Graph 2y^2+x+20y+51=0

$2 y^{2} + x + 20 y + 51 = 0$

Move all terms not containing $x$ to the right side of the equation.

Subtract $2 y^{2}$ from both sides of the equation.

$x + 20 y + 51 = - 2 y^{2}$

Subtract $20 y$ from both sides of the equation.

$x + 51 = - 2 y^{2} - 20 y$

Subtract $51$ from both sides of the equation.

$x = - 2 y^{2} - 20 y - 51$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 2 y^{2} - 20 y - 51$ .

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

$a = - 2$

$b = - 20$

$c = - 51$

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

$d = \frac{- 20}{2 \cdot - 2}$

Simplify the right side.

Cancel the common factor of $- 20$ and $2$ .

Factor $2$ out of $- 20$ .

$d = \frac{2 \cdot - 10}{2 \cdot - 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 10}{2 (- 2)}$

Cancel the common factor.

$d = \frac{2 \cdot - 10}{2 \cdot - 2}$

Rewrite the expression.

$d = \frac{- 10}{- 2}$

Cancel the common factor of $- 10$ and $- 2$ .

Factor $- 2$ out of $- 10$ .

$d = \frac{- 2 (5)}{- 2}$

Cancel the common factors.

Factor $- 2$ out of $- 2$ .

$d = \frac{- 2 \cdot 5}{- 2 \cdot 1}$

Cancel the common factor.

$d = \frac{- 2 \cdot 5}{- 2 \cdot 1}$

Rewrite the expression.

$d = \frac{5}{1}$

Divide $5$ by $1$ .

$d = 5$

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

$e = - 51 - \frac{{(- 20)}^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raise $- 20$ to the power of $2$ .

$e = - 51 - \frac{400}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = - 51 - \frac{400}{- 8}$

Divide $400$ by $- 8$ .

$e = - 51 - - 50$

Multiply $- 1$ by $- 50$ .

$e = - 51 + 50$

Add $- 51$ and $50$ .

$e = - 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(y + 5)}^{2} - 1$ .

$- 2 {(y + 5)}^{2} - 1$

Set $x$ equal to the new right side.

$x = - 2 {(y + 5)}^{2} - 1$

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

$a = - 2$

$h = - 1$

$k = - 5$

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

Opens Left

Find the vertex $(h, k)$ .

$(- 1, - 5)$

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.

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

Simplify.

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(- \frac{9}{8}, - 5)$

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

$y = - 5$

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.

$x = - \frac{7}{8}$

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

Direction: Opens Left

Vertex: $(- 1, - 5)$

Focus: $(- \frac{9}{8}, - 5)$

Axis of Symmetry: $y = - 5$

Directrix: $x = - \frac{7}{8}$

Direction: Opens Left

Vertex: $(- 1, - 5)$

Focus: $(- \frac{9}{8}, - 5)$

Axis of Symmetry: $y = - 5$

Directrix: $x = - \frac{7}{8}$

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 $- 3$ into $f (x) = - \frac{10 - \sqrt{2 (- x - 1)}}{2}$ . In this case, the point is $(- 3, - 4)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - \frac{10 - \sqrt{2 (- (- 3) - 1)}}{2}$

Simplify the result.

Simplify the numerator.

Multiply $- 1$ by $- 3$ .

$f (- 3) = - \frac{10 - \sqrt{2 (3 - 1)}}{2}$

Subtract $1$ from $3$ .

$f (- 3) = - \frac{10 - \sqrt{2 \cdot 2}}{2}$

Multiply $2$ by $2$ .

$f (- 3) = - \frac{10 - \sqrt{4}}{2}$

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

$f (- 3) = - \frac{10 - \sqrt{2^{2}}}{2}$

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

$f (- 3) = - \frac{10 - 1 \cdot 2}{2}$

Multiply $- 1$ by $2$ .

$f (- 3) = - \frac{10 - 2}{2}$

Subtract $2$ from $10$ .

$f (- 3) = - \frac{8}{2}$

Simplify the expression.

Divide $8$ by $2$ .

$f (- 3) = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$f (- 3) = - 4$

The final answer is $- 4$ .

$y = - 4$

Convert $- 4$ to decimal.

$= - 4$

Substitute the $x$ value $- 3$ into $f (x) = - \frac{10 + \sqrt{2 (- x - 1)}}{2}$ . In this case, the point is $(- 3, - 6)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - \frac{10 + \sqrt{2 (- (- 3) - 1)}}{2}$

Simplify the result.

Simplify the numerator.

Multiply $- 1$ by $- 3$ .

$f (- 3) = - \frac{10 + \sqrt{2 (3 - 1)}}{2}$

Subtract $1$ from $3$ .

$f (- 3) = - \frac{10 + \sqrt{2 \cdot 2}}{2}$

Multiply $2$ by $2$ .

$f (- 3) = - \frac{10 + \sqrt{4}}{2}$

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

$f (- 3) = - \frac{10 + \sqrt{2^{2}}}{2}$

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

$f (- 3) = - \frac{10 + 2}{2}$

Add $10$ and $2$ .

$f (- 3) = - \frac{12}{2}$

Simplify the expression.

Divide $12$ by $2$ .

$f (- 3) = - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$f (- 3) = - 6$

The final answer is $- 6$ .

$y = - 6$

Convert $- 6$ to decimal.

$= - 6$

Substitute the $x$ value $- 2$ into $f (x) = - \frac{10 - \sqrt{2 (- x - 1)}}{2}$ . In this case, the point is $(- 2, - 4.29289321)$ .

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

$f (- 2) = - \frac{10 - \sqrt{2 (- (- 2) - 1)}}{2}$

Simplify the result.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$f (- 2) = - \frac{10 - \sqrt{2 (2 - 1)}}{2}$

Subtract $1$ from $2$ .

$f (- 2) = - \frac{10 - \sqrt{2 \cdot 1}}{2}$

Multiply $2$ by $1$ .

$f (- 2) = - \frac{10 - \sqrt{2}}{2}$

The final answer is $- \frac{10 - \sqrt{2}}{2}$ .

$y = - \frac{10 - \sqrt{2}}{2}$

Convert $- \frac{10 - \sqrt{2}}{2}$ to decimal.

$= - 4.29289321$

Substitute the $x$ value $- 2$ into $f (x) = - \frac{10 + \sqrt{2 (- x - 1)}}{2}$ . In this case, the point is $(- 2, - 5.70710678)$ .

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

$f (- 2) = - \frac{10 + \sqrt{2 (- (- 2) - 1)}}{2}$

Simplify the result.

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$f (- 2) = - \frac{10 + \sqrt{2 (2 - 1)}}{2}$

Subtract $1$ from $2$ .

$f (- 2) = - \frac{10 + \sqrt{2 \cdot 1}}{2}$

Multiply $2$ by $1$ .

$f (- 2) = - \frac{10 + \sqrt{2}}{2}$

The final answer is $- \frac{10 + \sqrt{2}}{2}$ .

$y = - \frac{10 + \sqrt{2}}{2}$

Convert $- \frac{10 + \sqrt{2}}{2}$ to decimal.

$= - 5.70710678$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & - 4 \\ - 3 & - 6 \\ - 2 & - 4.29 \\ - 2 & - 5.71 \\ - 1 & - 5 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(- 1, - 5)$

Focus: $(- \frac{9}{8}, - 5)$

Axis of Symmetry: $y = - 5$

Directrix: $x = - \frac{7}{8}$

$\begin{array}{c} x & y \\ - 3 & - 4 \\ - 3 & - 6 \\ - 2 & - 4.29 \\ - 2 & - 5.71 \\ - 1 & - 5 \end{array}$

Graph 2y^2+x+20y+51=0

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