Precalculus Examples

$f (x) = 9 + 4 x - 4 x^{2}$ ; $[0, 5]$

Step 1

Write $f (x) = 9 + 4 x - 4 x^{2}$ as an equation.

$y = 9 + 4 x - 4 x^{2}$

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.

$\frac{f (5) - f (0)}{(5) - (0)}$

Step 2.2

Substitute the equation $y = 9 + 4 x - 4 x^{2}$ for $f (5)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(9 + 4 (5) - 4 {(5)}^{2}) - (9 + 4 (0) - 4 {(0)}^{2})}{(5) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $4$ by $5$ .

$\frac{9 + 20 - 4 \cdot 5^{2} - (9 + 4 (0) - 4 \cdot 0^{2})}{5 - (0)}$

Step 3.1.2

Raise $5$ to the power of $2$ .

$\frac{9 + 20 - 4 \cdot 25 - (9 + 4 (0) - 4 \cdot 0^{2})}{5 - (0)}$

Step 3.1.3

Multiply $- 4$ by $25$ .

$\frac{9 + 20 - 100 - (9 + 4 (0) - 4 \cdot 0^{2})}{5 - (0)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Multiply $4$ by $0$ .

$\frac{9 + 20 - 100 - (9 + 0 - 4 \cdot 0^{2})}{5 - (0)}$

Step 3.1.4.2

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

$\frac{9 + 20 - 100 - (9 + 0 - 4 \cdot 0)}{5 - (0)}$

Step 3.1.4.3

Multiply $- 4$ by $0$ .

$\frac{9 + 20 - 100 - (9 + 0 + 0)}{5 - (0)}$

Step 3.1.5

Add $9$ and $0$ .

$\frac{9 + 20 - 100 - (9 + 0)}{5 - (0)}$

Step 3.1.6

Add $9$ and $0$ .

$\frac{9 + 20 - 100 - 1 \cdot 9}{5 - (0)}$

Step 3.1.7

Multiply $- 1$ by $9$ .

$\frac{9 + 20 - 100 - 9}{5 - (0)}$

Step 3.1.8

Add $9$ and $20$ .

$\frac{29 - 100 - 9}{5 - (0)}$

Step 3.1.9

Subtract $100$ from $29$ .

$\frac{- 71 - 9}{5 - (0)}$

Step 3.1.10

Subtract $9$ from $- 71$ .

$\frac{- 80}{5 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{- 80}{5 + 0}$

Step 3.2.2

Add $5$ and $0$ .

$\frac{- 80}{5}$

Step 3.3

Divide $- 80$ by $5$ .

$- 16$