Finite Math Examples

$f (x) = - 2 x^{2} + x + 1$ , $[- 1, 0]$

Step 1

Write $f (x) = - 2 x^{2} + x + 1$ as an equation.

$y = - 2 x^{2} + x + 1$

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.

$\frac{f (0) - f (- 1)}{(0) - (- 1)}$

Step 2.2

Substitute the equation $y = - 2 x^{2} + x + 1$ for $f (0)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{- 2 {(0)}^{2} + 0 + 1 - (- 2 {(- 1)}^{2} - 1 + 1)}{(0) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Remove parentheses.

$\frac{- 2 {(0)}^{2} + 0 + 1 - (- 2 {(- 1)}^{2} - 1 + 1)}{(0) - (- 1)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

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

$\frac{- 2 \cdot 0 + 0 + 1 - (- 2 {(- 1)}^{2} - 1 + 1)}{0 - (- 1)}$

Step 3.2.2

Multiply $- 2$ by $0$ .

$\frac{0 + 0 + 1 - (- 2 {(- 1)}^{2} - 1 + 1)}{0 - (- 1)}$

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

Raise $- 1$ to the power of $2$ .

$\frac{0 + 0 + 1 - (- 2 \cdot 1 - 1 + 1)}{0 - (- 1)}$

Step 3.2.3.2

Multiply $- 2$ by $1$ .

$\frac{0 + 0 + 1 - (- 2 - 1 + 1)}{0 - (- 1)}$

Step 3.2.4

Subtract $1$ from $- 2$ .

$\frac{0 + 0 + 1 - (- 3 + 1)}{0 - (- 1)}$

Step 3.2.5

Add $- 3$ and $1$ .

$\frac{0 + 0 + 1 - - 2}{0 - (- 1)}$

Step 3.2.6

Multiply $- 1$ by $- 2$ .

$\frac{0 + 0 + 1 + 2}{0 - (- 1)}$

Step 3.2.7

Add $0$ and $0$ .

$\frac{0 + 1 + 2}{0 - (- 1)}$

Step 3.2.8

Add $0$ and $1$ .

$\frac{1 + 2}{0 - (- 1)}$

Step 3.2.9

Add $1$ and $2$ .

$\frac{3}{0 - (- 1)}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{0 + 1}$

Step 3.3.2

Add $0$ and $1$ .

$\frac{3}{1}$

Step 3.4

Divide $3$ by $1$ .

$3$