Finite Math Examples

$f (x) = 20 x^{2} - 80 x + 20$ , $[0, 3]$

Step 1

Write $f (x) = 20 x^{2} - 80 x + 20$ as an equation.

$y = 20 x^{2} - 80 x + 20$

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.

$\frac{f (3) - f (0)}{(3) - (0)}$

Step 2.2

Substitute the equation $y = 20 x^{2} - 80 x + 20$ for $f (3)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(20 {(3)}^{2} - 80 \cdot 3 + 20) - (20 {(0)}^{2} - 80 \cdot 0 + 20)}{(3) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $3$ to the power of $2$ .

$\frac{20 \cdot 9 - 80 \cdot 3 + 20 - (20 \cdot 0^{2} - 80 \cdot 0 + 20)}{3 - (0)}$

Step 3.1.2

Multiply $20$ by $9$ .

$\frac{180 - 80 \cdot 3 + 20 - (20 \cdot 0^{2} - 80 \cdot 0 + 20)}{3 - (0)}$

Step 3.1.3

Multiply $- 80$ by $3$ .

$\frac{180 - 240 + 20 - (20 \cdot 0^{2} - 80 \cdot 0 + 20)}{3 - (0)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

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

$\frac{180 - 240 + 20 - (20 \cdot 0 - 80 \cdot 0 + 20)}{3 - (0)}$

Step 3.1.4.2

Multiply $20$ by $0$ .

$\frac{180 - 240 + 20 - (0 - 80 \cdot 0 + 20)}{3 - (0)}$

Step 3.1.4.3

Multiply $- 80$ by $0$ .

$\frac{180 - 240 + 20 - (0 + 0 + 20)}{3 - (0)}$

Step 3.1.5

Add $0$ and $0$ .

$\frac{180 - 240 + 20 - (0 + 20)}{3 - (0)}$

Step 3.1.6

Add $0$ and $20$ .

$\frac{180 - 240 + 20 - 1 \cdot 20}{3 - (0)}$

Step 3.1.7

Multiply $- 1$ by $20$ .

$\frac{180 - 240 + 20 - 20}{3 - (0)}$

Step 3.1.8

Subtract $240$ from $180$ .

$\frac{- 60 + 20 - 20}{3 - (0)}$

Step 3.1.9

Add $- 60$ and $20$ .

$\frac{- 40 - 20}{3 - (0)}$

Step 3.1.10

Subtract $20$ from $- 40$ .

$\frac{- 60}{3 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{- 60}{3 + 0}$

Step 3.2.2

Add $3$ and $0$ .

$\frac{- 60}{3}$

Step 3.3

Divide $- 60$ by $3$ .

$- 20$