Finite Math Examples

$f (x) = 8 x^{2} - 8 x + 9$ , $[- 2, 4]$

Step 1

Write $f (x) = 8 x^{2} - 8 x + 9$ as an equation.

$y = 8 x^{2} - 8 x + 9$

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 $y = 8 x^{2} - 8 x + 9$ for $f (4)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(8 {(4)}^{2} - 8 \cdot 4 + 9) - (8 {(- 2)}^{2} - 8 \cdot - 2 + 9)}{(4) - (- 2)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{8 \cdot 16 - 8 \cdot 4 + 9 - (8 {(- 2)}^{2} - 8 \cdot - 2 + 9)}{4 - (- 2)}$

Step 3.1.2

Multiply $8$ by $16$ .

$\frac{128 - 8 \cdot 4 + 9 - (8 {(- 2)}^{2} - 8 \cdot - 2 + 9)}{4 - (- 2)}$

Step 3.1.3

Multiply $- 8$ by $4$ .

$\frac{128 - 32 + 9 - (8 {(- 2)}^{2} - 8 \cdot - 2 + 9)}{4 - (- 2)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

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

$\frac{128 - 32 + 9 - (8 \cdot 4 - 8 \cdot - 2 + 9)}{4 - (- 2)}$

Step 3.1.4.2

Multiply $8$ by $4$ .

$\frac{128 - 32 + 9 - (32 - 8 \cdot - 2 + 9)}{4 - (- 2)}$

Step 3.1.4.3

Multiply $- 8$ by $- 2$ .

$\frac{128 - 32 + 9 - (32 + 16 + 9)}{4 - (- 2)}$

Step 3.1.5

Add $32$ and $16$ .

$\frac{128 - 32 + 9 - (48 + 9)}{4 - (- 2)}$

Step 3.1.6

Add $48$ and $9$ .

$\frac{128 - 32 + 9 - 1 \cdot 57}{4 - (- 2)}$

Step 3.1.7

Multiply $- 1$ by $57$ .

$\frac{128 - 32 + 9 - 57}{4 - (- 2)}$

Step 3.1.8

Subtract $32$ from $128$ .

$\frac{96 + 9 - 57}{4 - (- 2)}$

Step 3.1.9

Add $96$ and $9$ .

$\frac{105 - 57}{4 - (- 2)}$

Step 3.1.10

Subtract $57$ from $105$ .

$\frac{48}{4 - (- 2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

$\frac{48}{4 + 2}$

Step 3.2.2

Add $4$ and $2$ .

$\frac{48}{6}$

Step 3.3

Divide $48$ by $6$ .

$8$