Algebra Examples

$f (x) = 2 {(\frac{1}{4})}^{x} - 9$

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 $x$ .

$0 = 2 {(\frac{1}{4})}^{x} - 9$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 {(\frac{1}{4})}^{x} - 9 = 0$ .

$2 {(\frac{1}{4})}^{x} - 9 = 0$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Apply the product rule to $\frac{1}{4}$ .

$2 \frac{1^{x}}{4^{x}} - 9 = 0$

Step 1.2.2.2

One to any power is one.

$2 \frac{1}{4^{x}} - 9 = 0$

Step 1.2.2.3

Combine $2$ and $\frac{1}{4^{x}}$ .

$\frac{2}{4^{x}} - 9 = 0$

Step 1.2.3

Add $9$ to both sides of the equation.

$\frac{2}{4^{x}} = 9$

Step 1.2.4

Multiply both sides by $4^{x}$ .

$\frac{2}{4^{x}} \cdot 4^{x} = 9 \cdot 4^{x}$

Step 1.2.5

Simplify the left side.

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Step 1.2.5.1

Cancel the common factor of $4^{x}$ .

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Step 1.2.5.1.1

Cancel the common factor.

$\frac{2}{4^{x}} \cdot 4^{x} = 9 \cdot 4^{x}$

Step 1.2.5.1.2

Rewrite the expression.

$2 = 9 \cdot 4^{x}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Rewrite the equation as $9 \cdot 4^{x} = 2$ .

$9 \cdot 4^{x} = 2$

Step 1.2.6.2

Divide each term in $9 \cdot 4^{x} = 2$ by $9$ and simplify.

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Step 1.2.6.2.1

Divide each term in $9 \cdot 4^{x} = 2$ by $9$ .

$\frac{9 \cdot 4^{x}}{9} = \frac{2}{9}$

Step 1.2.6.2.2

Simplify the left side.

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Step 1.2.6.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.6.2.2.1.1

Cancel the common factor.

$\frac{9 \cdot 4^{x}}{9} = \frac{2}{9}$

Step 1.2.6.2.2.1.2

Divide $4^{x}$ by $1$ .

$4^{x} = \frac{2}{9}$

Step 1.2.6.3

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

$\ln (4^{x}) = \ln (\frac{2}{9})$

Step 1.2.6.4

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

$x \ln (4) = \ln (\frac{2}{9})$

Step 1.2.6.5

Divide each term in $x \ln (4) = \ln (\frac{2}{9})$ by $\ln (4)$ and simplify.

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Step 1.2.6.5.1

Divide each term in $x \ln (4) = \ln (\frac{2}{9})$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{\ln (\frac{2}{9})}{\ln (4)}$

Step 1.2.6.5.2

Simplify the left side.

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Step 1.2.6.5.2.1

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

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Step 1.2.6.5.2.1.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{\ln (\frac{2}{9})}{\ln (4)}$

Step 1.2.6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{2}{9})}{\ln (4)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\ln (\frac{2}{9})}{\ln (4)}, 0)$

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 $y$ .

$y = 2 {(\frac{1}{4})}^{0} - 9$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 {(\frac{1}{4})}^{0} - 9$

Step 2.2.2

Simplify $2 {(\frac{1}{4})}^{0} - 9$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Apply the product rule to $\frac{1}{4}$ .

$y = 2 \frac{1^{0}}{4^{0}} - 9$

Step 2.2.2.1.2

Anything raised to $0$ is $1$ .

$y = 2 \frac{1}{4^{0}} - 9$

Step 2.2.2.1.3

Anything raised to $0$ is $1$ .

$y = 2 (\frac{1}{1}) - 9$

Step 2.2.2.1.4

Cancel the common factor of $1$ .

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Step 2.2.2.1.4.1

Cancel the common factor.

$y = 2 (\frac{1}{1}) - 9$

Step 2.2.2.1.4.2

Rewrite the expression.

$y = 2 \cdot 1 - 9$

Step 2.2.2.1.5

Multiply $2$ by $1$ .

$y = 2 - 9$

Step 2.2.2.2

Subtract $9$ from $2$ .

$y = - 7$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 7)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\ln (\frac{2}{9})}{\ln (4)}, 0)$

y-intercept(s): $(0, - 7)$

Step 4