Precalculus Examples

$x^{2} + 2 x - 3$ ; $[- 2, 0]$

Step 1

Write $x^{2} + 2 x - 3$ as an equation.

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

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.

$\frac{f (0) - f (- 2)}{(0) - (- 2)}$

Step 2.2

Substitute the equation $y = x^{2} + 2 x - 3$ for $f (0)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(0)}^{2} + 2 (0) - 3) - ({(- 2)}^{2} + 2 (- 2) - 3)}{(0) - (- 2)}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{0 + 2 (0) - 3 - ({(- 2)}^{2} + 2 (- 2) - 3)}{0 - (- 2)}$

Step 3.1.2

Multiply $2$ by $0$ .

$\frac{0 + 0 - 3 - ({(- 2)}^{2} + 2 (- 2) - 3)}{0 - (- 2)}$

Step 3.1.3

Simplify each term.

Tap for more steps...

Step 3.1.3.1

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

$\frac{0 + 0 - 3 - (4 + 2 (- 2) - 3)}{0 - (- 2)}$

Step 3.1.3.2

Multiply $2$ by $- 2$ .

$\frac{0 + 0 - 3 - (4 - 4 - 3)}{0 - (- 2)}$

Step 3.1.4

Subtract $4$ from $4$ .

$\frac{0 + 0 - 3 - (0 - 3)}{0 - (- 2)}$

Step 3.1.5

Subtract $3$ from $0$ .

$\frac{0 + 0 - 3 - - 3}{0 - (- 2)}$

Step 3.1.6

Multiply $- 1$ by $- 3$ .

$\frac{0 + 0 - 3 + 3}{0 - (- 2)}$

Step 3.1.7

Add $0$ and $0$ .

$\frac{0 - 3 + 3}{0 - (- 2)}$

Step 3.1.8

Subtract $3$ from $0$ .

$\frac{- 3 + 3}{0 - (- 2)}$

Step 3.1.9

Add $- 3$ and $3$ .

$\frac{0}{0 - (- 2)}$

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Multiply $- 1$ by $- 2$ .

$\frac{0}{0 + 2}$

Step 3.2.2

Add $0$ and $2$ .

$\frac{0}{2}$

Step 3.3

Divide $0$ by $2$ .

$0$