Respuesta :

Check the picture below.

so let's use those two points the line passes through to get its slope.

[tex]\bf (\stackrel{x_1}{2}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{-12}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{-12-0}{4-2}\implies \cfrac{-12}{2}\implies -6[/tex]

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The function with an average rate of change of -6 over the interval [2, 4] is h(x) which is shown in the graph i.e. the 3rd option is correct.

What is the average rate of change?

The slope of a graphed function is determined using the average rate of change formula. It is especially useful for determining changes in measurable values such as average speed or velocity. Divide the change in y-values by the change in x-values to find the average rate of change.

How to solve this problem?

Since the interval is [2, 4], we get x₁ = 2, x₂ = 4 and x₂ - x₁ = 4 - 2 = 2.

First, we check the function f(x).

Here, f(x) = -(5/2)(3)^x

y₁ = f(2) = -(5/2)(3)^2 = -(5/2)*9 = -45/2

y₂ = f(4) = -(5/2)(3)^4 = -(5/2)*81 = -405/2

y₂ - y₁ = f(4)-f(2) = -405/2 - (-45/2) = -405/2 + 45/2 = -360/2 = -180

So, the average rate of change = (y₂ - y₁)/(x₂ - x₁) = -180/2 = -90 ≠ -6

From the table, y₁ = g(2) = -5

y₂ = g(4) = -77

y₂ - y₁ = g(4)-g(2) = -77 - (-5) = -77 + 5 = -72

So, the average rate of change = (y₂ - y₁)/(x₂ - x₁) = -72/2 = -36 ≠ -6

From the graph, y₁ = h(2) = 0

y₂ = h(4) = -12

y₂ - y₁ = h(4)-h(2) = -12 - 0 = -12

So, the average rate of change = (y₂ - y₁)/(x₂ - x₁) = -12/2 = -6

Therefore, the function with an average rate of change of -6 over the interval [2, 4] is h(x) which is shown in the graph i.e. the 3rd option is correct.

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