Precalculus Examples

$y = \sqrt{9 - x^{2}}$

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 $x$ .

$0 = \sqrt{9 - x^{2}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\sqrt{9 - x^{2}} = 0$ .

$\sqrt{9 - x^{2}} = 0$

Step 1.2.2

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

${\sqrt{9 - x^{2}}}^{2} = 0^{2}$

Step 1.2.3

Simplify each side of the equation.

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Step 1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x^{2}}$ as ${(9 - x^{2})}^{\frac{1}{2}}$ .

${({(9 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify ${({(9 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.1

Multiply the exponents in ${({(9 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.1.1

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

${(9 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1.2.1

Cancel the common factor.

${(9 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.3.2.1.1.2.2

Rewrite the expression.

${(9 - x^{2})}^{1} = 0^{2}$

Step 1.2.3.2.1.2

Simplify.

$9 - x^{2} = 0^{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Raising $0$ to any positive power yields $0$ .

$9 - x^{2} = 0$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Subtract $9$ from both sides of the equation.

$- x^{2} = - 9$

Step 1.2.4.2

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

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Step 1.2.4.2.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

Step 1.2.4.2.2.2

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

$x^{2} = \frac{- 9}{- 1}$

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

Step 1.2.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Step 1.2.4.4

Simplify $\pm \sqrt{9}$ .

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Step 1.2.4.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 1.2.4.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 1.2.4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 1.2.4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 1.2.4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = \sqrt{9 - {(0)}^{2}}$

Step 2.2

Simplify $\sqrt{9 - {(0)}^{2}}$ .

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Step 2.2.1

Raising $0$ to any positive power yields $0$ .

$y = \sqrt{9 - 0}$

Step 2.2.2

Multiply $- 1$ by $0$ .

$y = \sqrt{9 + 0}$

Step 2.2.3

Add $9$ and $0$ .

$y = \sqrt{9}$

Step 2.2.4

Rewrite $9$ as $3^{2}$ .

$y = \sqrt{3^{2}}$

Step 2.2.5

Pull terms out from under the radical, assuming positive real numbers.

$y = 3$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4