Precalculus Examples

$f (x) = x^{4} - 8 x^{2} + 2$ on $- 3$ , $1$

Step 1

Write $f (x) = x^{4} - 8 x^{2} + 2$ as an equation.

$y = x^{4} - 8 x^{2} + 2$

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.

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

Step 2.2

Substitute the equation $y = x^{4} - 8 x^{2} + 2$ for $f (1)$ and $f (- 3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(1)}^{4} - 8 {(1)}^{2} + 2) - ({(- 3)}^{4} - 8 {(- 3)}^{2} + 2)}{(1) - (- 3)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

$\frac{1 - 8 \cdot 1^{2} + 2 - ({(- 3)}^{4} - 8 {(- 3)}^{2} + 2)}{1 - (- 3)}$

Step 3.1.2

One to any power is one.

$\frac{1 - 8 \cdot 1 + 2 - ({(- 3)}^{4} - 8 {(- 3)}^{2} + 2)}{1 - (- 3)}$

Step 3.1.3

Multiply $- 8$ by $1$ .

$\frac{1 - 8 + 2 - ({(- 3)}^{4} - 8 {(- 3)}^{2} + 2)}{1 - (- 3)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Raise $- 3$ to the power of $4$ .

$\frac{1 - 8 + 2 - (81 - 8 {(- 3)}^{2} + 2)}{1 - (- 3)}$

Step 3.1.4.2

Raise $- 3$ to the power of $2$ .

$\frac{1 - 8 + 2 - (81 - 8 \cdot 9 + 2)}{1 - (- 3)}$

Step 3.1.4.3

Multiply $- 8$ by $9$ .

$\frac{1 - 8 + 2 - (81 - 72 + 2)}{1 - (- 3)}$

Step 3.1.5

Subtract $72$ from $81$ .

$\frac{1 - 8 + 2 - (9 + 2)}{1 - (- 3)}$

Step 3.1.6

Add $9$ and $2$ .

$\frac{1 - 8 + 2 - 1 \cdot 11}{1 - (- 3)}$

Step 3.1.7

Multiply $- 1$ by $11$ .

$\frac{1 - 8 + 2 - 11}{1 - (- 3)}$

Step 3.1.8

Subtract $8$ from $1$ .

$\frac{- 7 + 2 - 11}{1 - (- 3)}$

Step 3.1.9

Add $- 7$ and $2$ .

$\frac{- 5 - 11}{1 - (- 3)}$

Step 3.1.10

Subtract $11$ from $- 5$ .

$\frac{- 16}{1 - (- 3)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 3$ .

$\frac{- 16}{1 + 3}$

Step 3.2.2

Add $1$ and $3$ .

$\frac{- 16}{4}$

Step 3.3

Divide $- 16$ by $4$ .

$- 4$