptotes: $x = \frac{3}{2}, x = -\frac{3}{2}$

Question 13

While on top of a $24$-ft building, an object is thrown into the air. The height of the object is given by $h(t) = -16t^2 + 24t + 24$, where $t$ denotes the time (in seconds) after the ball was thrown. What is the maximum height of the ball and when does it occur?

Accepted Answer

Maximum height: $42$ ft at $t = 0.75$ sec

Question 14

Given the point $(1, 5)$ on the graph of $y = f(x)$, find the corresponding point for the graph of $y = f(x - 2) + 3$.

Accepted Answer

$(3, 8)$

Question 15

Simplify $(3 + 4i)(4 - 3i)$ and express your answer in standard form $(a + bi)$.

Accepted Answer

$24 + 7i$

Question 16

Find the domain of $h(y) = \frac{9}{y - 4}$.

Accepted Answer

All real $y$ except $y = 4$

Question 17

A parabola has a vertex at the point $(3, 4)$ and passes through the point $(-1, 7)$. Write its equation in vertex form.

Accepted Answer

$y = \frac{1}{4}(x - 3)^2 + 4$

Question 18

If the point $(7, -1)$ is on the graph of $y = f(x)$, what is the corresponding point on the graph of $y = -2f(x - 3) + 5$?

Accepted Answer

$(10, 7)$

Question 19

Turnip yield $T$ is related to job data inversely as the number of people $P$ working on that job. If it takes $5$ hours for $7$ bricklayers to build a wall, how long will it take $10$ bricklayers to complete the same wall?

Accepted Answer

$3.5$ hours

Precalculus Midterm Exam Review: Key Concepts and Practice Problems

Study Guide - Practice Questions

Study Guide - Flashcards