Q: A company tracks the cost $$C of $$producing $$x$$ units as $$C(x) = 5x + 200. $$What is the average rate of change of cost as production increases from $$x = 10 to$$ $$x = 20$$?

$$\frac{C(20) - C(10)}{20 - 10} = \frac{(5 \times 20 + 200) - (5 \times 10 + 200)}{10} = \frac{300 - 250}{10} = 5$$

Question 1

Given the points $$A(2, 3)$$ and $$B(-4, 7)$$, what is the distance between them?

Accepted Answer

$$\sqrt{(2 - (-4))^2$ + (3 - 7)^2$} = \sqrt{36 + 16} = \sqrt{52}$$

Question 2

What is the midpoint between the points $$P(5, -2)$$ and $$Q(-1, 4)$$?

Accepted Answer

$$(\frac{5 + (-1)}{2}, \frac{-2 + 4}{2}) = (2, 1)$$

Question 3

A hospital is planning to build a circular helipad. If the center is at $$(6, 8)$$ and the radius is $$5$$ meters, what is the equation of the circle in standard form?

Accepted Answer

$$(x - 6)^2$ + (y - 8)^2$ = 25$$

Question 4

Given the equation $$x^2$$ + $$y^2 - 4x + 6y = 12$, what is the center and radius of the circle?

Accepted Answer

Center: $$(2, -3)$$, Radius: $$\sqrt{25} = 5$$

Question 5

Which of the following equations is symmetric with respect to the y-axis?

Accepted Answer

$$y = x^2$

Question 6

A line passes through the points $$(1, 2)$$ and $$(4, 8). $$What is its slope?

Accepted Answer

$$\frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$$

Question 7

Given the line $$y = 3x - 5$$, what are its slope and y-intercept?

Accepted Answer

Slope: $$3$$, y-intercept: $$-5$$

Question 8

Which of the following pairs of lines are perpendicular?

Accepted Answer

$$y = 2x + 1$$ and $$y = -\frac{1}{2}x + 3$$

Question 9

A company tracks the cost $$C of $$producing $$x$$ units as $$C(x) = 5x + 200. $$What is the average rate of change of cost as production increases from $$x = 10 to$$ $$x = 20$$?

Accepted Answer

$$\frac{C(20) - C(10)}{20 - 10} = \frac{(5 	imes 20 + 200) - (5 	imes 10 + 200)}{10} = \frac{300 - 250}{10} = 5$$

Question 10

Given $$f(x) = x^2$ - 3x + 2$$, what is the difference quotient $$\frac{f(x + h) - f(x)}{h}$$?

Accepted Answer

$$\frac{(x + h)^2$ - 3(x + h) + 2 - (x^2$ - 3x + 2)}{h} = \frac{2xh + h^2$ - 3h}{h} = 2x - 3 + h$$

Question 11

Which of the following is NOT a function?

Accepted Answer

$$x^2$$ + $$y^2 = 1$

Question 12

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

Accepted Answer

All real numbers except $$x = 2$$

Question 13

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

Accepted Answer

$$(f + g)(x) = x^2$ + x + 1$$, domain: all real numbers

Question 14

Which of the following equations is odd?

Accepted Answer

$$y = x^3$

Question 15

A line has equation $$y = -4x + 7. $$What is the equation of a line perpendicular to it passing through $$(1, 2)$$?

Accepted Answer

$$y - 2 = \frac{1}{4}(x - 1)$$

Question 16

Given $$f(x) = x^2$ - 4$$, what are the x-intercepts of the graph?

Accepted Answer

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

Question 17

A function $$f(x) is $$increasing on the interval $$(-\infty, 2)$$ and decreasing on $$(2, \infty). $$What can be said about $$x = 2$$?

Accepted Answer

$$x = 2 is a $$relative maximum

Question 18

Given $$f(x) = x^2$ + 2x + 1$$, evaluate the difference quotient $$\frac{f(x + h) - f(x)}{h}.$$

Accepted Answer

$$2x + h + 2$$

Question 19

A secant line passes through the points $$(2, f(2))$$ and $$(5, f(5))$$ for $$f(x) = x^2$. What is the equation of the secant line?

Accepted Answer

Slope: $$\frac{f(5) - f(2)}{5 - 2} = \frac{25 - 4}{3} = 7$$, Equation: $$y - 4 = 7(x - 2)$$

Precalculus Study Guide: Graphs, Lines, Circles, and Functions (Sections 1.1–2.3)

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