Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2^{3 (x + 1)} - 4 = 0$ .

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Apply the distributive property.

$2^{3 x + 3 \cdot 1} - 4 = 0$

Step 1.2.2.2

Multiply $3$ by $1$ .

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

Step 1.2.3

Add $4$ to both sides of the equation.

$2^{3 x + 3} = 4$

Step 1.2.4

Create equivalent expressions in the equation that all have equal bases.

$2^{3 x + 3} = 2^{2}$

Step 1.2.5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 x + 3 = 2$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.6.1.1

Subtract $3$ from both sides of the equation.

$3 x = 2 - 3$

Step 1.2.6.1.2

Subtract $3$ from $2$ .

$3 x = - 1$

Step 1.2.6.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 1.2.6.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

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.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 1.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 1.2.6.2.3

Simplify the right side.

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Step 1.2.6.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{3}, 0)$

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 $y$ .

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

Step 2.2

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

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Add $0$ and $1$ .

$y = 2^{3 \cdot 1} - 4$

Step 2.2.1.2

Multiply $3$ by $1$ .

$y = 2^{3} - 4$

Step 2.2.1.3

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

$y = 8 - 4$

Step 2.2.2

Subtract $4$ from $8$ .

$y = 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{3}, 0)$

y-intercept(s): $(0, 4)$

Step 4