Algebra Examples

$f (x) = - \frac{1}{8} {(4)}^{x + 3} + 6$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \frac{1}{8} \cdot {(4)}^{x + 3} + 6$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{8} \cdot 4^{x + 3} + 6 = 0$ .

$- \frac{1}{8} \cdot 4^{x + 3} + 6 = 0$

Step 1.2.2

Simplify $- \frac{1}{8} \cdot 4^{x + 3} + 6$ .

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Step 1.2.2.1

Combine $4^{x + 3}$ and $\frac{1}{8}$ .

$- \frac{4^{x + 3}}{8} + 6 = 0$

Step 1.2.2.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{4^{x + 3}}{8} + 6 \cdot \frac{8}{8} = 0$

Step 1.2.2.3

Combine $6$ and $\frac{8}{8}$ .

$- \frac{4^{x + 3}}{8} + \frac{6 \cdot 8}{8} = 0$

Step 1.2.2.4

Combine the numerators over the common denominator.

$\frac{- 4^{x + 3} + 6 \cdot 8}{8} = 0$

Step 1.2.2.5

Multiply $6$ by $8$ .

$\frac{- 4^{x + 3} + 48}{8} = 0$

Step 1.2.2.6

Factor $- 1$ out of $- 4^{x + 3}$ .

$\frac{- (4^{x + 3}) + 48}{8} = 0$

Step 1.2.2.7

Rewrite $48$ as $- 1 (- 48)$ .

$\frac{- (4^{x + 3}) - 1 (- 48)}{8} = 0$

Step 1.2.2.8

Factor $- 1$ out of $- (4^{x + 3}) - 1 (- 48)$ .

$\frac{- (4^{x + 3} - 48)}{8} = 0$

Step 1.2.2.9

Simplify the expression.

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Step 1.2.2.9.1

Rewrite $- (4^{x + 3} - 48)$ as $- 1 (4^{x + 3} - 48)$ .

$\frac{- 1 (4^{x + 3} - 48)}{8} = 0$

Step 1.2.2.9.2

Move the negative in front of the fraction.

$- \frac{4^{x + 3} - 48}{8} = 0$

Step 1.2.3

Set the numerator equal to zero.

$4^{x + 3} - 48 = 0$

Step 1.2.4

Solve the equation for $x$ .

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Step 1.2.4.1

Add $48$ to both sides of the equation.

$4^{x + 3} = 48$

Step 1.2.4.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (4^{x + 3}) = \ln (48)$

Step 1.2.4.3

Expand $\ln (4^{x + 3})$ by moving $x + 3$ outside the logarithm.

$(x + 3) \ln (4) = \ln (48)$

Step 1.2.4.4

Simplify the left side.

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Step 1.2.4.4.1

Apply the distributive property.

$x \ln (4) + 3 \ln (4) = \ln (48)$

Step 1.2.4.5

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (4) + 3 \ln (4) - \ln (48) = 0$

Step 1.2.4.6

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.4.6.1

Subtract $3 \ln (4)$ from both sides of the equation.

$x \ln (4) - \ln (48) = - 3 \ln (4)$

Step 1.2.4.6.2

Add $\ln (48)$ to both sides of the equation.

$x \ln (4) = - 3 \ln (4) + \ln (48)$

Step 1.2.4.7

Divide each term in $x \ln (4) = - 3 \ln (4) + \ln (48)$ by $\ln (4)$ and simplify.

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Step 1.2.4.7.1

Divide each term in $x \ln (4) = - 3 \ln (4) + \ln (48)$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{- 3 \ln (4)}{\ln (4)} + \frac{\ln (48)}{\ln (4)}$

Step 1.2.4.7.2

Simplify the left side.

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Step 1.2.4.7.2.1

Cancel the common factor of $\ln (4)$ .

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Step 1.2.4.7.2.1.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{- 3 \ln (4)}{\ln (4)} + \frac{\ln (48)}{\ln (4)}$

Step 1.2.4.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (4)}{\ln (4)} + \frac{\ln (48)}{\ln (4)}$

Step 1.2.4.7.3

Simplify the right side.

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Step 1.2.4.7.3.1

Cancel the common factor of $\ln (4)$ .

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Step 1.2.4.7.3.1.1

Cancel the common factor.

$x = \frac{- 3 \ln (4)}{\ln (4)} + \frac{\ln (48)}{\ln (4)}$

Step 1.2.4.7.3.1.2

Divide $- 3$ by $1$ .

$x = - 3 + \frac{\ln (48)}{\ln (4)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 3 + \frac{\ln (48)}{\ln (4)}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \frac{1}{8} \cdot {(4)}^{(0) + 3} + 6$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - \frac{1}{8} \cdot 4^{0 + 3} + 6$

Step 2.2.2

Remove parentheses.

$y = - \frac{1}{8} \cdot {(4)}^{(0) + 3} + 6$

Step 2.2.3

Simplify $- \frac{1}{8} \cdot {(4)}^{(0) + 3} + 6$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Add $0$ and $3$ .

$y = - \frac{1}{8} \cdot 4^{3} + 6$

Step 2.2.3.1.2

Raise $4$ to the power of $3$ .

$y = - \frac{1}{8} \cdot 64 + 6$

Step 2.2.3.1.3

Cancel the common factor of $8$ .

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Step 2.2.3.1.3.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$y = \frac{- 1}{8} \cdot 64 + 6$

Step 2.2.3.1.3.2

Factor $8$ out of $64$ .

$y = \frac{- 1}{8} \cdot (8 (8)) + 6$

Step 2.2.3.1.3.3

Cancel the common factor.

$y = \frac{- 1}{8} \cdot (8 \cdot 8) + 6$

Step 2.2.3.1.3.4

Rewrite the expression.

$y = - 1 \cdot 8 + 6$

Step 2.2.3.1.4

Multiply $- 1$ by $8$ .

$y = - 8 + 6$

Step 2.2.3.2

Add $- 8$ and $6$ .

$y = - 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

x-intercept(s): $(- 3 + \frac{\ln (48)}{\ln (4)}, 0)$

y-intercept(s): $(0, - 2)$

Step 4