Graph -2y^2+x-4y+1=0

$- 2 y^{2} + x - 4 y + 1 = 0$

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

Add $2 y^{2}$ to both sides of the equation.

$x - 4 y + 1 = 2 y^{2}$

Add $4 y$ to both sides of the equation.

$x + 1 = 2 y^{2} + 4 y$

Subtract $1$ from both sides of the equation.

$x = 2 y^{2} + 4 y - 1$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $2 y^{2} + 4 y - 1$ .

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

$a = 2$

$b = 4$

$c = - 1$

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

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

Simplify the right side.

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

Factor $2$ out of $4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

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

Simplify the right side.

Simplify each term.

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$e = - 1 - \frac{4}{2}$

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

Factor $2$ out of $4$ .

$e = - 1 - \frac{2 \cdot 2}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

$e = - 1 - \frac{2 \cdot 2}{2 \cdot 1}$

Rewrite the expression.

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

Divide $2$ by $1$ .

$e = - 1 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = - 1 - 2$

Subtract $2$ from $- 1$ .

$e = - 3$

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

$2 {(y + 1)}^{2} - 3$

Set $x$ equal to the new right side.

$x = 2 {(y + 1)}^{2} - 3$

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

$a = 2$

$h = - 3$

$k = - 1$

Since the value of $a$ is positive, the parabola opens right.

Opens Right

Find the vertex $(h, k)$ .

$(- 3, - 1)$

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.

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

Multiply $4$ by $2$ .

$\frac{1}{8}$

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.

$(- \frac{23}{8}, - 1)$

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

$y = - 1$

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.

$x = - \frac{25}{8}$

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

Direction: Opens Right

Vertex: $(- 3, - 1)$

Focus: $(- \frac{23}{8}, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - \frac{25}{8}$

Direction: Opens Right

Vertex: $(- 3, - 1)$

Focus: $(- \frac{23}{8}, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - \frac{25}{8}$

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 $- 2$ into $f (x) = - \frac{2 + \sqrt{2 (x + 3)}}{2}$ . In this case, the point is $(- 2, - 1.70710678)$ .

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

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

Simplify the result.

Simplify the numerator.

Add $- 2$ and $3$ .

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

Multiply $2$ by $1$ .

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

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

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

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

$= - 1.70710678$

Substitute the $x$ value $- 2$ into $f (x) = - \frac{2 - \sqrt{2 (x + 3)}}{2}$ . In this case, the point is $(- 2, - 0.29289321)$ .

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

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

Simplify the result.

Simplify the numerator.

Add $- 2$ and $3$ .

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

Multiply $2$ by $1$ .

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

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

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

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

$= - 0.29289321$

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

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

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

Simplify the result.

Simplify the numerator.

Add $- 1$ and $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

Add $2$ and $2$ .

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

Simplify the expression.

Divide $4$ by $2$ .

$f (- 1) = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$f (- 1) = - 2$

The final answer is $- 2$ .

$y = - 2$

Convert $- 2$ to decimal.

$= - 2$

Substitute the $x$ value $- 1$ into $f (x) = - \frac{2 - \sqrt{2 (x + 3)}}{2}$ . In this case, the point is $(- 1, 0)$ .

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

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

Simplify the result.

Simplify the numerator.

Add $- 1$ and $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $2$ .

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

Simplify the expression.

Divide $0$ by $2$ .

$f (- 1) = - 0$

Multiply $- 1$ by $0$ .

$f (- 1) = 0$

The final answer is $0$ .

$y = 0$

Convert $0$ to decimal.

$= 0$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & - 1 \\ - 2 & - 1.71 \\ - 2 & - 0.29 \\ - 1 & - 2 \\ - 1 & 0 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 3, - 1)$

Focus: $(- \frac{23}{8}, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - \frac{25}{8}$

$\begin{array}{c} x & y \\ - 3 & - 1 \\ - 2 & - 1.71 \\ - 2 & - 0.29 \\ - 1 & - 2 \\ - 1 & 0 \end{array}$

Graph -2y^2+x-4y+1=0

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