Question

calculate the average rate of change from \\(x = -2\\) to \\(x = 1\\) of the function below.

\\f(x) = x^2 + 5x - 12\\

first, find the change in \\(y\\) by evaluating the function at \\(-2\\) and \\(1\\):

\\(f(-2) =\\)

\\(f(1) =\\)

the change in \\(y\\) is

-24
12
-16
24

Explanation:

Evaluate the function at \(x = -2\)

Using the Function Evaluation knowledge point, we substitute \(x = -2\) into the given quadratic function \(f(x) = x^2 + 5x - 12\):

$$ f(-2) = (-2)^2 + 5(-2) - 12 $$

$$ f(-2) = 4 - 10 - 12 = -18 $$

Evaluate the function at \(x = 1\)

Using the Function Evaluation knowledge point, we substitute \(x = 1\) into the function:

$$ f(1) = (1)^2 + 5(1) - 12 $$

$$ f(1) = 1 + 5 - 12 = -6 $$

Calculate the change in \(y\)

The change in \(y\) (often denoted as \(\Delta y\)) is the difference between the function values at the end point and the starting point:

$$ \Delta y = f(1) - f(-2) $$

$$ \Delta y = -6 - (-18) $$

$$ \Delta y = -6 + 18 = 12 $$

Calculate the average rate of change

Using the Average Rate of Change knowledge point, we divide the change in \(y\) by the change in \(x\):

$$ \text{Average Rate of Change} = \frac{f(1) - f(-2)}{1 - (-2)} $$

$$ \text{Average Rate of Change} = \frac{12}{1 - (-2)} = \frac{12}{3} = 4 $$

Answer:

Question 1

\(f(-2) =\) <blank>-18</blank>

Question 2

\(f(1) =\) <blank>-6</blank>

Question 3

The change in \(y\) is <blank>12</blank>