Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

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

Step 1.2.2

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

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Apply the distributive property.

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

Step 1.2.2.1.2

Multiply $2$ by $1$ .

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

Step 1.2.2.1.3

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

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

Step 1.2.2.2

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

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

Step 1.2.2.3

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

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

Step 1.2.2.4

Combine the numerators over the common denominator.

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

Step 1.2.3

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

$x \approx - 3.79725096$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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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 $1$ .

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

Step 2.2.3.1.2

Multiply $2$ by $1$ .

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

Step 2.2.3.1.3

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

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

Step 2.2.3.1.4

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

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

Step 2.2.3.1.5

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Combine the numerators over the common denominator.

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

Step 2.2.3.5

Simplify the numerator.

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Step 2.2.3.5.1

Multiply $- 5$ by $16$ .

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

Step 2.2.3.5.2

Subtract $80$ from $9$ .

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

Step 2.2.3.6

Move the negative in front of the fraction.

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

Step 2.3

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