Question 1

Given the points $$A(2, -3)$$ and $$B(-4, 5)$$, what is the distance between them?

Accepted Answer

$$\sqrt{(2 - (-4))^2$ + ((-3) - 5)^2$} = \sqrt{36 + 64} = \sqrt{100} = 10$$

Question 2

Find the midpoint of the segment joining the points $$(-6, 8)$$ and $$(4, -2).$$

Accepted Answer

$$\left( \frac{-6 + 4}{2}, \frac{8 + (-2)}{2} \right) = (-1, 3)$$

Question 3

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

Accepted Answer

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

Question 4

Given the equation of a circle $$x^2$$ + $$y^2 - 6x + 4y + 9 = 0$, what is the center and radius?

Accepted Answer

Center: $$(3, -2)$$, Radius: $$2$$

Question 5

Which of the following relations is NOT a function?

Accepted Answer

$$\{(1,2), (2,3), (1,4)\}$$

Question 6

If $$f(x) = 2x^2$ - 3x + 1$$, what is $$f(-2)$$?

Accepted Answer

$$2(-2)^2 - 3(-2) + 1 = 8 + 6 + 1 = 15$

Question 7

What is the domain of the function $$f(x) = \frac{1}{x-4}$$?

Accepted Answer

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

Question 8

Find the slope of the line passing through the points $$(7, -1)$$ and $$(2, 4).$$

Accepted Answer

$$m = \frac{-1 - 4}{7 - 2} = \frac{-5}{5} = -1$$

Question 9

Given the linear equation $$3x - 2y = 6$$, what is the slope and y-intercept?

Accepted Answer

Question 10

Write the equation of a line with slope $$-2$$ and y-intercept $$5.$$

Accepted Answer

$$y = -2x + 5$$

Question 11

Find the equation of the line passing through $$(1, 2)$$ and $$(3, 6).$$

Accepted Answer

$$y = 2x$$

Question 12

Write the equation of a line passing through $$(2, -1)$$ and parallel to $$y = 3x + 4.$$

Accepted Answer

$$y = 3x - 7$$

Question 13

Write the equation of a line passing through $$(0, 5)$$ and perpendicular to $$y = -\frac{1}{2}x + 3.$$

Accepted Answer

$$y = 2x + 5$$

Question 14

A rectangular garden has a perimeter of $$60$$ meters. If the length is $$5$$ meters more than the width, what are the dimensions of the garden?

Accepted Answer

Length: $$17.5 m$$, Width: $$12.5 m$$

Question 15

Solve the inequality $$2x - 5 \u003c 7$$ and write the solution in interval notation.

Accepted Answer

$$(-\infty, 6)$$

Question 16

Given $$f(x) = x^2$ - 4$$ and $$g(x) = x + 2$$, find $$(f+g)(x)$$ and its domain.

Accepted Answer

$$(f+g)(x) = x^2$ + x - 2$$, domain: $$(-\infty, \infty)$$

Question 17

Given $$f(x) = \sqrt{x-1}$$ and $$g(x) = x^2$, find (f$$ \circ g)(x)$$ and its domain.

Accepted Answer

$$(f \circ g)(x) = \sqrt{x^2$ - 1}$$, domain: $$(-\infty, -1] \cup [1, \infty)$$

Question 18

A company’s profit $$P (in $$dollars) is modeled by $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold (in hundreds). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit: $$300$$, at $$x = 10 (i.e$$., 1000 units)

Question 19

Given the piecewise function $$f(x) = \begin{cases} x+2 & x \u003c 0 \ x^2$ & x \geq 0 \end{cases}$$, what is $$f(-3) + f(2)$$?

Accepted Answer

$$(-3 + 2) + (2^2$) = (-1) + 4 = 3$$

Question 20

Given $$f(x) = x^2$ - 1$$ and $$g(x) = x - 1$$, find $$\left(\frac{f}{g}\right)(x)$$ and its domain.

Accepted Answer

$$\frac{x^2$ - 1}{x - 1}$$, domain: $$(-\infty, 1) \cup (1, \infty)$$

College Algebra Exam 1 Study Guide: Graphs, Functions, and Linear Equations

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