Precalculus Examples

$g (x) = 5 - x^{2}$ on $2$ , $4$

Step 1

Write $g (x) = 5 - x^{2}$ as an equation.

$y = 5 - x^{2}$

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.

$\frac{f (4) - f (2)}{(4) - (2)}$

Step 2.2

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

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

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 $2$ .

$\frac{5 - 1 \cdot 16 - (5 - 2^{2})}{4 - (2)}$

Step 3.1.2

Multiply $- 1$ by $16$ .

$\frac{5 - 16 - (5 - 2^{2})}{4 - (2)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

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

$\frac{5 - 16 - (5 - 1 \cdot 4)}{4 - (2)}$

Step 3.1.3.2

Multiply $- 1$ by $4$ .

$\frac{5 - 16 - (5 - 4)}{4 - (2)}$

Step 3.1.4

Subtract $4$ from $5$ .

$\frac{5 - 16 - 1 \cdot 1}{4 - (2)}$

Step 3.1.5

Multiply $- 1$ by $1$ .

$\frac{5 - 16 - 1}{4 - (2)}$

Step 3.1.6

Subtract $16$ from $5$ .

$\frac{- 11 - 1}{4 - (2)}$

Step 3.1.7

Subtract $1$ from $- 11$ .

$\frac{- 12}{4 - (2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $2$ .

$\frac{- 12}{4 - 2}$

Step 3.2.2

Subtract $2$ from $4$ .

$\frac{- 12}{2}$

Step 3.3

Divide $- 12$ by $2$ .

$- 6$