Question 1

Which transformation is applied to the parent function $f(x) = \frac{1}{x}$ to obtain $g(x) = \frac{1}{x-2} + 3$?

Accepted Answer

Shift right by 2 units and up by 3 units

Question 2

Given $g(x) = \frac{1}{x+3} - 2$, what are the equations of the vertical and horizontal asymptotes?

Accepted Answer

Vertical: $x = -3$, Horizontal: $y = -2$

Question 3

For the rational function $f(x) = \frac{2x-3}{x-1}$, what is the domain?

Accepted Answer

$(-\infty, 1) \cup (1, \infty)$

Question 4

What is the $y$-intercept of $f(x) = \frac{2x-3}{x-1}$?

Accepted Answer

$(0, 3)$

Question 5

A rational function $f(x)$ has a hole at $x = 2$. Which of the following must be true?

Accepted Answer

Both the numerator and denominator have a factor of $(x-2)$

Question 6

Given $f(x) = \frac{x^2+3}{x^2+5x+6}$, what are the vertical asymptotes?

Accepted Answer

$x = -2$ and $x = -3$

Question 7

For $f(x) = \frac{2x^2}{x^2-4}$, what is the horizontal asymptote?

Accepted Answer

$y = 2$

Question 8

Which of the following describes the end behavior of $f(x) = \frac{2x^2}{x^2-4}$ as $x 	o \infty$?

Accepted Answer

$f(x) 	o 2$

Question 9

If $f(x) = \frac{x^2-9}{x^2-4}$, what are the $x$-intercepts?

Accepted Answer

$x = 3$ and $x = -3$

Question 10

For $f(x) = \frac{x^2+3}{x^2+5x+6}$, what is the $y$-intercept?

Accepted Answer

$(0, 0.5)$

Question 11

Which transformation is applied to $f(x) = \frac{1}{x}$ to obtain $g(x) = -\frac{1}{x+1} + 2$?

Accepted Answer

Reflect over $x$-axis, shift left by 1, up by 2

Question 12

A rational function $f(x)$ has a vertical asymptote at $x = 4$ and a horizontal asymptote at $y = 0$. Which could be its equation?

Accepted Answer

$f(x) = \frac{2x}{x-4}$

Question 13

For $f(x) = \frac{2x-3}{x-1}$, what is the equation of the vertical asymptote?

Accepted Answer

$x = 1$

Question 14

If $f(x) = \frac{x^2+3}{x^2+5x+6}$, what is the range of $f(x)$?

Accepted Answer

$(-\infty, \infty)$ except $y = 0$

Question 15

Which of the following is true about the graph of $f(x) = \frac{1}{x}$?

Accepted Answer

It has vertical asymptote at $x = 0$ and horizontal asymptote at $y = 0$

Question 16

For $f(x) = \frac{2x^2}{x^2-4}$, what are the vertical asymptotes?

Accepted Answer

$x = 2$ and $x = -2$

Question 17

If $f(x) = \frac{x^2-9}{x^2-4}$, what is the domain?

Accepted Answer

$(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$

Question 18

Which of the following rational functions has a hole at $x = 1$?

Accepted Answer

$f(x) = \frac{(x-1)(x+2)}{(x-1)(x-3)}$

Graphing Rational Functions and Transformations: Study Notes for College Algebra

Study Guide - Practice Questions