Question 1

A biologist is tracking the population of a species over time. She models the population $$P as a $$function of time $$t in $$years by $$P(t) = 500 + 30t. $$What is the slope of the graph of this function, and what does it represent in this context?

Accepted Answer

$$30$$, representing the population increase per year

Question 2

Which of the following equations represents a circle with center $$(2, -3)$$ and radius $$5$$?

Accepted Answer

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

Question 3

A company’s profit $$P (in $$thousands of dollars) is given by $$P(x) = 2x - 15$$, where $$x is $$the number of units sold. What is the $$y-$$intercept of the graph, and what does it mean in this context?

Accepted Answer

$$-15$$, representing the profit when no units are sold

Question 4

Given points $$(1, 2)$$ and $$(4, 8)$$, what is the equation of the line passing through them in point-slope form?

Accepted Answer

Question 5

A function $$f is $$defined by $$f(x) = x^2$ - 4x + 3. $$What is the $$y-$$coordinate of the vertex of its graph?

Accepted Answer

$$-1$$

Question 6

Which transformation is applied to the graph of $$f(x) to $$obtain the graph of $$g(x) = f(x-2) + 5$$?

Accepted Answer

Shift right by 2 units and up by 5 units

Question 7

A city engineer uses the distance formula to find the length of a new road between $$(3, 7)$$ and $$(11, 2). $$What is the length of the road?

Accepted Answer

$$\sqrt{(11-3)^2$ + (2-7)^2$} = \sqrt{64 + 25} = \sqrt{89}$$

Question 8

Which of the following is the correct formula for the slope $$m of a $$line passing through $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

$$m = \frac{y_2$ - y_1$}{x_2$ - x_1$}$$

Question 9

If $$f(x) = 3x + 2$$ and $$g(x) = x^2$, what is (f$$ \circ g)(x)$$?

Accepted Answer

$$3x^2 + 2$

Question 10

A medical researcher models the spread of a virus with $$V(t) = 100e$$^{0.05t}$$, where t$ is days since outbreak. What is the initial number of cases?

Accepted Answer

$$100$$

Question 11

Which of the following lines is perpendicular to the line $$y = -\frac{1}{2}x + 3$$?

Accepted Answer

$$y = 2x - 1$$

Question 12

A function $$f$$ has domain $$(-\infty, 2)$$ and range $$[0, \infty). $$Which of the following could be the graph of $$f$$?

Accepted Answer

A graph that exists only to the left of $$x=2$$ and is always above the $$x-$$axis

Question 13

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

Accepted Answer

$$(x+1)^2$$

Question 14

A chemical reaction’s temperature $$T (in$$ $$^	ext{o}C) is $$modeled by $$T(t) = 20 + 5t$$, where $$t is $$time in minutes. After how many minutes will the temperature reach $$45^	ext{o}C$$?

Accepted Answer

$$5$$

Question 15

Which of the following is the equation of a line with slope $$m = 4$$ passing through $$(1, -2) in $$point-slope form?

Accepted Answer

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

Question 16

If $$f(x) = 2x - 1$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x+1}{2}$$

Question 17

A function $$f is $$even if $$f(-x) = f(x)$$ for all $$x in $$its domain. Which of the following is an example of an even function?

Accepted Answer

$$f(x) = x^2$

Question 18

Which of the following describes the effect of replacing $$x$$ with $$-x in $$the function $$f(x)$$?

Accepted Answer

Reflects the graph of $$f(x)$$ across the $$y-$$axis

Question 19

A line passes through $$(2, 5)$$ and has slope $$-3. $$What is its equation in slope-intercept form?

Accepted Answer

$$y = -3x + 11$$

Precalculus Study Notes: Functions, Graphs, and Equations of Lines

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