Algebra Examples

$f (x) = - \frac{1}{4} \sqrt{8 - 4 x} + 1$

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 $x$ .

$0 = - \frac{1}{4} \cdot \sqrt{8 - 4 x} + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{4} \cdot \sqrt{8 - 4 x} + 1 = 0$ .

$- \frac{1}{4} \cdot \sqrt{8 - 4 x} + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$- \frac{1}{4} \cdot \sqrt{8 - 4 x} = - 1$

Step 1.2.3

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \frac{1}{4} \cdot \sqrt{8 - 4 x})}^{2} = {(- 1)}^{2}$

Step 1.2.4

Simplify each side of the equation.

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Step 1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 - 4 x}$ as ${(8 - 4 x)}^{\frac{1}{2}}$ .

${(- \frac{1}{4} \cdot {(8 - 4 x)}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Simplify ${(- \frac{1}{4} \cdot {(8 - 4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.1

Combine ${(8 - 4 x)}^{\frac{1}{2}}$ and $\frac{1}{4}$ .

${(- \frac{{(8 - 4 x)}^{\frac{1}{2}}}{4})}^{2} = {(- 1)}^{2}$

Step 1.2.4.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.2.4.2.1.2.1

Apply the product rule to $- \frac{{(8 - 4 x)}^{\frac{1}{2}}}{4}$ .

${(- 1)}^{2} {(\frac{{(8 - 4 x)}^{\frac{1}{2}}}{4})}^{2} = {(- 1)}^{2}$

Step 1.2.4.2.1.2.2

Apply the product rule to $\frac{{(8 - 4 x)}^{\frac{1}{2}}}{4}$ .

${(- 1)}^{2} \frac{{({(8 - 4 x)}^{\frac{1}{2}})}^{2}}{4^{2}} = {(- 1)}^{2}$

Step 1.2.4.2.1.3

Simplify the expression.

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Step 1.2.4.2.1.3.1

Raise $- 1$ to the power of $2$ .

$1 \frac{{({(8 - 4 x)}^{\frac{1}{2}})}^{2}}{4^{2}} = {(- 1)}^{2}$

Step 1.2.4.2.1.3.2

Multiply $\frac{{({(8 - 4 x)}^{\frac{1}{2}})}^{2}}{4^{2}}$ by $1$ .

$\frac{{({(8 - 4 x)}^{\frac{1}{2}})}^{2}}{4^{2}} = {(- 1)}^{2}$

Step 1.2.4.2.1.4

Simplify the numerator.

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Step 1.2.4.2.1.4.1

Multiply the exponents in ${({(8 - 4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(8 - 4 x)}^{\frac{1}{2} \cdot 2}}{4^{2}} = {(- 1)}^{2}$

Step 1.2.4.2.1.4.1.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.4.1.2.1

Cancel the common factor.

$\frac{{(8 - 4 x)}^{\frac{1}{2} \cdot 2}}{4^{2}} = {(- 1)}^{2}$

Step 1.2.4.2.1.4.1.2.2

Rewrite the expression.

$\frac{{(8 - 4 x)}^{1}}{4^{2}} = {(- 1)}^{2}$

Step 1.2.4.2.1.4.2

Simplify.

$\frac{8 - 4 x}{4^{2}} = {(- 1)}^{2}$

Step 1.2.4.2.1.4.3

Factor $4$ out of $8 - 4 x$ .

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Step 1.2.4.2.1.4.3.1

Factor $4$ out of $8$ .

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

Step 1.2.4.2.1.4.3.2

Factor $4$ out of $- 4 x$ .

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

Step 1.2.4.2.1.4.3.3

Factor $4$ out of $4 (2) + 4 (- x)$ .

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

Step 1.2.4.2.1.5

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

$\frac{4 (2 - x)}{16} = {(- 1)}^{2}$

Step 1.2.4.2.1.6

Cancel the common factors.

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Step 1.2.4.2.1.6.1

Factor $4$ out of $16$ .

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

Step 1.2.4.2.1.6.2

Cancel the common factor.

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

Step 1.2.4.2.1.6.3

Rewrite the expression.

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

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Raise $- 1$ to the power of $2$ .

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

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Multiply both sides by $4$ .

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

Step 1.2.5.2

Simplify.

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Step 1.2.5.2.1

Simplify the left side.

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Step 1.2.5.2.1.1

Simplify $\frac{2 - x}{4} \cdot 4$ .

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Step 1.2.5.2.1.1.1

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.1.1.1

Cancel the common factor.

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

Step 1.2.5.2.1.1.1.2

Rewrite the expression.

$2 - x = 1 \cdot 4$

Step 1.2.5.2.1.1.2

Reorder $2$ and $- x$ .

$- x + 2 = 1 \cdot 4$

Step 1.2.5.2.2

Simplify the right side.

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Step 1.2.5.2.2.1

Multiply $4$ by $1$ .

$- x + 2 = 4$

Step 1.2.5.3

Solve for $x$ .

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Step 1.2.5.3.1

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

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Step 1.2.5.3.1.1

Subtract $2$ from both sides of the equation.

$- x = 4 - 2$

Step 1.2.5.3.1.2

Subtract $2$ from $4$ .

$- x = 2$

Step 1.2.5.3.2

Divide each term in $- x = 2$ by $- 1$ and simplify.

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Step 1.2.5.3.2.1

Divide each term in $- x = 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{2}{- 1}$

Step 1.2.5.3.2.2

Simplify the left side.

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Step 1.2.5.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{2}{- 1}$

Step 1.2.5.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{2}{- 1}$

Step 1.2.5.3.2.3

Simplify the right side.

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Step 1.2.5.3.2.3.1

Divide $2$ by $- 1$ .

$x = - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0)$

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 $y$ .

$y = - \frac{1}{4} \cdot \sqrt{8 - 4 \cdot 0} + 1$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $- 4$ by $0$ .

$y = - \frac{1}{4} \cdot \sqrt{8 + 0} + 1$

Step 2.2.2

Add $8$ and $0$ .

$y = - \frac{1}{4} \cdot \sqrt{8} + 1$

Step 2.2.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.2.3.1

Factor $4$ out of $8$ .

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

Step 2.2.3.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.2.4

Pull terms out from under the radical.

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

Step 2.2.5

Cancel the common factor of $2$ .

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Step 2.2.5.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

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

Step 2.2.5.2

Factor $2$ out of $4$ .

$y = \frac{- 1}{2 (2)} \cdot (2 \sqrt{2}) + 1$

Step 2.2.5.3

Factor $2$ out of $2 \sqrt{2}$ .

$y = \frac{- 1}{2 (2)} \cdot (2 (\sqrt{2})) + 1$

Step 2.2.5.4

Cancel the common factor.

$y = \frac{- 1}{2 \cdot 2} \cdot (2 \sqrt{2}) + 1$

Step 2.2.5.5

Rewrite the expression.

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

Step 2.2.6

Combine $\frac{- 1}{2}$ and $\sqrt{2}$ .

$y = \frac{- \sqrt{2}}{2} + 1$

Step 2.2.7

Move the negative in front of the fraction.

$y = - \frac{\sqrt{2}}{2} + 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{\sqrt{2}}{2} + 1)$

Step 3

List the intersections.

x-intercept(s): $(- 2, 0)$

y-intercept(s): $(0, - \frac{\sqrt{2}}{2} + 1)$

Step 4