Precalculus Examples

$f (x) = x^{2} - 7$ on $0$ , $9$

Step 1

Write $f (x) = x^{2} - 7$ as an equation.

$y = x^{2} - 7$

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.

$\frac{f (9) - f (0)}{(9) - (0)}$

Step 2.2

Substitute the equation $y = x^{2} - 7$ for $f (9)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(9)}^{2} - 7) - ({(0)}^{2} - 7)}{(9) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $9$ to the power of $2$ .

$\frac{81 - 7 - (0^{2} - 7)}{9 - (0)}$

Step 3.1.2

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

$\frac{81 - 7 - (0 - 7)}{9 - (0)}$

Step 3.1.3

Subtract $7$ from $0$ .

$\frac{81 - 7 - - 7}{9 - (0)}$

Step 3.1.4

Multiply $- 1$ by $- 7$ .

$\frac{81 - 7 + 7}{9 - (0)}$

Step 3.1.5

Subtract $7$ from $81$ .

$\frac{74 + 7}{9 - (0)}$

Step 3.1.6

Add $74$ and $7$ .

$\frac{81}{9 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{81}{9 + 0}$

Step 3.2.2

Add $9$ and $0$ .

$\frac{81}{9}$

Step 3.3

Divide $81$ by $9$ .

$9$