Question 1

Given the linear function $$f(x) = -5x + 2$$, what is the slope and what does it represent in terms of rate of change?

Accepted Answer

Slope is $$-5$$, representing a decrease of 5 units in $$f(x)$$ for every 1 unit increase in $$x.$$

Question 2

Which of the following is the correct formula for a linear function with slope $$\frac{2}{5}$$ and $$y-$$intercept $$4$$?

Accepted Answer

$$f(x) = \frac{2}{5}x + 4$$

Question 3

If a function is given by $$f(x) = b$$, where $$b is a $$constant, what is the slope and what does the graph look like?

Accepted Answer

Slope is $$0$$; the graph is a horizontal line crossing the $$y-$$axis at $$(0, b).$$

Question 4

Given two points $$(2, 5)$$ and $$(6, 4)$$, what is the slope of the line passing through them?

Accepted Answer

$$m = \frac{4-5}{6-2} = -\frac{1}{4}$$

Question 5

A line passes through the points $$(1, 2)$$ and $$(1, 3). $$What can you conclude about its slope and orientation?

Accepted Answer

Slope is undefined; the line is vertical.

Question 6

Which of the following best describes the graph of $$f(x) = 3x + 1$$?

Accepted Answer

A straight line with slope $$3$$ and $$y-$$intercept $$(0, 1).$$

Question 7

Suppose the cost $$C (in $$dollars) of buying $$x$$ square yards of carpet is represented by a linear function whose graph passes through $$(0, 0)$$ and $$(2, 50). $$What is the meaning of the slope in this context?

Accepted Answer

The cost increases by $25 for each additional square yard of carpet.

Question 8

A driver's distance from home after $$x$$ hours is given by $$D(x) = 60x. $$What does the slope represent, and how far is the driver after 3 hours?

Accepted Answer

Slope is $$60 ($$miles per hour); after 3 hours, the driver is $$180$$ miles from home.

Question 9

Given the graph of a linear function with $$y-$$intercept $$(0, 2)$$ and $$x-$$intercept $$(-3, 0)$$, what is the equation of the function?

Accepted Answer

$$f(x) = \frac{2}{3}x + 2$$

Question 10

If a function is increasing on the interval $$(0, 5)$$ and decreasing on $$(-4, 0)$$, which of the following statements is true?

Accepted Answer

The function's graph goes up from left to right on $$(0, 5)$$ and down from left to right on $$(-4, 0).$$

Question 11

Which of the following best describes a function that is constant on an interval?

Accepted Answer

The graph is a horizontal line on that interval.

Question 12

Given the linear function $$f(x) = -2x + 1$$, what is the slope and what does it mean for the graph?

Accepted Answer

Slope is $$-2$$; the graph decreases by 2 units for every 1 unit increase in $$x.$$

Question 13

If the slope of a line is undefined, what can you say about the line?

Accepted Answer

The line is vertical.

Question 14

Which of the following is the correct formula for the slope $$m$$ given two points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

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

Question 15

A linear function has a $$y-$$intercept at $$(0, 4)$$ and a slope of $$\frac{1}{3}. $$What is its equation?

Accepted Answer

$$f(x) = \frac{1}{3}x + 4$$

Question 16

If a linear function has a negative slope, what does this indicate about the function's graph?

Accepted Answer

The graph decreases from left to right.

Linear Functions and Their Rates of Change

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