Precalculus Examples

$f (x) = 3 x^{2} + 8 x$ , $[- 1, 5]$

Step 1

Write $f (x) = 3 x^{2} + 8 x$ as an equation.

$y = 3 x^{2} + 8 x$

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 $y = 3 x^{2} + 8 x$ for $f (5)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(3 {(5)}^{2} + 8 (5)) - (3 {(- 1)}^{2} + 8 (- 1))}{(5) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{3 \cdot 25 + 8 (5) - (3 {(- 1)}^{2} + 8 (- 1))}{5 - (- 1)}$

Step 3.1.2

Multiply $3$ by $25$ .

$\frac{75 + 8 (5) - (3 {(- 1)}^{2} + 8 (- 1))}{5 - (- 1)}$

Step 3.1.3

Multiply $8$ by $5$ .

$\frac{75 + 40 - (3 {(- 1)}^{2} + 8 (- 1))}{5 - (- 1)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

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

$\frac{75 + 40 - (3 \cdot 1 + 8 (- 1))}{5 - (- 1)}$

Step 3.1.4.2

Multiply $3$ by $1$ .

$\frac{75 + 40 - (3 + 8 (- 1))}{5 - (- 1)}$

Step 3.1.4.3

Multiply $8$ by $- 1$ .

$\frac{75 + 40 - (3 - 8)}{5 - (- 1)}$

Step 3.1.5

Subtract $8$ from $3$ .

$\frac{75 + 40 - - 5}{5 - (- 1)}$

Step 3.1.6

Multiply $- 1$ by $- 5$ .

$\frac{75 + 40 + 5}{5 - (- 1)}$

Step 3.1.7

Add $75$ and $40$ .

$\frac{115 + 5}{5 - (- 1)}$

Step 3.1.8

Add $115$ and $5$ .

$\frac{120}{5 - (- 1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{120}{5 + 1}$

Step 3.2.2

Add $5$ and $1$ .

$\frac{120}{6}$

Step 3.3

Divide $120$ by $6$ .

$20$