Precalculus Examples

$f (x) = 3 x^{2} - x - 2$ on $- 1$ , $4$

Step 1

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

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

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.

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

Step 2.2

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{3 \cdot 16 - (4) - 2 - (3 {(- 1)}^{2} - (- 1) - 2)}{4 - (- 1)}$

Step 3.1.2

Multiply $3$ by $16$ .

$\frac{48 - (4) - 2 - (3 {(- 1)}^{2} - (- 1) - 2)}{4 - (- 1)}$

Step 3.1.3

Multiply $- 1$ by $4$ .

$\frac{48 - 4 - 2 - (3 {(- 1)}^{2} - (- 1) - 2)}{4 - (- 1)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

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

$\frac{48 - 4 - 2 - (3 \cdot 1 - (- 1) - 2)}{4 - (- 1)}$

Step 3.1.4.2

Multiply $3$ by $1$ .

$\frac{48 - 4 - 2 - (3 - (- 1) - 2)}{4 - (- 1)}$

Step 3.1.4.3

Multiply $- 1$ by $- 1$ .

$\frac{48 - 4 - 2 - (3 + 1 - 2)}{4 - (- 1)}$

Step 3.1.5

Add $3$ and $1$ .

$\frac{48 - 4 - 2 - (4 - 2)}{4 - (- 1)}$

Step 3.1.6

Subtract $2$ from $4$ .

$\frac{48 - 4 - 2 - 1 \cdot 2}{4 - (- 1)}$

Step 3.1.7

Multiply $- 1$ by $2$ .

$\frac{48 - 4 - 2 - 2}{4 - (- 1)}$

Step 3.1.8

Subtract $4$ from $48$ .

$\frac{44 - 2 - 2}{4 - (- 1)}$

Step 3.1.9

Subtract $2$ from $44$ .

$\frac{42 - 2}{4 - (- 1)}$

Step 3.1.10

Subtract $2$ from $42$ .

$\frac{40}{4 - (- 1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{40}{4 + 1}$

Step 3.2.2

Add $4$ and $1$ .

$\frac{40}{5}$

Step 3.3

Divide $40$ by $5$ .

$8$