Precalculus Examples

$y = 1200 {(0.86)}^{4 x}$

Step 1

Write $y = 1200 {(0.86)}^{4 x}$ as a function.

$f (x) = 1200 {(0.86)}^{4 x}$

Step 2

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = x + h$ .

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Step 3.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 1200 {(0.86)}^{4 (x + h)}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (x + h) = 1200 \cdot {0.86}^{4 x + 4 h}$

Step 3.1.2.2

The final answer is $1200 \cdot {0.86}^{4 x + 4 h}$ .

$1200 \cdot {0.86}^{4 x + 4 h}$

Step 3.2

Find the components of the definition.

$f (x + h) = 1200 \cdot {0.86}^{4 x + 4 h}$

$f (x) = 1200 \cdot {0.86}^{4 x}$

$f (x + h) = 1200 \cdot {0.86}^{4 x + 4 h}$

$f (x) = 1200 \cdot {0.86}^{4 x}$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{1200 \cdot {0.86}^{4 x + 4 h} - (1200 \cdot {0.86}^{4 x})}{h}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $1200$ .

$\frac{1200 \cdot {0.86}^{4 x + 4 h} - 1200 \cdot {0.86}^{4 x}}{h}$

Step 5.2

Rewrite $1200 \cdot {0.86}^{4 x + 4 h} - 1200 \cdot {0.86}^{4 x}$ in a factored form.

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Step 5.2.1

Factor $1200$ out of $1200 \cdot {0.86}^{4 x + 4 h} - 1200 \cdot {0.86}^{4 x}$ .

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Step 5.2.1.1

Factor $1200$ out of $1200 \cdot {0.86}^{4 x + 4 h}$ .

$\frac{1200 \cdot {0.86}^{4 x + 4 h} - 1200 \cdot {0.86}^{4 x}}{h}$

Step 5.2.1.2

Factor $1200$ out of $- 1200 \cdot {0.86}^{4 x}$ .

$\frac{1200 \cdot {0.86}^{4 x + 4 h} + 1200 (- {0.86}^{4 x})}{h}$

Step 5.2.1.3

Factor $1200$ out of $1200 \cdot {0.86}^{4 x + 4 h} + 1200 (- {0.86}^{4 x})$ .

$\frac{1200 ({0.86}^{4 x + 4 h} - {0.86}^{4 x})}{h}$

Step 5.2.2

Rewrite ${0.86}^{4 x + 4 h}$ as ${({0.86}^{2 x + 2 h})}^{2}$ .

$\frac{1200 ({({0.86}^{2 x + 2 h})}^{2} - {0.86}^{4 x})}{h}$

Step 5.2.3

Rewrite ${0.86}^{4 x}$ as ${({0.86}^{2 x})}^{2}$ .

$\frac{1200 ({({0.86}^{2 x + 2 h})}^{2} - {({0.86}^{2 x})}^{2})}{h}$

Step 5.2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {0.86}^{2 x + 2 h}$ and $b = {0.86}^{2 x}$ .

$\frac{1200 (({0.86}^{2 x + 2 h} + {0.86}^{2 x}) ({0.86}^{2 x + 2 h} - {0.86}^{2 x}))}{h}$

Step 5.2.5

Simplify.

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Step 5.2.5.1

Rewrite ${0.86}^{2 x + 2 h}$ as ${({0.86}^{x + h})}^{2}$ .

$\frac{1200 (({0.86}^{2 x + 2 h} + {0.86}^{2 x}) ({({0.86}^{x + h})}^{2} - {0.86}^{2 x}))}{h}$

Step 5.2.5.2

Rewrite ${0.86}^{2 x}$ as ${({0.86}^{x})}^{2}$ .

$\frac{1200 (({0.86}^{2 x + 2 h} + {0.86}^{2 x}) ({({0.86}^{x + h})}^{2} - {({0.86}^{x})}^{2}))}{h}$

Step 5.2.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {0.86}^{x + h}$ and $b = {0.86}^{x}$ .

$\frac{1200 (({0.86}^{2 x + 2 h} + {0.86}^{2 x}) ({0.86}^{x + h} + {0.86}^{x}) ({0.86}^{x + h} - {0.86}^{x}))}{h}$

$\frac{1200 ({0.86}^{2 x + 2 h} + {0.86}^{2 x}) ({0.86}^{x + h} + {0.86}^{x}) ({0.86}^{x + h} - {0.86}^{x})}{h}$

Step 6