Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{2} \cdot 2^{x} - 4 = 0$ .

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Combine $\frac{1}{2}$ and $2^{x}$ .

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

Step 1.2.2.2

Cancel the common factor of $2^{x}$ and $2$ .

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Step 1.2.2.2.1

Factor $2$ out of $2^{x}$ .

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

Step 1.2.2.2.2

Cancel the common factors.

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Step 1.2.2.2.2.1

Factor $2$ out of $2$ .

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

Step 1.2.2.2.2.2

Cancel the common factor.

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

Step 1.2.2.2.2.3

Rewrite the expression.

$\frac{2^{x - 1}}{1} - 4 = 0$

Step 1.2.2.2.2.4

Divide $2^{x - 1}$ by $1$ .

$2^{x - 1} - 4 = 0$

Step 1.2.3

Add $4$ to both sides of the equation.

$2^{x - 1} = 4$

Step 1.2.4

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

$2^{x - 1} = 2^{2}$

Step 1.2.5

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

$x - 1 = 2$

Step 1.2.6

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

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Step 1.2.6.1

Add $1$ to both sides of the equation.

$x = 2 + 1$

Step 1.2.6.2

Add $2$ and $1$ .

$x = 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0)$

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 $y$ .

$y = \frac{1}{2} \cdot 2^{0} - 4$

Step 2.2

Simplify $\frac{1}{2} \cdot 2^{0} - 4$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Anything raised to $0$ is $1$ .

$y = \frac{1}{2} \cdot 1 - 4$

Step 2.2.1.2

Multiply $\frac{1}{2}$ by $1$ .

$y = \frac{1}{2} - 4$

Step 2.2.2

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

$y = \frac{1}{2} - 4 \cdot \frac{2}{2}$

Step 2.2.3

Combine $- 4$ and $\frac{2}{2}$ .

$y = \frac{1}{2} + \frac{- 4 \cdot 2}{2}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{1 - 4 \cdot 2}{2}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 4$ by $2$ .

$y = \frac{1 - 8}{2}$

Step 2.2.5.2

Subtract $8$ from $1$ .

$y = \frac{- 7}{2}$

Step 2.2.6

Move the negative in front of the fraction.

$y = - \frac{7}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{7}{2})$

Step 3

List the intersections.

x-intercept(s): $(3, 0)$

y-intercept(s): $(0, - \frac{7}{2})$

Step 4