Find the Average Rate of Change f(x)=2x^2-x+6 ; [-2,3]

$f (x) = 2 x^{2} - x + 6$ ; $[- 2, 3]$

Step 1

Write $f (x) = 2 x^{2} - x + 6$ as an equation.

$y = 2 x^{2} - x + 6$

Step 2

Substitute using the average rate of change formula.

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.

$\frac{f (3) - f (- 2)}{(3) - (- 2)}$

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

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

Step 3

Simplify the expression.

Simplify the numerator.

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

$\frac{2 \cdot 9 - (3) + 6 - (2 {(- 2)}^{2} - (- 2) + 6)}{3 - (- 2)}$

Multiply $2$ by $9$ .

$\frac{18 - (3) + 6 - (2 {(- 2)}^{2} - (- 2) + 6)}{3 - (- 2)}$

Multiply $- 1$ by $3$ .

$\frac{18 - 3 + 6 - (2 {(- 2)}^{2} - (- 2) + 6)}{3 - (- 2)}$

Simplify each term.

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

$\frac{18 - 3 + 6 - (2 \cdot 4 - (- 2) + 6)}{3 - (- 2)}$

Multiply $2$ by $4$ .

$\frac{18 - 3 + 6 - (8 - (- 2) + 6)}{3 - (- 2)}$

Multiply $- 1$ by $- 2$ .

$\frac{18 - 3 + 6 - (8 + 2 + 6)}{3 - (- 2)}$

Add $8$ and $2$ .

$\frac{18 - 3 + 6 - (10 + 6)}{3 - (- 2)}$

Add $10$ and $6$ .

$\frac{18 - 3 + 6 - 1 \cdot 16}{3 - (- 2)}$

Multiply $- 1$ by $16$ .

$\frac{18 - 3 + 6 - 16}{3 - (- 2)}$

Subtract $3$ from $18$ .

$\frac{15 + 6 - 16}{3 - (- 2)}$

Add $15$ and $6$ .

$\frac{21 - 16}{3 - (- 2)}$

Subtract $16$ from $21$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 2$ .

$\frac{5}{3 + 2}$

Add $3$ and $2$ .

$\frac{5}{5}$

Divide $5$ by $5$ .

$1$

Find the Average Rate of Change f(x)=2x^2-x+6 ; [-2,3]

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