Precalculus Examples

$f (x) = 2 x^{2} + 3 x - 3$ ; intervals $[- 2, 2]$

Step 1

Write $f (x) = 2 x^{2} + 3 x - 3$ as an equation.

$y = 2 x^{2} + 3 x - 3$

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.

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

Step 2.2

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 3.1.1.1

Multiply $2$ by $2^{2}$ .

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Step 3.1.1.1.1

Raise $2$ to the power of $1$ .

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

Step 3.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.1.2

Add $1$ and $2$ .

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

Step 3.1.2

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

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

Step 3.1.3

Multiply $3$ by $2$ .

$\frac{8 + 6 - 3 - (2 {(- 2)}^{2} + 3 (- 2) - 3)}{2 - (- 2)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

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

$\frac{8 + 6 - 3 - (2 \cdot 4 + 3 (- 2) - 3)}{2 - (- 2)}$

Step 3.1.4.2

Multiply $2$ by $4$ .

$\frac{8 + 6 - 3 - (8 + 3 (- 2) - 3)}{2 - (- 2)}$

Step 3.1.4.3

Multiply $3$ by $- 2$ .

$\frac{8 + 6 - 3 - (8 - 6 - 3)}{2 - (- 2)}$

Step 3.1.5

Subtract $6$ from $8$ .

$\frac{8 + 6 - 3 - (2 - 3)}{2 - (- 2)}$

Step 3.1.6

Subtract $3$ from $2$ .

$\frac{8 + 6 - 3 - - 1}{2 - (- 2)}$

Step 3.1.7

Multiply $- 1$ by $- 1$ .

$\frac{8 + 6 - 3 + 1}{2 - (- 2)}$

Step 3.1.8

Add $8$ and $6$ .

$\frac{14 - 3 + 1}{2 - (- 2)}$

Step 3.1.9

Subtract $3$ from $14$ .

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

Step 3.1.10

Add $11$ and $1$ .

$\frac{12}{2 - (- 2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

$\frac{12}{2 + 2}$

Step 3.2.2

Add $2$ and $2$ .

$\frac{12}{4}$

Step 3.3

Divide $12$ by $4$ .

$3$