Precalculus Examples

$f (x) = 2 {(x - 2)}^{2} + 1$ over the interval $[- 2, 4]$

Step 1

Write $f (x) = 2 {(x - 2)}^{2} + 1$ as an equation.

$y = 2 {(x - 2)}^{2} + 1$

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Subtract $2$ from $4$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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

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

Step 3.1.2.2

Add $1$ and $2$ .

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

Step 3.1.3

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

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

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Subtract $2$ from $- 2$ .

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

Step 3.1.4.2

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

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

Step 3.1.4.3

Multiply $2$ by $16$ .

$\frac{8 + 1 - (32 + 1)}{4 - (- 2)}$

Step 3.1.5

Add $32$ and $1$ .

$\frac{8 + 1 - 1 \cdot 33}{4 - (- 2)}$

Step 3.1.6

Multiply $- 1$ by $33$ .

$\frac{8 + 1 - 33}{4 - (- 2)}$

Step 3.1.7

Add $8$ and $1$ .

$\frac{9 - 33}{4 - (- 2)}$

Step 3.1.8

Subtract $33$ from $9$ .

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

$\frac{- 24}{4 + 2}$

Step 3.2.2

Add $4$ and $2$ .

$\frac{- 24}{6}$

Step 3.3

Divide $- 24$ by $6$ .

$- 4$