Precalculus Examples

$f (x) = 5 - 2 x^{2}$ , $(- 2, 4)$

Step 1

Write $f (x) = 5 - 2 x^{2}$ as an equation.

$y = 5 - 2 x^{2}$

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.

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

Step 2.2

Substitute the equation $y = 5 - 2 x^{2}$ for $f (4)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(5 - 2 {(4)}^{2}) - (5 - 2 {(- 2)}^{2})}{(4) - (- 2)}$

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 $2$ .

$\frac{5 - 2 \cdot 16 - (5 - 2 {(- 2)}^{2})}{4 - (- 2)}$

Step 3.1.2

Multiply $- 2$ by $16$ .

$\frac{5 - 32 - (5 - 2 {(- 2)}^{2})}{4 - (- 2)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

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Step 3.1.3.1.1

Multiply $- 2$ by ${(- 2)}^{2}$ .

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Step 3.1.3.1.1.1

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

$\frac{5 - 32 - (5 + {(- 2)}^{1} {(- 2)}^{2})}{4 - (- 2)}$

Step 3.1.3.1.1.2

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

$\frac{5 - 32 - (5 + {(- 2)}^{1 + 2})}{4 - (- 2)}$

Step 3.1.3.1.2

Add $1$ and $2$ .

$\frac{5 - 32 - (5 + {(- 2)}^{3})}{4 - (- 2)}$

Step 3.1.3.2

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

$\frac{5 - 32 - (5 - 8)}{4 - (- 2)}$

Step 3.1.4

Subtract $8$ from $5$ .

$\frac{5 - 32 - - 3}{4 - (- 2)}$

Step 3.1.5

Multiply $- 1$ by $- 3$ .

$\frac{5 - 32 + 3}{4 - (- 2)}$

Step 3.1.6

Subtract $32$ from $5$ .

$\frac{- 27 + 3}{4 - (- 2)}$

Step 3.1.7

Add $- 27$ and $3$ .

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

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 $6$ .

$- 4$