Precalculus Examples

$h (x) = 6 - 4 x^{2}$ on $- 2$ , $4$

Step 1

Write $h (x) = 6 - 4 x^{2}$ as an equation.

$y = 6 - 4 x^{2}$

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.

$\frac{f (4) - f (- 2)}{(4) - (- 2)}$

Step 2.2

Substitute the equation $y = 6 - 4 x^{2}$ for $f (4)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(6 - 4 {(4)}^{2}) - (6 - 4 {(- 2)}^{2})}{(4) - (- 2)}$

Step 3

Simplify the expression.

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Step 3.1

Cancel the common factor of $(6 - 4 {(4)}^{2}) - (6 - 4 {(- 2)}^{2})$ and $(4) - (- 2)$ .

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Step 3.1.1

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{- 1 (- 6) - 4 {(4)}^{2} - (6 - 4 {(- 2)}^{2})}{(4) - (- 2)}$

Step 3.1.2

Factor $- 1$ out of $- 4 {(4)}^{2}$ .

$\frac{- 1 (- 6) - (4 {(4)}^{2}) - (6 - 4 {(- 2)}^{2})}{(4) - (- 2)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- 6) - (4 {(4)}^{2})$ .

$\frac{- 1 (- 6 + 4 {(4)}^{2}) - (6 - 4 {(- 2)}^{2})}{(4) - (- 2)}$

Step 3.1.4

Factor $- 1$ out of $- 1 (- 6 + 4 {(4)}^{2}) - (6 - 4 {(- 2)}^{2})$ .

$\frac{- 1 (- 6 + 4 {(4)}^{2} + 6 - 4 {(- 2)}^{2})}{(4) - (- 2)}$

Step 3.1.5

Factor $2$ out of $- 1 (- 6 + 4 \cdot 4^{2} + 6 - 4 {(- 2)}^{2})$ .

$\frac{2 (- 1 (- 3 + 2 \cdot 4^{2} + 3 - 2 {(- 2)}^{2}))}{4 - (- 2)}$

Step 3.1.6

Cancel the common factors.

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Step 3.1.6.1

Factor $2$ out of $4$ .

$\frac{2 (- 1 (- 3 + 2 \cdot 4^{2} + 3 - 2 {(- 2)}^{2}))}{2 (2) - (- 2)}$

Step 3.1.6.2

Factor $2$ out of $- (- 2)$ .

$\frac{2 (- 1 (- 3 + 2 \cdot 4^{2} + 3 - 2 {(- 2)}^{2}))}{2 (2) + 2 (- (- 1))}$

Step 3.1.6.3

Factor $2$ out of $2 (2) + 2 (- (- 1))$ .

$\frac{2 (- 1 (- 3 + 2 \cdot 4^{2} + 3 - 2 {(- 2)}^{2}))}{2 (2 - (- 1))}$

Step 3.1.6.4

Cancel the common factor.

$\frac{2 (- 1 (- 3 + 2 \cdot 4^{2} + 3 - 2 {(- 2)}^{2}))}{2 (2 - (- 1))}$

Step 3.1.6.5

Rewrite the expression.

$\frac{- 1 (- 3 + 2 \cdot 4^{2} + 3 - 2 {(- 2)}^{2})}{2 - (- 1)}$

Step 3.2

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

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Step 3.2.1

Multiply $- 2$ by ${(- 2)}^{2}$ .

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Step 3.2.1.1

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

$\frac{- 1 (- 3 + 2 \cdot 4^{2} + 3 + {(- 2)}^{1} {(- 2)}^{2})}{2 - (- 1)}$

Step 3.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 1 (- 3 + 2 \cdot 4^{2} + 3 + {(- 2)}^{1 + 2})}{2 - (- 1)}$

Step 3.2.2

Add $1$ and $2$ .

$\frac{- 1 (- 3 + 2 \cdot 4^{2} + 3 + {(- 2)}^{3})}{2 - (- 1)}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Raise $4$ to the power of $2$ .

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

Step 3.3.2

Multiply $2$ by $16$ .

$\frac{- 1 (- 3 + 32 + 3 + {(- 2)}^{3})}{2 - (- 1)}$

Step 3.3.3

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

$\frac{- 1 (- 3 + 32 + 3 - 8)}{2 - (- 1)}$

Step 3.3.4

Add $- 3$ and $32$ .

$\frac{- 1 (29 + 3 - 8)}{2 - (- 1)}$

Step 3.3.5

Add $29$ and $3$ .

$\frac{- 1 (32 - 8)}{2 - (- 1)}$

Step 3.3.6

Subtract $8$ from $32$ .

$\frac{- 1 \cdot 24}{2 - (- 1)}$

Step 3.4

Simplify the denominator.

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Step 3.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- 1 \cdot 24}{2 + 1}$

Step 3.4.2

Add $2$ and $1$ .

$\frac{- 1 \cdot 24}{3}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Multiply $- 1$ by $24$ .

$\frac{- 24}{3}$

Step 3.5.2

Divide $- 24$ by $3$ .

$- 8$