Graph y^2=-9/8x

$y^{2} = - \frac{9}{8} x$

Rewrite the equation as $- \frac{9}{8} x = y^{2}$ .

$- \frac{9}{8} x = y^{2}$

Multiply both sides of the equation by $- \frac{8}{9}$ .

$- \frac{8}{9} (- \frac{9}{8} x) = - \frac{8}{9} y^{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{8}{9} (- \frac{9}{8} x)$ .

Combine $x$ and $\frac{9}{8}$ .

$- \frac{8}{9} (- \frac{x \cdot 9}{8}) = - \frac{8}{9} y^{2}$

Cancel the common factor of $8$ .

Move the leading negative in $- \frac{8}{9}$ into the numerator.

$\frac{- 8}{9} (- \frac{x \cdot 9}{8}) = - \frac{8}{9} y^{2}$

Move the leading negative in $- \frac{x \cdot 9}{8}$ into the numerator.

$\frac{- 8}{9} \cdot \frac{- x \cdot 9}{8} = - \frac{8}{9} y^{2}$

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{9} \cdot \frac{- x \cdot 9}{8} = - \frac{8}{9} y^{2}$

Cancel the common factor.

$\frac{8 \cdot - 1}{9} \cdot \frac{- x \cdot 9}{8} = - \frac{8}{9} y^{2}$

Rewrite the expression.

$\frac{- 1}{9} (- x \cdot 9) = - \frac{8}{9} y^{2}$

Cancel the common factor of $9$ .

Factor $9$ out of $- x \cdot 9$ .

$\frac{- 1}{9} (9 \cdot (- x)) = - \frac{8}{9} y^{2}$

Cancel the common factor.

$\frac{- 1}{9} (9 \cdot (- x)) = - \frac{8}{9} y^{2}$

Rewrite the expression.

$- - x = - \frac{8}{9} y^{2}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{8}{9} y^{2}$

Multiply $x$ by $1$ .

$x = - \frac{8}{9} y^{2}$

Simplify the right side.

Simplify $- \frac{8}{9} y^{2}$ .

Combine $y^{2}$ and $\frac{8}{9}$ .

$x = - \frac{y^{2} \cdot 8}{9}$

Move $8$ to the left of $y^{2}$ .

$x = - \frac{8 y^{2}}{9}$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- \frac{8 y^{2}}{9}$ .

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

$a = - \frac{8}{9}$

$b = 0$

$c = 0$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{0}{2 (- \frac{8}{9})}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{8}{9})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{8}{9})}$

Rewrite the expression.

$d = \frac{0}{- \frac{8}{9}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- \frac{9}{8})$

Multiply $0 (- \frac{9}{8})$ .

Multiply $- 1$ by $0$ .

$d = 0 (\frac{9}{8})$

Multiply $0$ by $\frac{9}{8}$ .

$d = 0$

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

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

$e = 0 - \frac{0^{2}}{4 (- \frac{8}{9})}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{0}{4 (- \frac{8}{9})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 0 - \frac{0}{- 4 (\frac{8}{9})}$

Combine $- 4$ and $\frac{8}{9}$ .

$e = 0 - \frac{0}{\frac{- 4 \cdot 8}{9}}$

Multiply $- 4$ by $8$ .

$e = 0 - \frac{0}{\frac{- 32}{9}}$

Move the negative in front of the fraction.

$e = 0 - \frac{0}{- \frac{32}{9}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (- \frac{9}{32}))$

Multiply $0 (- \frac{9}{32})$ .

Multiply $- 1$ by $0$ .

$e = 0 - (0 (\frac{9}{32}))$

Multiply $0$ by $\frac{9}{32}$ .

$e = 0 - 0$

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Add $0$ and $0$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{8}{9} y^{2}$ .

$- \frac{8}{9} y^{2}$

Set $x$ equal to the new right side.

$x = - \frac{8}{9} y^{2}$

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

$a = - \frac{8}{9}$

$h = 0$

$k = 0$

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

Opens Left

Find the vertex $(h, k)$ .

$(0, 0)$

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

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

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{8}{9})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{8}{9})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{8}{9})}$

Combine $4$ and $\frac{8}{9}$ .

$- \frac{1}{\frac{4 \cdot 8}{9}}$

Multiply $4$ by $8$ .

$- \frac{1}{\frac{32}{9}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{9}{32}))$

Multiply $\frac{9}{32}$ by $1$ .

$- \frac{9}{32}$

Find the focus.

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

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

$(- \frac{9}{32}, 0)$

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

$y = 0$

Find the directrix.

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

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

$x = \frac{9}{32}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{9}{32}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{9}{32}$

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{9}{32}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{9}{32}$

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

Substitute the $x$ value $- 2$ into $f (x) = \frac{3 \sqrt{- \frac{x}{2}}}{2}$ . In this case, the point is $(- 2, 1.5)$ .

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \frac{3 \sqrt{- \frac{- 2}{2}}}{2}$

Simplify the result.

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

Factor $2$ out of $- 2$ .

$f (- 2) = \frac{3 \sqrt{- \frac{2 \cdot - 1}{2}}}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (- 2) = \frac{3 \sqrt{- \frac{2 \cdot - 1}{2 (1)}}}{2}$

Cancel the common factor.

$f (- 2) = \frac{3 \sqrt{- \frac{2 \cdot - 1}{2 \cdot 1}}}{2}$

Rewrite the expression.

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

Divide $- 1$ by $1$ .

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

Any root of $1$ is $1$ .

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

Multiply $3$ by $1$ .

$f (- 2) = \frac{3}{2}$

The final answer is $\frac{3}{2}$ .

$y = \frac{3}{2}$

Convert $\frac{3}{2}$ to decimal.

$= 1.5$

Substitute the $x$ value $- 2$ into $f (x) = - \frac{3 \sqrt{- \frac{x}{2}}}{2}$ . In this case, the point is $(- 2, - 1.5)$ .

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = - \frac{3 \sqrt{- \frac{- 2}{2}}}{2}$

Simplify the result.

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

Factor $2$ out of $- 2$ .

$f (- 2) = - \frac{3 \sqrt{- \frac{2 \cdot - 1}{2}}}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (- 2) = - \frac{3 \sqrt{- \frac{2 \cdot - 1}{2 (1)}}}{2}$

Cancel the common factor.

$f (- 2) = - \frac{3 \sqrt{- \frac{2 \cdot - 1}{2 \cdot 1}}}{2}$

Rewrite the expression.

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

Divide $- 1$ by $1$ .

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

Any root of $1$ is $1$ .

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

Multiply $3$ by $1$ .

$f (- 2) = - \frac{3}{2}$

The final answer is $- \frac{3}{2}$ .

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

Convert $- \frac{3}{2}$ to decimal.

$= - 1.5$

Substitute the $x$ value $- 1$ into $f (x) = \frac{3 \sqrt{- \frac{x}{2}}}{2}$ . In this case, the point is $(- 1, 1.06066017)$ .

Replace the variable $x$ with $- 1$ in the expression.

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

Simplify the result.

Simplify the numerator.

Move the negative in front of the fraction.

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

Combine exponents.

Factor out negative.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{1}{2}$ by $1$ .

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

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Any root of $1$ is $1$ .

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

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (- 1) = \frac{3 (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}{2}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 1) = \frac{3 (\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}})}{2}$

Add $1$ and $1$ .

$f (- 1) = \frac{3 (\frac{\sqrt{2}}{{\sqrt{2}}^{2}})}{2}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

$f (- 1) = \frac{3 (\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}{2}$

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

$f (- 1) = \frac{3 (\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}{2}$

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

$f (- 1) = \frac{3 (\frac{\sqrt{2}}{2^{\frac{2}{2}}})}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- 1) = \frac{3 (\frac{\sqrt{2}}{2^{\frac{2}{2}}})}{2}$

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Multiply the numerator by the reciprocal of the denominator.

$f (- 1) = \frac{3 \sqrt{2}}{2} \cdot \frac{1}{2}$

Multiply $\frac{3 \sqrt{2}}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$f (- 1) = \frac{3 \sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

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

The final answer is $\frac{3 \sqrt{2}}{4}$ .

$y = \frac{3 \sqrt{2}}{4}$

Convert $\frac{3 \sqrt{2}}{4}$ to decimal.

$= 1.06066017$

Substitute the $x$ value $- 1$ into $f (x) = - \frac{3 \sqrt{- \frac{x}{2}}}{2}$ . In this case, the point is $(- 1, - 1.06066017)$ .

Replace the variable $x$ with $- 1$ in the expression.

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

Simplify the result.

Simplify the numerator.

Move the negative in front of the fraction.

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

Combine exponents.

Factor out negative.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{1}{2}$ by $1$ .

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

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Any root of $1$ is $1$ .

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

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (- 1) = - \frac{3 (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}{2}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 1) = - \frac{3 (\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}})}{2}$

Add $1$ and $1$ .

$f (- 1) = - \frac{3 (\frac{\sqrt{2}}{{\sqrt{2}}^{2}})}{2}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

$f (- 1) = - \frac{3 (\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}{2}$

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

$f (- 1) = - \frac{3 (\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}{2}$

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

$f (- 1) = - \frac{3 (\frac{\sqrt{2}}{2^{\frac{2}{2}}})}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- 1) = - \frac{3 (\frac{\sqrt{2}}{2^{\frac{2}{2}}})}{2}$

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Multiply the numerator by the reciprocal of the denominator.

$f (- 1) = - (\frac{3 \sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $\frac{3 \sqrt{2}}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$f (- 1) = - \frac{3 \sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

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

The final answer is $- \frac{3 \sqrt{2}}{4}$ .

$y = - \frac{3 \sqrt{2}}{4}$

Convert $- \frac{3 \sqrt{2}}{4}$ to decimal.

$= - 1.06066017$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 1.5 \\ - 2 & - 1.5 \\ - 1 & 1.06 \\ - 1 & - 1.06 \\ 0 & 0 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{9}{32}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{9}{32}$

$\begin{array}{c} x & y \\ - 2 & 1.5 \\ - 2 & - 1.5 \\ - 1 & 1.06 \\ - 1 & - 1.06 \\ 0 & 0 \end{array}$

Graph y^2=-9/8x

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