Algebra Examples

$f (x) = - 2 \sqrt{x + 1} + 4$

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 $x$ .

$0 = - 2 \sqrt{x + 1} + 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 2 \sqrt{x + 1} + 4 = 0$ .

$- 2 \sqrt{x + 1} + 4 = 0$

Step 1.2.2

Subtract $4$ from both sides of the equation.

$- 2 \sqrt{x + 1} = - 4$

Step 1.2.3

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

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

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 $\sqrt{x + 1}$ as ${(x + 1)}^{\frac{1}{2}}$ .

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

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

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

Step 1.2.4.2.1.2

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

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

Step 1.2.4.2.1.3

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

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Step 1.2.4.2.1.3.1

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

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

Step 1.2.4.2.1.3.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.3.2.1

Cancel the common factor.

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

Step 1.2.4.2.1.3.2.2

Rewrite the expression.

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

Step 1.2.4.2.1.4

Simplify.

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

Step 1.2.4.2.1.5

Apply the distributive property.

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

Step 1.2.4.2.1.6

Multiply $4$ by $1$ .

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

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

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

$4 x + 4 = 16$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

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

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Step 1.2.5.1.1

Subtract $4$ from both sides of the equation.

$4 x = 16 - 4$

Step 1.2.5.1.2

Subtract $4$ from $16$ .

$4 x = 12$

Step 1.2.5.2

Divide each term in $4 x = 12$ by $4$ and simplify.

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Step 1.2.5.2.1

Divide each term in $4 x = 12$ by $4$ .

$\frac{4 x}{4} = \frac{12}{4}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{12}{4}$

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{4}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Divide $12$ by $4$ .

$x = 3$

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 $y$ .

$y = - 2 \sqrt{(0) + 1} + 4$

Step 2.2

Simplify $- 2 \sqrt{(0) + 1} + 4$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Add $0$ and $1$ .

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

Step 2.2.1.2

Any root of $1$ is $1$ .

$y = - 2 \cdot 1 + 4$

Step 2.2.1.3

Multiply $- 2$ by $1$ .

$y = - 2 + 4$

Step 2.2.2

Add $- 2$ and $4$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4