Precalculus Examples

$g (x) = - \frac{x^{2}}{4} + 7$ , $[- 2, 4]$

Step 1

Write $g (x) = - \frac{x^{2}}{4} + 7$ as an equation.

$y = - \frac{x^{2}}{4} + 7$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (4) - f (- 2)}{(4) - (- 2)}$

Step 2.2

Substitute the equation $y = - \frac{x^{2}}{4} + 7$ for $f (4)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- \frac{{(4)}^{2}}{4} + 7) - (- \frac{{(- 2)}^{2}}{4} + 7)}{(4) - (- 2)}$

Step 3

Simplify the expression.

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Step 3.1

Reduce the expression by cancelling the common factors.

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Step 3.1.1

Cancel the common factor of ${(4)}^{2}$ and $4$ .

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Step 3.1.1.1

Factor $4$ out of ${(4)}^{2}$ .

$\frac{- \frac{4 \cdot 4}{4} + 7 - (- \frac{{(- 2)}^{2}}{4} + 7)}{(4) - (- 2)}$

Step 3.1.1.2

Cancel the common factors.

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Step 3.1.1.2.1

Factor $4$ out of $4$ .

$\frac{- \frac{4 \cdot 4}{4 (1)} + 7 - (- \frac{{(- 2)}^{2}}{4} + 7)}{(4) - (- 2)}$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{- \frac{4 \cdot 4}{4 \cdot 1} + 7 - (- \frac{{(- 2)}^{2}}{4} + 7)}{(4) - (- 2)}$

Step 3.1.1.2.3

Rewrite the expression.

$\frac{- \frac{4}{1} + 7 - (- \frac{{(- 2)}^{2}}{4} + 7)}{(4) - (- 2)}$

Step 3.1.1.2.4

Divide $4$ by $1$ .

$\frac{- 1 \cdot 4 + 7 - (- \frac{{(- 2)}^{2}}{4} + 7)}{(4) - (- 2)}$

Step 3.1.2

Raise $- 2$ to the power of $2$ .

$\frac{- 1 \cdot 4 + 7 - (- \frac{4}{4} + 7)}{4 - (- 2)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Multiply $- 1$ by $4$ .

$\frac{- 4 + 7 - (- \frac{4}{4} + 7)}{4 - (- 2)}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 4 + 7 - (- \frac{4}{4} + 7)}{4 - (- 2)}$

Step 3.2.2.1.2

Rewrite the expression.

$\frac{- 4 + 7 - (- 1 \cdot 1 + 7)}{4 - (- 2)}$

Step 3.2.2.2

Multiply $- 1$ by $1$ .

$\frac{- 4 + 7 - (- 1 + 7)}{4 - (- 2)}$

Step 3.2.3

Add $- 1$ and $7$ .

$\frac{- 4 + 7 - 1 \cdot 6}{4 - (- 2)}$

Step 3.2.4

Multiply $- 1$ by $6$ .

$\frac{- 4 + 7 - 6}{4 - (- 2)}$

Step 3.2.5

Add $- 4$ and $7$ .

$\frac{3 - 6}{4 - (- 2)}$

Step 3.2.6

Subtract $6$ from $3$ .

$\frac{- 3}{4 - (- 2)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Multiply $- 1$ by $- 2$ .

$\frac{- 3}{4 + 2}$

Step 3.3.2

Add $4$ and $2$ .

$\frac{- 3}{6}$

Step 3.4

Reduce the expression by cancelling the common factors.

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Step 3.4.1

Cancel the common factor of $- 3$ and $6$ .

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Step 3.4.1.1

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{6}$

Step 3.4.1.2

Cancel the common factors.

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Step 3.4.1.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot - 1}{3 \cdot 2}$

Step 3.4.1.2.2

Cancel the common factor.

$\frac{3 \cdot - 1}{3 \cdot 2}$

Step 3.4.1.2.3

Rewrite the expression.

$\frac{- 1}{2}$

Step 3.4.2

Move the negative in front of the fraction.

$- \frac{1}{2}$