Algebra Examples

$f (x) = 3 \cdot {(\frac{1}{4})}^{x} - 3$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 \cdot {(\frac{1}{4})}^{x} - 3 = 0$ .

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

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

Step 1.2.2.2

One to any power is one.

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

Step 1.2.2.3

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

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

Step 1.2.3

Add $3$ to both sides of the equation.

$\frac{3}{4^{x}} = 3$

Step 1.2.4

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

$\frac{3}{4^{x}} \cdot 4^{x} = 3 \cdot 4^{x}$

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.

$\frac{3}{4^{x}} \cdot 4^{x} = 3 \cdot 4^{x}$

Step 1.2.5.1.2

Rewrite the expression.

$3 = 3 \cdot 4^{x}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Rewrite the equation as $3 \cdot 4^{x} = 3$ .

$3 \cdot 4^{x} = 3$

Step 1.2.6.2

Divide each term in $3 \cdot 4^{x} = 3$ by $3$ and simplify.

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Step 1.2.6.2.1

Divide each term in $3 \cdot 4^{x} = 3$ by $3$ .

$\frac{3 \cdot 4^{x}}{3} = \frac{3}{3}$

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 $3$ .

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Step 1.2.6.2.2.1.1

Cancel the common factor.

$\frac{3 \cdot 4^{x}}{3} = \frac{3}{3}$

Step 1.2.6.2.2.1.2

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

$4^{x} = \frac{3}{3}$

Step 1.2.6.2.3

Simplify the right side.

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Step 1.2.6.2.3.1

Divide $3$ by $3$ .

$4^{x} = 1$

Step 1.2.6.3

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

$\ln (4^{x}) = \ln (1)$

Step 1.2.6.4

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

$x \ln (4) = \ln (1)$

Step 1.2.6.5

Simplify the right side.

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Step 1.2.6.5.1

The natural logarithm of $1$ is $0$ .

$x \ln (4) = 0$

Step 1.2.6.6

Divide each term in $x \ln (4) = 0$ by $\ln (4)$ and simplify.

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Step 1.2.6.6.1

Divide each term in $x \ln (4) = 0$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{0}{\ln (4)}$

Step 1.2.6.6.2

Simplify the left side.

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Step 1.2.6.6.2.1

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

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Step 1.2.6.6.2.1.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{0}{\ln (4)}$

Step 1.2.6.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{\ln (4)}$

Step 1.2.6.6.3

Simplify the right side.

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Step 1.2.6.6.3.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$x = \frac{0}{\ln (2^{2})}$

Step 1.2.6.6.3.2

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$x = \frac{0}{2 \ln (2)}$

Step 1.2.6.6.3.3

Cancel the common factor of $0$ and $2$ .

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Step 1.2.6.6.3.3.1

Factor $2$ out of $0$ .

$x = \frac{2 (0)}{2 \ln (2)}$

Step 1.2.6.6.3.3.2

Cancel the common factors.

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Step 1.2.6.6.3.3.2.1

Factor $2$ out of $2 \ln (2)$ .

$x = \frac{2 (0)}{2 (\ln (2))}$

Step 1.2.6.6.3.3.2.2

Cancel the common factor.

$x = \frac{2 \cdot 0}{2 \ln (2)}$

Step 1.2.6.6.3.3.2.3

Rewrite the expression.

$x = \frac{0}{\ln (2)}$

Step 1.2.6.6.3.4

Divide $0$ by $\ln (2)$ .

$x = 0$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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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 $\frac{1}{4}$ .

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

Step 2.2.2.1.2

Anything raised to $0$ is $1$ .

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

Step 2.2.2.1.3

Anything raised to $0$ is $1$ .

$y = 3 \cdot \frac{1}{1} - 3$

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.

$y = 3 \cdot \frac{1}{1} - 3$

Step 2.2.2.1.4.2

Rewrite the expression.

$y = 3 \cdot 1 - 3$

Step 2.2.2.1.5

Multiply $3$ by $1$ .

$y = 3 - 3$

Step 2.2.2.2

Subtract $3$ from $3$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0)$

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

Step 4