Q: A student is analyzing the distance between two points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$) on a $$map. Which formula should they use?

$$d = \\sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$}$$

Q: Which of the following is the inverse of the function $$f(x) = 2x + 5$$?

$$f^{-1}(x$$) = \\frac{x - 5}{2}$$

Question 1

A biologist is tracking the population of a rare plant species over time. She models the population with the function $$P(t) = 200 - 5t$$, where $$t is $$the number of years since the study began. What does the slope of this function represent in the context of the problem?

Accepted Answer

The rate at which the population decreases each year.

Question 2

Given two points $$A(2, 5)$$ and $$B(6, 13)$$, what is the equation of the line passing through these points in point-slope form?

Accepted Answer

$$y - 5 = 4(x - 2)$$

Question 3

A city plans to build a circular park with center at $$(4, -3)$$ and a radius of $$7$$ units. Which equation correctly models the boundary of the park?

Accepted Answer

$$(x - 4)^2$ + (y + 3)^2$ = 49$$

Question 4

Which of the following best describes the graph of a function $$f(x)$$?

Accepted Answer

A set of points where each $$x-$$value corresponds to exactly one $$y-$$value.

Question 5

A company’s profit $$P (in $$thousands of dollars) is modeled by $$P(x) = 3x + 12$$, where $$x is $$the number of products sold (in hundreds). What is the $$y-$$intercept of this function and what does it represent?

Accepted Answer

$$12$$, representing the profit when no products are sold.

Question 6

Which transformation is applied to the graph of $$f(x) to $$obtain the graph of $$f(x) - 4$$?

Accepted Answer

Vertical shift down by 4 units.

Question 7

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

Accepted Answer

$$(x + 1)^2$

Question 8

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

Accepted Answer

$$rac{8 - 2}{4 - 1} = 2$$

Question 9

Which of the following equations represents a line with slope $$-3$$ and $$y-$$intercept $$5$$?

Accepted Answer

$$y = -3x + 5$$

Question 10

A student is analyzing the distance between two points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$) on a $$map. Which formula should they use?

Accepted Answer

$$d = \sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$}$$

Question 11

Which of the following lines is perpendicular to the line $$y = 2x + 1$$?

Accepted Answer

$$y = -rac{1}{2}x + 5$$

Question 12

A function $$f(x)$$ has the property that $$f(-x) = -f(x)$$ for all $$x in $$its domain. What type of function is $$f(x)$$?

Accepted Answer

Odd function

Question 13

If the graph of $$f(x) is $$shifted horizontally to the right by $$3$$ units, which equation represents the new function?

Accepted Answer

$$f(x - 3)$$

Question 14

A medical researcher models the spread of a virus with $$V(t) = 100e$$^{0.2t}$$, where t$ is days since outbreak. What does the 0.2$ in the exponent represent?

Accepted Answer

The rate of exponential growth per day.

Question 15

Which of the following is the inverse of the function $$f(x) = 2x + 5$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 5}{2}$$

Question 16

A function $$f(x) is $$defined only for $$x \u003e 0. $$What is the domain of $$f(x)$$?

Accepted Answer

$$(0,  5c	ext{infty})$$

Question 17

Which of the following is the correct slope-intercept form for a line passing through $$(0, 4)$$ with slope $$-2$$?

Accepted Answer

$$y = -2x + 4$$

Question 18

A function $$h(x) is $$defined as $$h(x) = |x|. $$Which statement is true about $$h(x)$$?

Accepted Answer

$$h(x) is an $$even function.

Question 19

A student wants to find the equation of a circle with center $$(2, -5)$$ and radius $$6. $$Which equation should they use?

Accepted Answer

$$(x - 2)^2$ + (y + 5)^2$ = 36$$

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

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