Precalculus Examples

Given the function $h (x) = x^{2} + x - 3$ , determine the average rate of change of the function over the interval $- 4 \leq x \leq 1$

Step 1

Write $h (x) = x^{2} + x - 3$ 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.

$\frac{f (1) - f (- 4)}{(1) - (- 4)}$

Step 2.2

Substitute the equation $y = x^{2} + x - 3$ for $f (1)$ and $f (- 4)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{{(1)}^{2} + 1 - 3 - ({(- 4)}^{2} - 4 - 3)}{(1) - (- 4)}$

Step 3

Simplify the expression.

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Step 3.1

Remove parentheses.

$\frac{{(1)}^{2} + 1 - 3 - ({(- 4)}^{2} - 4 - 3)}{(1) - (- 4)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

One to any power is one.

$\frac{1 + 1 - 3 - ({(- 4)}^{2} - 4 - 3)}{1 - (- 4)}$

Step 3.2.2

Raise $- 4$ to the power of $2$ .

$\frac{1 + 1 - 3 - (16 - 4 - 3)}{1 - (- 4)}$

Step 3.2.3

Subtract $4$ from $16$ .

$\frac{1 + 1 - 3 - (12 - 3)}{1 - (- 4)}$

Step 3.2.4

Subtract $3$ from $12$ .

$\frac{1 + 1 - 3 - 1 \cdot 9}{1 - (- 4)}$

Step 3.2.5

Multiply $- 1$ by $9$ .

$\frac{1 + 1 - 3 - 9}{1 - (- 4)}$

Step 3.2.6

Add $1$ and $1$ .

$\frac{2 - 3 - 9}{1 - (- 4)}$

Step 3.2.7

Subtract $3$ from $2$ .

$\frac{- 1 - 9}{1 - (- 4)}$

Step 3.2.8

Subtract $9$ from $- 1$ .

$\frac{- 10}{1 - (- 4)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Multiply $- 1$ by $- 4$ .

$\frac{- 10}{1 + 4}$

Step 3.3.2

Add $1$ and $4$ .

$\frac{- 10}{5}$

Step 3.4

Divide $- 10$ by $5$ .

$- 2$