Algebra Examples

$y = - \frac{2}{3} x^{2} - 1$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x^{2}$ and $\frac{2}{3}$ .

$y = - \frac{x^{2} \cdot 2}{3} - 1$

Step 1.1.2

Move $2$ to the left of $x^{2}$ .

$y = - \frac{2 x^{2}}{3} - 1$

Step 1.2

Complete the square for $- \frac{2 x^{2}}{3} - 1$ .

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Step 1.2.1

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

$a = - \frac{2}{3}$

$b = 0$

$c = - 1$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{0}{2 (- \frac{2}{3})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{2}{3})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{2}{3})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{2}{3}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- \frac{3}{2})$

Step 1.2.3.2.3

Multiply $0 (- \frac{3}{2})$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $0$ .

$d = 0 (\frac{3}{2})$

Step 1.2.3.2.3.2

Multiply $0$ by $\frac{3}{2}$ .

$d = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = - 1 - \frac{0^{2}}{4 (- \frac{2}{3})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

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

$e = - 1 - \frac{0}{4 (- \frac{2}{3})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - 1 - \frac{0}{- 4 (\frac{2}{3})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{2}{3}$ .

$e = - 1 - \frac{0}{\frac{- 4 \cdot 2}{3}}$

Step 1.2.4.2.1.3

Multiply $- 4$ by $2$ .

$e = - 1 - \frac{0}{\frac{- 8}{3}}$

Step 1.2.4.2.1.4

Move the negative in front of the fraction.

$e = - 1 - \frac{0}{- \frac{8}{3}}$

Step 1.2.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = - 1 - (0 (- \frac{3}{8}))$

Step 1.2.4.2.1.6

Multiply $0 (- \frac{3}{8})$ .

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Step 1.2.4.2.1.6.1

Multiply $- 1$ by $0$ .

$e = - 1 - (0 (\frac{3}{8}))$

Step 1.2.4.2.1.6.2

Multiply $0$ by $\frac{3}{8}$ .

$e = - 1 - 0$

Step 1.2.4.2.1.7

Multiply $- 1$ by $0$ .

$e = - 1 + 0$

Step 1.2.4.2.2

Add $- 1$ and $0$ .

$e = - 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{2}{3} {(x + 0)}^{2} - 1$ .

$- \frac{2}{3} {(x + 0)}^{2} - 1$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{2}{3} \cdot {(x + 0)}^{2} - 1$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{2}{3}$

$h = 0$

$k = - 1$

Step 3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(0, - 1)$

Step 5

Find $p$ , the distance from the vertex to the focus.

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{2}{3})}$

Step 5.3

Simplify.

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Step 5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{2}{3})}$

Step 5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{2}{3})}$

Step 5.3.2

Combine $4$ and $\frac{2}{3}$ .

$- \frac{1}{\frac{4 \cdot 2}{3}}$

Step 5.3.3

Multiply $4$ by $2$ .

$- \frac{1}{\frac{8}{3}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{8}))$

Step 5.3.5

Multiply $\frac{3}{8}$ by $1$ .

$- \frac{3}{8}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

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

$(0, - \frac{11}{8})$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = 0$

Step 8