Question 1

For the given solution set $$\{2, -1\}$$, write a quadratic equation in general form.

Accepted Answer

$$x^2 - x - 2 = 0$

Question 2

Find the solution set for the following inequality. Express the answer in interval notation. $$(9x + 2)/(x - 5) \leq 3x + 2$$

Accepted Answer

$$(-\infty, -1/2]$$

Question 3

Determine if the following relation is a function. If so, find its domain and range: $$\{(1, 11), (1, 22), (2, 8), (3, 10)\}$$

Accepted Answer

Not a function

Question 4

Identify if the given table of $$x$$ and $$y$$ values represents a function. Also, find the domain and range.

\begin{align*}
\begin{array}{c|c}
x & y \
\hline
-5 & 9 \
-4 & 17 \
-17 & 17 \
\end{array}
\end{align*}

Accepted Answer

Function; Domain: $$\{-5, -4, -17\}$$; Range: $$\{9, 17\}$$

Question 5

For the given conditions: Slope $$m = 2$$, passing through $$(-1, -\frac{1}{2}). $$Write an equation for the line in slope-intercept form by first going through the point-slope form.

Accepted Answer

$$y = 2x + 3/2$$

Question 6

For a circle with the following center and radius, find its equation (in standard form).

Center $$= (8, -16)$$, $$r = \sqrt{8}$$

Accepted Answer

$$(x - 8)^2$ + (y + 16)^2$ = 8$$

Question 7

Given the two functions $$f(x) = 71x^2$ - 2$$, $$g(x) = x + 10$$, find $$f(g(x))$$ and identify the domain.

Accepted Answer

$$f(g(x)) = 71x^2$ + 712x - 20$$; Domain: $$(-\infty, \infty)$$

Question 8

Find out the equation of the parabola in vertex form for the given condition:
(a) Shape of the parabola same as $$f(x) = 13x^2$
(b) Vertex of the parabola (-1$$, -1)$$

Accepted Answer

$$y = 13(x + 1)^2$ - 1$$

Question 9

If $$1 is a $$zero of the following polynomial function, solve for all the other zeros. $$f(x) = x^3$ - 13x^2$ - 14.$$

Accepted Answer

$$x = 7 + \sqrt{35}$$ and $$x = 7 - \sqrt{35}$$

Question 10

Determine the end behavior of the graph of the following polynomial function using the Leading Coefficient Test: $$f(x) = 2x^4$ + 19x^2$ - 3x - 2$$

Accepted Answer

Graph rises to the left and right

Question 11

A company’s profit $$P (in $$thousands of dollars) is modeled by the quadratic function $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$18$$ thousand dollars at $$x = 2$$ thousand units

Question 12

A population of bacteria triples every 4 hours. If the initial population is 500, what will the population be after 12 hours?

Accepted Answer

$$500 	imes 3^3$ = 13,500$$

Question 13

A quadratic equation $$ax^2 + bx + c = 0$ has real and equal roots. Which condition must be satisfied?

Accepted Answer

$$b^2 - 4ac = 0$

Question 14

A car rental company charges a flat fee of $$\50$ plus $$\$$0.20$$ per mile driven. If a customer has $$\90$ to spend, what is the maximum number of miles they can drive?

Accepted Answer

$$200$$ miles

Question 15

Which of the following is the correct equation for a line perpendicular to $$y = -\frac{1}{2}x + 3$$ and passing through $$(4, -2)$$?

Accepted Answer

$$y = 2x - 10$$

Question 16

If $$f(x) = x^2$ - 4$$ and $$g(x) = x + 2$$, find $$(f \circ g)(3).$$

Accepted Answer

$$f(g(3)) = 25$$

Question 17

A quadratic function $$f(x)$$ has its vertex at $$(2, -3)$$ and passes through the point $$(4, 5). $$What is its equation in vertex form?

Accepted Answer

$$f(x) = 2(x - 2)^2$ - 3$$

College Algebra Midterm Study Guide: Key Concepts and Practice

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