Precalculus Examples

Given the function $f (x) = - x^{2} - 6 x + 17$ , determine the average rate of change of the function over the interval $- 5 \leq x \leq 1$

Step 1

Write $f (x) = - x^{2} - 6 x + 17$ as an equation.

$y = - x^{2} - 6 x + 17$

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.

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

Step 2.2

Substitute the equation $y = - x^{2} - 6 x + 17$ for $f (1)$ and $f (- 5)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- {(1)}^{2} - 6 \cdot 1 + 17) - (- {(- 5)}^{2} - 6 \cdot - 5 + 17)}{(1) - (- 5)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

$\frac{- 1 \cdot 1 - 6 \cdot 1 + 17 - (- {(- 5)}^{2} - 6 \cdot - 5 + 17)}{1 - (- 5)}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{- 1 - 6 \cdot 1 + 17 - (- {(- 5)}^{2} - 6 \cdot - 5 + 17)}{1 - (- 5)}$

Step 3.1.3

Multiply $- 6$ by $1$ .

$\frac{- 1 - 6 + 17 - (- {(- 5)}^{2} - 6 \cdot - 5 + 17)}{1 - (- 5)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

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

$\frac{- 1 - 6 + 17 - (- 1 \cdot 25 - 6 \cdot - 5 + 17)}{1 - (- 5)}$

Step 3.1.4.2

Multiply $- 1$ by $25$ .

$\frac{- 1 - 6 + 17 - (- 25 - 6 \cdot - 5 + 17)}{1 - (- 5)}$

Step 3.1.4.3

Multiply $- 6$ by $- 5$ .

$\frac{- 1 - 6 + 17 - (- 25 + 30 + 17)}{1 - (- 5)}$

Step 3.1.5

Add $- 25$ and $30$ .

$\frac{- 1 - 6 + 17 - (5 + 17)}{1 - (- 5)}$

Step 3.1.6

Add $5$ and $17$ .

$\frac{- 1 - 6 + 17 - 1 \cdot 22}{1 - (- 5)}$

Step 3.1.7

Multiply $- 1$ by $22$ .

$\frac{- 1 - 6 + 17 - 22}{1 - (- 5)}$

Step 3.1.8

Subtract $6$ from $- 1$ .

$\frac{- 7 + 17 - 22}{1 - (- 5)}$

Step 3.1.9

Add $- 7$ and $17$ .

$\frac{10 - 22}{1 - (- 5)}$

Step 3.1.10

Subtract $22$ from $10$ .

$\frac{- 12}{1 - (- 5)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 5$ .

$\frac{- 12}{1 + 5}$

Step 3.2.2

Add $1$ and $5$ .

$\frac{- 12}{6}$

Step 3.3

Divide $- 12$ by $6$ .

$- 2$