Question

Determine if the following functions are increasing or decreasing, and compare their rates of change. A. Both functions are increasing and have different rates of change. B. Both functions are decreasing and have different rates of change. C. Both functions are increasing and have the same rate of change. D. Both functions are decreasing and have the same rate of change.

Answer 1

Answer: Choice D

Both functions are decreasing and have the same rate of change.

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Explanation:

The slope intercept form is y = mx+b

• m = slope
• b = y intercept

The first equation y = -2x-3 has slope m = -2, which is the rate of change.

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Let's find the slope of the line through (0,0) and (3,-6)

$(x_1,y_1) = (0,0) \text{ and } (x_2,y_2) = (3,-6)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{-6 - 0}{3 - 0}\\\\m = \frac{-6}{3}\\\\m = -2$

Or you can use the shortcut that m = y/x = -6/3 = -2. This shortcut only works when the line goes through the origin. We divide the nonzero x,y values.

The slope of the second line is the same as the first line's slope.

Both lines have the same rate of change.

Both lines are decreasing due to the negative slope value.

Visually, each line goes downhill as we move from left to right.

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Side note: Both lines have the same slope (-2), but different y intercepts. This makes the lines parallel.