Precalculus Examples

$g (x) = 2 x^{2} - 4 x$ on $[- 1, 3]$

Step 1

Write $g (x) = 2 x^{2} - 4 x$ as an equation.

$y = 2 x^{2} - 4 x$

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.

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

Step 2.2

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

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

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Multiply $2$ by $9$ .

$\frac{18 - 4 \cdot 3 - (2 {(- 1)}^{2} - 4 \cdot - 1)}{3 - (- 1)}$

Step 3.1.3

Multiply $- 4$ by $3$ .

$\frac{18 - 12 - (2 {(- 1)}^{2} - 4 \cdot - 1)}{3 - (- 1)}$

Step 3.1.4

Simplify each term.

Tap for more steps...

Step 3.1.4.1

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

$\frac{18 - 12 - (2 \cdot 1 - 4 \cdot - 1)}{3 - (- 1)}$

Step 3.1.4.2

Multiply $2$ by $1$ .

$\frac{18 - 12 - (2 - 4 \cdot - 1)}{3 - (- 1)}$

Step 3.1.4.3

Multiply $- 4$ by $- 1$ .

$\frac{18 - 12 - (2 + 4)}{3 - (- 1)}$

Step 3.1.5

Add $2$ and $4$ .

$\frac{18 - 12 - 1 \cdot 6}{3 - (- 1)}$

Step 3.1.6

Multiply $- 1$ by $6$ .

$\frac{18 - 12 - 6}{3 - (- 1)}$

Step 3.1.7

Subtract $12$ from $18$ .

$\frac{6 - 6}{3 - (- 1)}$

Step 3.1.8

Subtract $6$ from $6$ .

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

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{0}{3 + 1}$

Step 3.2.2

Add $3$ and $1$ .

$\frac{0}{4}$

Step 3.3

Divide $0$ by $4$ .

$0$