Precalculus Examples

$f (x) = {(x - 3)}^{2}$ , $[2, 5]$

Step 1

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

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

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.

$\frac{f (5) - f (2)}{(5) - (2)}$

Step 2.2

Substitute the equation $y = {(x - 3)}^{2}$ for $f (5)$ and $f (2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({((5) - 3)}^{2}) - ({((2) - 3)}^{2})}{(5) - (2)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Subtract $3$ from $5$ .

$\frac{2^{2} - {(2 - 3)}^{2}}{5 - (2)}$

Step 3.1.2

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

$\frac{4 - {(2 - 3)}^{2}}{5 - (2)}$

Step 3.1.3

Subtract $3$ from $2$ .

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

Step 3.1.4

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.1.4.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.1.4.1.1

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

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

Step 3.1.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.4.2

Add $1$ and $2$ .

$\frac{4 + {(- 1)}^{3}}{5 - (2)}$

Step 3.1.5

Raise $- 1$ to the power of $3$ .

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

Step 3.1.6

Subtract $1$ from $4$ .

$\frac{3}{5 - (2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $2$ .

$\frac{3}{5 - 2}$

Step 3.2.2

Subtract $2$ from $5$ .

$\frac{3}{3}$

Step 3.3

Divide $3$ by $3$ .

$1$