Precalculus Examples

Given the function $g (x) = x^{2} + x - 3$ , determine the average rate of change of the function over the interval $- 3 \leq x \leq 6$

Step 1

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

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

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.

$\frac{f (6) - f (- 3)}{(6) - (- 3)}$

Step 2.2

Substitute the equation $y = x^{2} + x - 3$ for $f (6)$ and $f (- 3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{{(6)}^{2} + 6 - 3 - ({(- 3)}^{2} - 3 - 3)}{(6) - (- 3)}$

Step 3

Simplify the expression.

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Step 3.1

Remove parentheses.

$\frac{{(6)}^{2} + 6 - 3 - ({(- 3)}^{2} - 3 - 3)}{(6) - (- 3)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Raise $6$ to the power of $2$ .

$\frac{36 + 6 - 3 - ({(- 3)}^{2} - 3 - 3)}{6 - (- 3)}$

Step 3.2.2

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

$\frac{36 + 6 - 3 - (9 - 3 - 3)}{6 - (- 3)}$

Step 3.2.3

Subtract $3$ from $9$ .

$\frac{36 + 6 - 3 - (6 - 3)}{6 - (- 3)}$

Step 3.2.4

Subtract $3$ from $6$ .

$\frac{36 + 6 - 3 - 1 \cdot 3}{6 - (- 3)}$

Step 3.2.5

Multiply $- 1$ by $3$ .

$\frac{36 + 6 - 3 - 3}{6 - (- 3)}$

Step 3.2.6

Add $36$ and $6$ .

$\frac{42 - 3 - 3}{6 - (- 3)}$

Step 3.2.7

Subtract $3$ from $42$ .

$\frac{39 - 3}{6 - (- 3)}$

Step 3.2.8

Subtract $3$ from $39$ .

$\frac{36}{6 - (- 3)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Multiply $- 1$ by $- 3$ .

$\frac{36}{6 + 3}$

Step 3.3.2

Add $6$ and $3$ .

$\frac{36}{9}$

Step 3.4

Divide $36$ by $9$ .

$4$