Find the x and y Intercepts f(x)=(3/4)^(2(x+1))-5

$f (x) = {(\frac{3}{4})}^{2 (x + 1)} - 5$

Step 1

Find the x-intercepts.

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

$0 = {(\frac{3}{4})}^{2 (x + 1)} - 5$

Solve the equation.

Remove parentheses.

$0 = {(\frac{3}{4})}^{2 (x + 1)} - 5$

Simplify ${(\frac{3}{4})}^{2 (x + 1)} - 5$ .

Simplify each term.

Apply the distributive property.

$0 = {(\frac{3}{4})}^{2 x + 2 \cdot 1} - 5$

Multiply $2$ by $1$ .

$0 = {(\frac{3}{4})}^{2 x + 2} - 5$

Apply the product rule to $\frac{3}{4}$ .

$0 = \frac{3^{2 x + 2}}{4^{2 x + 2}} - 5$

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4^{2 x + 2}}{4^{2 x + 2}}$ .

$0 = \frac{3^{2 x + 2}}{4^{2 x + 2}} - 5 \cdot \frac{4^{2 x + 2}}{4^{2 x + 2}}$

Combine $- 5$ and $\frac{4^{2 x + 2}}{4^{2 x + 2}}$ .

$0 = \frac{3^{2 x + 2}}{4^{2 x + 2}} + \frac{- 5 \cdot 4^{2 x + 2}}{4^{2 x + 2}}$

Combine the numerators over the common denominator.

$0 = \frac{3^{2 x + 2} - 5 \cdot 4^{2 x + 2}}{4^{2 x + 2}}$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 3.79725096$

x-intercept(s) in point form.

x-intercept(s): $(- 3.79725096, 0)$

Step 2

Find the y-intercepts.

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

$y = {(\frac{3}{4})}^{2 ((0) + 1)} - 5$

Solve the equation.

Remove parentheses.

$y = {(\frac{3}{4})}^{2 (0 + 1)} - 5$

Remove parentheses.

$y = {(\frac{3}{4})}^{2 ((0) + 1)} - 5$

Simplify ${(\frac{3}{4})}^{2 ((0) + 1)} - 5$ .

Simplify each term.

Add $0$ and $1$ .

$y = {(\frac{3}{4})}^{2 \cdot 1} - 5$

Multiply $2$ by $1$ .

$y = {(\frac{3}{4})}^{2} - 5$

Apply the product rule to $\frac{3}{4}$ .

$y = \frac{3^{2}}{4^{2}} - 5$

Raise $3$ to the power of $2$ .

$y = \frac{9}{4^{2}} - 5$

Raise $4$ to the power of $2$ .

$y = \frac{9}{16} - 5$

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$y = \frac{9}{16} - 5 \cdot \frac{16}{16}$

Combine $- 5$ and $\frac{16}{16}$ .

$y = \frac{9}{16} + \frac{- 5 \cdot 16}{16}$

Combine the numerators over the common denominator.

$y = \frac{9 - 5 \cdot 16}{16}$

Simplify the numerator.

Multiply $- 5$ by $16$ .

$y = \frac{9 - 80}{16}$

Subtract $80$ from $9$ .

$y = \frac{- 71}{16}$

Move the negative in front of the fraction.

$y = - \frac{71}{16}$

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{71}{16})$

Step 3

List the intersections.

x-intercept(s): $(- 3.79725096, 0)$

y-intercept(s): $(0, - \frac{71}{16})$

Step 4

Find the x and y Intercepts f(x)=(3/4)^(2(x+1))-5

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