Precalculus Examples

$(\frac{y}{2}) - 1 = 4^{x - 1}$

Step 1

Simplify.

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Step 1.1

Add $1$ to both sides of the equation.

$\frac{y}{2} = 4^{x - 1} + 1$

Step 1.2

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 (4^{x - 1} + 1)$

Step 1.3

Simplify both sides of the equation.

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Step 1.3.1

Simplify the left side.

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Step 1.3.1.1

Cancel the common factor of $2$ .

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Step 1.3.1.1.1

Cancel the common factor.

$2 \frac{y}{2} = 2 (4^{x - 1} + 1)$

Step 1.3.1.1.2

Rewrite the expression.

$y = 2 (4^{x - 1} + 1)$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Simplify $2 (4^{x - 1} + 1)$ .

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Step 1.3.2.1.1

Apply the distributive property.

$y = 2 \cdot 4^{x - 1} + 2 \cdot 1$

Step 1.3.2.1.2

Multiply $2 \cdot 4^{x - 1}$ .

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Step 1.3.2.1.2.1

Rewrite $4$ as $2^{2}$ .

$y = 2 \cdot {(2^{2})}^{x - 1} + 2 \cdot 1$

Step 1.3.2.1.2.2

Multiply the exponents in ${(2^{2})}^{x - 1}$ .

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Step 1.3.2.1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = 2 \cdot 2^{2 (x - 1)} + 2 \cdot 1$

Step 1.3.2.1.2.2.2

Apply the distributive property.

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

Step 1.3.2.1.2.2.3

Multiply $2$ by $- 1$ .

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

Step 1.3.2.1.2.3

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

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

Step 1.3.2.1.2.4

Subtract $2$ from $1$ .

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

Step 1.3.2.1.3

Multiply $2$ by $1$ .

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

Step 2

Exponential functions have a horizontal asymptote. The equation of the horizontal asymptote is $y = 2$ .

Horizontal Asymptote: $y = 2$

Step 3