Precalculus Examples

Consider the function $f (x) = x^{2} + x$ and the point $(1, 2)$

Step 1

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

$y = x^{2} + x$

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.

$\frac{f (2) - f (1)}{(2) - (1)}$

Step 2.2

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

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

Step 3

Simplify the expression.

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Step 3.1

Remove parentheses.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

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

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

Step 3.2.2

One to any power is one.

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

Step 3.2.3

Add $1$ and $1$ .

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

Step 3.2.4

Multiply $- 1$ by $2$ .

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

Step 3.2.5

Add $4$ and $2$ .

$\frac{6 - 2}{2 - (1)}$

Step 3.2.6

Subtract $2$ from $6$ .

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

Multiply $- 1$ by $1$ .

$\frac{4}{2 - 1}$

Step 3.3.2

Subtract $1$ from $2$ .

$\frac{4}{1}$

Step 3.4

Divide $4$ by $1$ .

$4$