Precalculus Examples

$g (x) = x^{4} + 2 x^{2} - 5$ ; $[- 4, - 2]$

Step 1

Write $g (x) = x^{4} + 2 x^{2} - 5$ as an equation.

$y = x^{4} + 2 x^{2} - 5$

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.

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

Step 2.2

Substitute the equation $y = x^{4} + 2 x^{2} - 5$ for $f (- 2)$ and $f (- 4)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(- 2)}^{4} + 2 {(- 2)}^{2} - 5) - ({(- 4)}^{4} + 2 {(- 4)}^{2} - 5)}{(- 2) - (- 4)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 2$ to the power of $4$ .

$\frac{16 + 2 {(- 2)}^{2} - 5 - ({(- 4)}^{4} + 2 {(- 4)}^{2} - 5)}{- 2 - (- 4)}$

Step 3.1.2

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

$\frac{16 + 2 \cdot 4 - 5 - ({(- 4)}^{4} + 2 {(- 4)}^{2} - 5)}{- 2 - (- 4)}$

Step 3.1.3

Multiply $2$ by $4$ .

$\frac{16 + 8 - 5 - ({(- 4)}^{4} + 2 {(- 4)}^{2} - 5)}{- 2 - (- 4)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Raise $- 4$ to the power of $4$ .

$\frac{16 + 8 - 5 - (256 + 2 {(- 4)}^{2} - 5)}{- 2 - (- 4)}$

Step 3.1.4.2

Raise $- 4$ to the power of $2$ .

$\frac{16 + 8 - 5 - (256 + 2 \cdot 16 - 5)}{- 2 - (- 4)}$

Step 3.1.4.3

Multiply $2$ by $16$ .

$\frac{16 + 8 - 5 - (256 + 32 - 5)}{- 2 - (- 4)}$

Step 3.1.5

Add $256$ and $32$ .

$\frac{16 + 8 - 5 - (288 - 5)}{- 2 - (- 4)}$

Step 3.1.6

Subtract $5$ from $288$ .

$\frac{16 + 8 - 5 - 1 \cdot 283}{- 2 - (- 4)}$

Step 3.1.7

Multiply $- 1$ by $283$ .

$\frac{16 + 8 - 5 - 283}{- 2 - (- 4)}$

Step 3.1.8

Add $16$ and $8$ .

$\frac{24 - 5 - 283}{- 2 - (- 4)}$

Step 3.1.9

Subtract $5$ from $24$ .

$\frac{19 - 283}{- 2 - (- 4)}$

Step 3.1.10

Subtract $283$ from $19$ .

$\frac{- 264}{- 2 - (- 4)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 4$ .

$\frac{- 264}{- 2 + 4}$

Step 3.2.2

Add $- 2$ and $4$ .

$\frac{- 264}{2}$

Step 3.3

Divide $- 264$ by $2$ .

$- 132$