Graph y^2=-24x

$y^{2} = - 24 x$

Step 1

Rewrite the equation as $- 24 x = y^{2}$ .

$- 24 x = y^{2}$

Step 2

Divide each term in $- 24 x = y^{2}$ by $- 24$ and simplify.

Divide each term in $- 24 x = y^{2}$ by $- 24$ .

$\frac{- 24 x}{- 24} = \frac{y^{2}}{- 24}$

Simplify the left side.

Cancel the common factor of $- 24$ .

Cancel the common factor.

$\frac{- 24 x}{- 24} = \frac{y^{2}}{- 24}$

Divide $x$ by $1$ .

$x = \frac{y^{2}}{- 24}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{24}$

Step 3

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- \frac{y^{2}}{24}$ .

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

$a = - \frac{1}{24}$

$b = 0$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 (- \frac{1}{24})}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{24})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{24})}$

Rewrite the expression.

$d = \frac{0}{- \frac{1}{24}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 24)$

Multiply $0 (- 1 \cdot 24)$ .

Multiply $- 1$ by $24$ .

$d = 0 \cdot - 24$

Multiply $0$ by $- 24$ .

$d = 0$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{0^{2}}{4 (- \frac{1}{24})}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{0}{4 (- \frac{1}{24})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 0 - \frac{0}{- 4 (\frac{1}{24})}$

Combine $- 4$ and $\frac{1}{24}$ .

$e = 0 - \frac{0}{\frac{- 4}{24}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $24$ .

Factor $4$ out of $- 4$ .

$e = 0 - \frac{0}{\frac{4 (- 1)}{24}}$

Cancel the common factors.

Factor $4$ out of $24$ .

$e = 0 - \frac{0}{\frac{4 \cdot - 1}{4 \cdot 6}}$

Cancel the common factor.

$e = 0 - \frac{0}{\frac{4 \cdot - 1}{4 \cdot 6}}$

Rewrite the expression.

$e = 0 - \frac{0}{\frac{- 1}{6}}$

Move the negative in front of the fraction.

$e = 0 - \frac{0}{- \frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (- 1 \cdot 6))$

Multiply $- (0 (- 1 \cdot 6))$ .

Multiply $- 1$ by $6$ .

$e = 0 - (0 \cdot - 6)$

Multiply $0$ by $- 6$ .

$e = 0 - 0$

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 $- \frac{1}{24} y^{2}$ .

$- \frac{1}{24} y^{2}$

Set $x$ equal to the new right side.

$x = - \frac{1}{24} y^{2}$

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

$a = - \frac{1}{24}$

$h = 0$

$k = 0$

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.

$\frac{1}{4 \cdot (- \frac{1}{24})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{24})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{24})}$

Combine $4$ and $\frac{1}{24}$ .

$- \frac{1}{\frac{4}{24}}$

Cancel the common factor of $4$ and $24$ .

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{24}}$

Cancel the common factors.

Factor $4$ out of $24$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Rewrite the expression.

$- \frac{1}{\frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 6)$

Multiply $- (1 \cdot 6)$ .

Multiply $6$ by $1$ .

$- 1 \cdot 6$

Multiply $- 1$ by $6$ .

$- 6$

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.

$(- 6, 0)$

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.

$x = 6$

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

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- 6, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = 6$

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- 6, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = 6$

Step 4

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 $f (x) = 2 \sqrt{- 6 x}$ . In this case, the point is $(- 2, 6.92820323)$ .

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

$f (- 2) = 2 \sqrt{- 6 \cdot - 2}$

Simplify the result.

Multiply $- 6$ by $- 2$ .

$f (- 2) = 2 \sqrt{12}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

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

$y = 4 \sqrt{3}$

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

$= 6.92820323$

Substitute the $x$ value $- 2$ into $f (x) = - 2 \sqrt{- 6 x}$ . In this case, the point is $(- 2, - 6.92820323)$ .

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

$f (- 2) = - 2 \sqrt{- 6 \cdot - 2}$

Simplify the result.

Multiply $- 6$ by $- 2$ .

$f (- 2) = - 2 \sqrt{12}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 2$ .

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

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

$y = - 4 \sqrt{3}$

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

$= - 6.92820323$

Substitute the $x$ value $- 1$ into $f (x) = 2 \sqrt{- 6 x}$ . In this case, the point is $(- 1, 4.89897948)$ .

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

$f (- 1) = 2 \sqrt{- 6 \cdot - 1}$

Simplify the result.

Multiply $- 6$ by $- 1$ .

$f (- 1) = 2 \sqrt{6}$

The final answer is $2 \sqrt{6}$ .

$y = 2 \sqrt{6}$

Convert $2 \sqrt{6}$ to decimal.

$= 4.89897948$

Substitute the $x$ value $- 1$ into $f (x) = - 2 \sqrt{- 6 x}$ . In this case, the point is $(- 1, - 4.89897948)$ .

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

$f (- 1) = - 2 \sqrt{- 6 \cdot - 1}$

Simplify the result.

Multiply $- 6$ by $- 1$ .

$f (- 1) = - 2 \sqrt{6}$

The final answer is $- 2 \sqrt{6}$ .

$y = - 2 \sqrt{6}$

Convert $- 2 \sqrt{6}$ to decimal.

$= - 4.89897948$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 6.93 \\ - 2 & - 6.93 \\ - 1 & 4.9 \\ - 1 & - 4.9 \\ 0 & 0 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- 6, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = 6$

$\begin{array}{c} x & y \\ - 2 & 6.93 \\ - 2 & - 6.93 \\ - 1 & 4.9 \\ - 1 & - 4.9 \\ 0 & 0 \end{array}$

Step 6

Graph y^2=-24x

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