Find the x and y Intercepts f(x)=3*(1/4)^x-3

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

Simplify each term.

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

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

One to any power is one.

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

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

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

Add $3$ to both sides of the equation.

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Rewrite the expression.

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

Solve for $x$ .

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Divide $3$ by $3$ .

$4^{x} = 1$

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

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

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

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

Simplify the right side.

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

$x \ln (4) = 0$

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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

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

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

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

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

Factor $2$ out of $0$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$x = 0$

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

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

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

Anything raised to $0$ is $1$ .

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

Cancel the common factor of $1$ .

Cancel the common factor.

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

Rewrite the expression.

$y = 3 \cdot 1 - 3$

Multiply $3$ by $1$ .

$y = 3 - 3$

Subtract $3$ from $3$ .

$y = 0$

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

Find the x and y Intercepts f(x)=3*(1/4)^x-3

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