Trigonometry Examples

$4 x^{2} + 4 x + y^{2} = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Complete the square for $4 x^{2} + 4 x$ .

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Step 1.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 4$

$b = 4$

$c = 0$

Step 1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{4}{2 \cdot 4}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of $4$ and $2$ .

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Step 1.1.3.2.1.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2 \cdot 4}$

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot 2}{2 (4)}$

Step 1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 2}{2 \cdot 4}$

Step 1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{2}{4}$

Step 1.1.3.2.2

Cancel the common factor of $2$ and $4$ .

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Step 1.1.3.2.2.1

Factor $2$ out of $2$ .

$d = \frac{2 (1)}{4}$

Step 1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.3.2.2.2.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 1}{2 \cdot 2}$

Step 1.1.3.2.2.2.3

Rewrite the expression.

$d = \frac{1}{2}$

Step 1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Cancel the common factor of $4^{2}$ and $4$ .

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Step 1.1.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot 4}$

Step 1.1.4.2.1.1.2

Cancel the common factors.

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Step 1.1.4.2.1.1.2.1

Factor $4$ out of $4 \cdot 4$ .

$e = 0 - \frac{4 \cdot 4}{4 (4)}$

Step 1.1.4.2.1.1.2.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot 4}$

Step 1.1.4.2.1.1.2.3

Rewrite the expression.

$e = 0 - \frac{4}{4}$

Step 1.1.4.2.1.2

Cancel the common factor of $4$ .

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Step 1.1.4.2.1.2.1

Cancel the common factor.

$e = 0 - \frac{4}{4}$

Step 1.1.4.2.1.2.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 1.1.4.2.1.3

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 1.1.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + \frac{1}{2})}^{2} - 1$ .

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

Step 1.2

Substitute $4 {(x + \frac{1}{2})}^{2} - 1$ for $4 x^{2} + 4 x$ in the equation $4 x^{2} + 4 x + y^{2} = 0$ .

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

Step 1.3

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

$4 {(x + \frac{1}{2})}^{2} + y^{2} = 0 + 1$

Step 1.4

Add $0$ and $1$ .

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

Step 1.5

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

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

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

$\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1$

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = 1$

$b = \frac{1}{2}$

$k = 0$

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

Step 4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(- \frac{1}{2}, 0)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(1)}^{2} - {(\frac{1}{2})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

One to any power is one.

$\sqrt{1 - {(\frac{1}{2})}^{2}}$

Step 5.3.2

Apply the product rule to $\frac{1}{2}$ .

$\sqrt{1 - \frac{1^{2}}{2^{2}}}$

Step 5.3.3

One to any power is one.

$\sqrt{1 - \frac{1}{2^{2}}}$

Step 5.3.4

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

$\sqrt{1 - \frac{1}{4}}$

Step 5.3.5

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{4}{4} - \frac{1}{4}}$

Step 5.3.6

Combine the numerators over the common denominator.

$\sqrt{\frac{4 - 1}{4}}$

Step 5.3.7

Subtract $1$ from $4$ .

$\sqrt{\frac{3}{4}}$

Step 5.3.8

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\frac{\sqrt{3}}{\sqrt{4}}$

Step 5.3.9

Simplify the denominator.

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Step 5.3.9.1

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

$\frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 5.3.9.2

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

$\frac{\sqrt{3}}{2}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Step 6.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(- \frac{1}{2}, 0 + 1)$

Step 6.3

Simplify.

$(- \frac{1}{2}, 1)$

Step 6.4

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Step 6.5

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(- \frac{1}{2}, 0 - (1))$

Step 6.6

Simplify.

$(- \frac{1}{2}, - 1)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(- \frac{1}{2}, 1)$

${Vertex}_{2}$ : $(- \frac{1}{2}, - 1)$

${Vertex}_{1}$ : $(- \frac{1}{2}, 1)$

${Vertex}_{2}$ : $(- \frac{1}{2}, - 1)$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Step 7.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(- \frac{1}{2}, 0 + \frac{\sqrt{3}}{2})$

Step 7.3

Simplify.

$(- \frac{1}{2}, \frac{\sqrt{3}}{2})$

Step 7.4

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.5

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(- \frac{1}{2}, 0 - (\frac{\sqrt{3}}{2}))$

Step 7.6

Simplify.

$(- \frac{1}{2}, - \frac{\sqrt{3}}{2})$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(- \frac{1}{2}, \frac{\sqrt{3}}{2})$

${Focus}_{2}$ : $(- \frac{1}{2}, - \frac{\sqrt{3}}{2})$

${Focus}_{1}$ : $(- \frac{1}{2}, \frac{\sqrt{3}}{2})$

${Focus}_{2}$ : $(- \frac{1}{2}, - \frac{\sqrt{3}}{2})$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Step 8.2

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(1)}^{2} - {(\frac{1}{2})}^{2}}}{1}$

Step 8.3

Simplify.

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Step 8.3.1

Divide $\sqrt{{(1)}^{2} - {(\frac{1}{2})}^{2}}$ by $1$ .

$\sqrt{{(1)}^{2} - {(\frac{1}{2})}^{2}}$

Step 8.3.2

One to any power is one.

$\sqrt{1 - {(\frac{1}{2})}^{2}}$

Step 8.3.3

Apply the product rule to $\frac{1}{2}$ .

$\sqrt{1 - \frac{1^{2}}{2^{2}}}$

Step 8.3.4

One to any power is one.

$\sqrt{1 - \frac{1}{2^{2}}}$

Step 8.3.5

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

$\sqrt{1 - \frac{1}{4}}$

Step 8.3.6

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{4}{4} - \frac{1}{4}}$

Step 8.3.7

Combine the numerators over the common denominator.

$\sqrt{\frac{4 - 1}{4}}$

Step 8.3.8

Subtract $1$ from $4$ .

$\sqrt{\frac{3}{4}}$

Step 8.3.9

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\frac{\sqrt{3}}{\sqrt{4}}$

Step 8.3.10

Simplify the denominator.

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Step 8.3.10.1

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

$\frac{\sqrt{3}}{\sqrt{2^{2}}}$

Step 8.3.10.2

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

$\frac{\sqrt{3}}{2}$

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(- \frac{1}{2}, 0)$

${Vertex}_{1}$ : $(- \frac{1}{2}, 1)$

${Vertex}_{2}$ : $(- \frac{1}{2}, - 1)$

${Focus}_{1}$ : $(- \frac{1}{2}, \frac{\sqrt{3}}{2})$

${Focus}_{2}$ : $(- \frac{1}{2}, - \frac{\sqrt{3}}{2})$

Eccentricity: $\frac{\sqrt{3}}{2}$

Step 10