Precalculus Examples

$h (x) = x^{2} - 1$ , $- 3 \leq x \leq - 1$

Step 1

Write $h (x) = x^{2} - 1$ as an equation.

$y = x^{2} - 1$

Step 2

Substitute using the average rate of change formula.

Tap for more steps...

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.

$\frac{f (- 1) - f (- 3)}{(- 1) - (- 3)}$

Step 2.2

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

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

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Subtract $1$ from $9$ .

$\frac{1 - 1 - 1 \cdot 8}{- 1 - (- 3)}$

Step 3.1.4

Multiply $- 1$ by $8$ .

$\frac{1 - 1 - 8}{- 1 - (- 3)}$

Step 3.1.5

Subtract $1$ from $1$ .

$\frac{0 - 8}{- 1 - (- 3)}$

Step 3.1.6

Subtract $8$ from $0$ .

$\frac{- 8}{- 1 - (- 3)}$

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Multiply $- 1$ by $- 3$ .

$\frac{- 8}{- 1 + 3}$

Step 3.2.2

Add $- 1$ and $3$ .

$\frac{- 8}{2}$

Step 3.3

Divide $- 8$ by $2$ .

$- 4$