Question 1

Given the function $$f(x) = 3x + 5$$, compute the average rate of change of $$f$$ from $$x_1 = -6$ to x_2$ = -3. $$Round your answer to two decimal places. What does this value represent graphically?

Accepted Answer

The average rate of change is $$3.00$$, which represents the slope of the secant line connecting the points $$(-6, f(-6))$$ and $$(-3, f(-3)) on $$the graph.

Question 2

For the function $$f(x) = 3x + 5$$, what is the value of $$f(-6)$$?

Accepted Answer

$$-13$$

Question 3

For the function $$f(x) = 3x + 5$$, what is the value of $$f(-3)$$?

Accepted Answer

$$-4$$

Question 4

Which formula correctly calculates the average rate of change of a function $$f(x)$$ from $$x_1 to$$ $$x_2$$?

Accepted Answer

$$\frac{f(x_2$) - f(x_1$)}{x_2$ - x_1$}$$

Question 5

If the average rate of change of $$f(x)$$ from $$x_1 to$$ $$x_2 is $$positive, what does this indicate about the graph of $$f(x)$$ between $$x_1$$ and $$x_2$$?

Accepted Answer

The graph is increasing between $$x_1$$ and $$x_2.$$

Question 6

Given $$f(x) = 3x + 5$$, what is the slope of the line represented by $$f(x)$$?

Accepted Answer

$$3$$

Question 7

If a function $$f(x)$$ has a constant average rate of change over any interval, what type of function is $$f(x)$$?

Accepted Answer

Linear function

Question 8

Suppose a car's position over time is modeled by $$f(x) = 3x + 5$$, where $$x is $$time in hours and $$f(x) is $$distance in miles. What does the average rate of change from $$x_1 = -6$ to x_2$ = -3$$ represent in this context?

Accepted Answer

The car's constant speed over the interval, which is $$3$$ miles per hour.

Question 9

If the average rate of change of $$f(x)$$ from $$x_1 to$$ $$x_2 is$$ $$3$$, and $$f(x) is $$linear, what is the instantaneous rate of change at any point $$x$$?

Accepted Answer

$$3$$

Question 10

For $$f(x) = 3x + 5$$, what is the equation of the secant line passing through the points $$(-6, f(-6))$$ and $$(-3, f(-3))$$?

Accepted Answer

$$y = 3x + 5$$

Question 11

If $$f(x) = 3x + 5$$ models the cost in dollars to produce $$x$$ items, what does the average rate of change from $$x_1 = -6$ to x_2$ = -3$$ tell you about the cost per item?

Accepted Answer

The cost increases by $$3$$ dollars for each additional item produced.

Question 12

Which of the following best describes the graphical interpretation of the average rate of change for $$f(x) = 3x + 5$$ between $$x_1 = -6$ and x_2$ = -3$$?

Accepted Answer

It is the slope of the line connecting the points $$(-6, -13)$$ and $$(-3, -4).$$

Average Rate of Change of a Linear Function

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