$x \in \mathbb{R}, x \neq 2, x \neq -2$

$f(3) = 3^2 - 4 \times 3 + 1 = 9 - 12 + 1 = -2$

$f(0) = 0^2 - 4 \times 0 + 1 = 1$

Question 1

Given the points $A(2, 3)$ and $B(5, 7)$, what is the distance between them?

Accepted Answer

$\sqrt{(5-2)^2 + (7-3)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Question 2

Find the slope of the line passing through the points $(-1, 4)$ and $(3, -2)$.

Accepted Answer

$m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}$

Question 3

What is the equation of the circle with center $(2, -3)$ and radius $5$?

Accepted Answer

$(x - 2)^2 + (y + 3)^2 = 25$

Question 4

Which of the following is the equation of a line parallel to $y = 2x + 1$ and passing through $(4, -3)$?

Accepted Answer

$y = 2x - 11$

Question 5

Given the function $f(x) = \frac{1}{x^2 - 4}$, what is its domain?

Accepted Answer

$x \in \mathbb{R}, x 
eq 2, x 
eq -2$

Question 6

For the function $f(x) = x^2 - 4x + 1$, what is the value of $f(3)$?

Accepted Answer

$f(3) = 3^2 - 4 	imes 3 + 1 = 9 - 12 + 1 = -2$

Question 7

Given $f(x) = x^2 - 4x + 1$, find all $x$ such that $f(x) = 13$.

Accepted Answer

$x = 7$ or $x = -2$

Question 8

Which of the following describes the transformation from $y = f(x)$ to $y = f(x) + 2$?

Accepted Answer

Vertical shift up by 2 units

Question 9

Given $f(x) = \frac{1}{x}$, which of the following statements is true about its symmetry?

Accepted Answer

$f(x)$ is an odd function

Question 10

Find the $y$-intercept of the function $f(x) = x^2 - 4x + 1$.

Accepted Answer

$f(0) = 0^2 - 4 	imes 0 + 1 = 1$

Question 11

Given the function $f(x) = \sqrt{x}$, which transformation results in $y = -2\sqrt{x}$?

Accepted Answer

Vertical stretch by a factor of 2 and reflection over the $x$-axis

Question 12

For the function $f(x) = x^2 - 4x + 1$, what are the $x$-intercepts?

Accepted Answer

$x = 2 + \sqrt{3}$ and $x = 2 - \sqrt{3}$

Question 13

Which of the following is the domain of $f(x) = \frac{1}{x-3}$?

Accepted Answer

$x \in \mathbb{R}, x 
eq 3$

Question 14

If $f(x) = x^2$, what is the effect of the transformation $g(x) = f(x-2) + 3$?

Accepted Answer

Shift right by 2 units and up by 3 units

Question 15

Given $f(x) = x^2 - 4x + 1$, what is the vertex of the parabola?

Accepted Answer

$(2, -3)$

Question 16

Which of the following is the equation of a circle with center $(-1, 4)$ and radius $3$?

Accepted Answer

$(x + 1)^2 + (y - 4)^2 = 9$

Question 17

For the function $f(x) = \frac{1}{x^2 + 1}$, what is its range?

Accepted Answer

$0 < f(x) \leq 1$

Question 18

Which of the following lines is perpendicular to $y = \frac{1}{2}x + 3$ and passes through $(0, 1)$?

Accepted Answer

$y = -2x + 1$

Question 19

Given $f(x) = x^2 - 4x + 1$, for which values of $x$ is the function decreasing?

Accepted Answer

$x < 2$

College Algebra Study Guide: Functions, Graphs, and Equations

Study Guide - Practice Questions