Precalculus Examples

Given the function $h (x) = - x^{2} + x + 4$ , determine the average rate of change of the function over the interval $- 1 \leq x \leq 5$

Step 1

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

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

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.

$\frac{f (5) - f (- 1)}{(5) - (- 1)}$

Step 2.2

Substitute the equation $y = - x^{2} + x + 4$ for $f (5)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Remove parentheses.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Raise $5$ to the power of $2$ .

$\frac{- 1 \cdot 25 + 5 + 4 - (- {(- 1)}^{2} - 1 + 4)}{5 - (- 1)}$

Step 3.2.2

Multiply $- 1$ by $25$ .

$\frac{- 25 + 5 + 4 - (- {(- 1)}^{2} - 1 + 4)}{5 - (- 1)}$

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.2.3.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.2.3.1.1.1

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

$\frac{- 25 + 5 + 4 - ({(- 1)}^{1} {(- 1)}^{2} - 1 + 4)}{5 - (- 1)}$

Step 3.2.3.1.1.2

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

$\frac{- 25 + 5 + 4 - ({(- 1)}^{1 + 2} - 1 + 4)}{5 - (- 1)}$

Step 3.2.3.1.2

Add $1$ and $2$ .

$\frac{- 25 + 5 + 4 - ({(- 1)}^{3} - 1 + 4)}{5 - (- 1)}$

Step 3.2.3.2

Raise $- 1$ to the power of $3$ .

$\frac{- 25 + 5 + 4 - (- 1 - 1 + 4)}{5 - (- 1)}$

Step 3.2.4

Subtract $1$ from $- 1$ .

$\frac{- 25 + 5 + 4 - (- 2 + 4)}{5 - (- 1)}$

Step 3.2.5

Add $- 2$ and $4$ .

$\frac{- 25 + 5 + 4 - 1 \cdot 2}{5 - (- 1)}$

Step 3.2.6

Multiply $- 1$ by $2$ .

$\frac{- 25 + 5 + 4 - 2}{5 - (- 1)}$

Step 3.2.7

Add $- 25$ and $5$ .

$\frac{- 20 + 4 - 2}{5 - (- 1)}$

Step 3.2.8

Add $- 20$ and $4$ .

$\frac{- 16 - 2}{5 - (- 1)}$

Step 3.2.9

Subtract $2$ from $- 16$ .

$\frac{- 18}{5 - (- 1)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- 18}{5 + 1}$

Step 3.3.2

Add $5$ and $1$ .

$\frac{- 18}{6}$

Step 3.4

Divide $- 18$ by $6$ .

$- 3$