Precalculus Examples

$f (x) = 2 x^{2} + 4 x - 6$ ; interval $- 3$ , $3$

Step 1

Write $f (x) = 2 x^{2} + 4 x - 6$ as an equation.

$y = 2 x^{2} + 4 x - 6$

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.

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

Step 2.2

Substitute the equation $y = 2 x^{2} + 4 x - 6$ for $f (3)$ and $f (- 3)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{2 \cdot 9 + 4 (3) - 6 - (2 {(- 3)}^{2} + 4 (- 3) - 6)}{3 - (- 3)}$

Step 3.1.2

Multiply $2$ by $9$ .

$\frac{18 + 4 (3) - 6 - (2 {(- 3)}^{2} + 4 (- 3) - 6)}{3 - (- 3)}$

Step 3.1.3

Multiply $4$ by $3$ .

$\frac{18 + 12 - 6 - (2 {(- 3)}^{2} + 4 (- 3) - 6)}{3 - (- 3)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

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

$\frac{18 + 12 - 6 - (2 \cdot 9 + 4 (- 3) - 6)}{3 - (- 3)}$

Step 3.1.4.2

Multiply $2$ by $9$ .

$\frac{18 + 12 - 6 - (18 + 4 (- 3) - 6)}{3 - (- 3)}$

Step 3.1.4.3

Multiply $4$ by $- 3$ .

$\frac{18 + 12 - 6 - (18 - 12 - 6)}{3 - (- 3)}$

Step 3.1.5

Subtract $12$ from $18$ .

$\frac{18 + 12 - 6 - (6 - 6)}{3 - (- 3)}$

Step 3.1.6

Subtract $6$ from $6$ .

$\frac{18 + 12 - 6 - 0}{3 - (- 3)}$

Step 3.1.7

Multiply $- 1$ by $0$ .

$\frac{18 + 12 - 6 + 0}{3 - (- 3)}$

Step 3.1.8

Add $18 + 12 - 6$ and $0$ .

$\frac{18 + 12 - 6}{3 - (- 3)}$

Step 3.1.9

Add $18$ and $12$ .

$\frac{30 - 6}{3 - (- 3)}$

Step 3.1.10

Subtract $6$ from $30$ .

$\frac{24}{3 - (- 3)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 3$ .

$\frac{24}{3 + 3}$

Step 3.2.2

Add $3$ and $3$ .

$\frac{24}{6}$

Step 3.3

Divide $24$ by $6$ .

$4$