Q: Find the equation of the line passing through $(3,4)$ and perpendicular to the line $y = 2x$.

$y = -\frac{1}{2}x + \frac{11}{2}$

Q: Given $f(x) = x^2$ and $g(x) = -x^2 + 7$, which graph represents the transformation of $f(x)$ to $g(x)$?

Reflect over x-axis, shift 7 units up

Q: Solve the rational inequality $\frac{x-4}{x-2} \geq 0$ and express the solution in interval notation.

$(-\infty, 2) \cup [4, \infty)$

Q: Find the zeros and their multiplicities for $f(x) = (x-1)^2(x+3)$ and state whether the graph crosses or touches the x-axis at each zero.

Zero at $x=1$ (multiplicity 2, touches), zero at $x=-3$ (multiplicity 1, crosses)

Q: Which graph best represents the polynomial $f(x) = -2x^6 + 3x^3(x+1)$?

Graph with end behavior down on both sides, crosses at $x=0$ and $x=-1$, touches at $x=3$

Q: Solve the system of inequalities and graph the solution set: $y > 2x - 4$ and $y < -x + 3$.

Region above $y = 2x - 4$ and below $y = -x + 3$

Question 1

Solve for $x$ in the equation $9 - 2(3 - x) = 5x$.

Accepted Answer

$x = \frac{15}{7}$

Question 2

Which of the following is the solution to the quadratic equation $x^2 - 5x + 6 = 0$?

Accepted Answer

$x = 2$ and $x = 3$

Question 3

Given $f(x) = x^2 + 2x + 1$, find $f(x+h) - f(x)$ and simplify.

Accepted Answer

$2xh + h^2 + 2h$

Question 4

Solve the equation by completing the square: $x^2 - 6x + 5 = 0$.

Accepted Answer

$x = 1$ and $x = 5$

Question 5

Which of the following is the correct solution set for the compound inequality $8 < 4x < 16$?

Accepted Answer

$x \in (2, 4)$

Question 6

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

Accepted Answer

$(-1, 3)$

Question 7

Find the equation of the line passing through $(3,4)$ and perpendicular to the line $y = 2x$.

Accepted Answer

$y = -\frac{1}{2}x + \frac{11}{2}$

Question 8

Given $f(x) = x^2$ and $g(x) = -x^2 + 7$, which graph represents the transformation of $f(x)$ to $g(x)$?

Accepted Answer

Reflect over x-axis, shift 7 units up

Question 9

Find the domain of the function $f(x) = \frac{1}{x^2 - 6}$.

Accepted Answer

$x 
eq \pm \sqrt{6}$

Question 10

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

Accepted Answer

$(-2, 2)$

Question 11

Solve the exponential equation $e^{2x} = 8$ for $x$.

Accepted Answer

$x = \frac{1}{2} \ln 8$

Question 12

Which of the following is the correct solution to the logarithmic equation $\log(x^2 - 4x) = 2$?

Accepted Answer

$x = 5$ and $x = -1$

Question 13

A population is growing at a rate of 2.5% per year. If the current population is 17,638,000, what will the population be in 5 years? Use the formula $P = P_0(1 + r)^t$.

Accepted Answer

$P \approx 19,999,600$

Question 14

Solve the rational inequality $\frac{x-4}{x-2} \geq 0$ and express the solution in interval notation.

Accepted Answer

$(-\infty, 2) \cup [4, \infty)$

Question 15

Find the zeros and their multiplicities for $f(x) = (x-1)^2(x+3)$ and state whether the graph crosses or touches the x-axis at each zero.

Accepted Answer

Zero at $x=1$ (multiplicity 2, touches), zero at $x=-3$ (multiplicity 1, crosses)

Question 16

Which graph best represents the polynomial $f(x) = -2x^6 + 3x^3(x+1)$?

Accepted Answer

Graph with end behavior down on both sides, crosses at $x=0$ and $x=-1$, touches at $x=3$

Question 17

Solve the system of inequalities and graph the solution set: $y > 2x - 4$ and $y < -x + 3$.

Accepted Answer

Region above $y = 2x - 4$ and below $y = -x + 3$

Question 18

If $f(x) = x^2$ and $g(x) = x + 1$, what is $(f \circ g)(x)$?

Accepted Answer

$(x+1)^2$

Question 19

Given $f(x) = \sqrt{x+6}$, what is the domain of $f^{-1}(x)$?

Accepted Answer

$[0, \infty)$

College Algebra Final Exam Review: Comprehensive Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards