Precalculus Examples

$f (x) = - 3 x^{2} - x$ from $x_{1} = 5$ to $x_{2} = 6$

Step 1

Substitute using the average rate of change formula.

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Step 1.1

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.

$\frac{f (6) - f (5)}{(6) - (5)}$

Step 1.2

Substitute the equation $y = - 3 x^{2} - x$ for $f (6)$ and $f (5)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- 3 {(6)}^{2} - (6)) - (- 3 {(5)}^{2} - (5))}{(6) - (5)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Raise $6$ to the power of $2$ .

$\frac{- 3 \cdot 36 - (6) - (- 3 \cdot 5^{2} - (5))}{6 - (5)}$

Step 2.1.2

Multiply $- 3$ by $36$ .

$\frac{- 108 - (6) - (- 3 \cdot 5^{2} - (5))}{6 - (5)}$

Step 2.1.3

Multiply $- 1$ by $6$ .

$\frac{- 108 - 6 - (- 3 \cdot 5^{2} - (5))}{6 - (5)}$

Step 2.1.4

Simplify each term.

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Step 2.1.4.1

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

$\frac{- 108 - 6 - (- 3 \cdot 25 - (5))}{6 - (5)}$

Step 2.1.4.2

Multiply $- 3$ by $25$ .

$\frac{- 108 - 6 - (- 75 - (5))}{6 - (5)}$

Step 2.1.4.3

Multiply $- 1$ by $5$ .

$\frac{- 108 - 6 - (- 75 - 5)}{6 - (5)}$

Step 2.1.5

Subtract $5$ from $- 75$ .

$\frac{- 108 - 6 - - 80}{6 - (5)}$

Step 2.1.6

Multiply $- 1$ by $- 80$ .

$\frac{- 108 - 6 + 80}{6 - (5)}$

Step 2.1.7

Subtract $6$ from $- 108$ .

$\frac{- 114 + 80}{6 - (5)}$

Step 2.1.8

Add $- 114$ and $80$ .

$\frac{- 34}{6 - (5)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $5$ .

$\frac{- 34}{6 - 5}$

Step 2.2.2

Subtract $5$ from $6$ .

$\frac{- 34}{1}$

Step 2.3

Divide $- 34$ by $1$ .

$- 34$