Finite Math Examples

$y = | x^{2} + 2 x - 24 |$ , $[0, 6]$

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.

$\frac{f (6) - f (0)}{(6) - (0)}$

Step 1.2

Substitute the equation $y = | x^{2} + 2 x - 24 |$ for $f (6)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(| {(6)}^{2} + 2 (6) - 24 |) - (| {(0)}^{2} + 2 (0) - 24 |)}{(6) - (0)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$\frac{| 36 + 2 (6) - 24 | - | 0^{2} + 2 (0) - 24 |}{6 - (0)}$

Step 2.1.1.2

Multiply $2$ by $6$ .

$\frac{| 36 + 12 - 24 | - | 0^{2} + 2 (0) - 24 |}{6 - (0)}$

Step 2.1.2

Add $36$ and $12$ .

$\frac{| 48 - 24 | - | 0^{2} + 2 (0) - 24 |}{6 - (0)}$

Step 2.1.3

Subtract $24$ from $48$ .

$\frac{| 24 | - | 0^{2} + 2 (0) - 24 |}{6 - (0)}$

Step 2.1.4

The absolute value is the distance between a number and zero. The distance between $0$ and $24$ is $24$ .

$\frac{24 - | 0^{2} + 2 (0) - 24 |}{6 - (0)}$

Step 2.1.5

Simplify each term.

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Step 2.1.5.1

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

$\frac{24 - | 0 + 2 (0) - 24 |}{6 - (0)}$

Step 2.1.5.2

Multiply $2$ by $0$ .

$\frac{24 - | 0 + 0 - 24 |}{6 - (0)}$

Step 2.1.6

Add $0$ and $0$ .

$\frac{24 - | 0 - 24 |}{6 - (0)}$

Step 2.1.7

Subtract $24$ from $0$ .

$\frac{24 - | - 24 |}{6 - (0)}$

Step 2.1.8

The absolute value is the distance between a number and zero. The distance between $- 24$ and $0$ is $24$ .

$\frac{24 - 1 \cdot 24}{6 - (0)}$

Step 2.1.9

Multiply $- 1$ by $24$ .

$\frac{24 - 24}{6 - (0)}$

Step 2.1.10

Subtract $24$ from $24$ .

$\frac{0}{6 - (0)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $0$ .

$\frac{0}{6 + 0}$

Step 2.2.2

Add $6$ and $0$ .

$\frac{0}{6}$

Step 2.3

Divide $0$ by $6$ .

$0$