Graph 2y^2=x-4

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

Step 1

Rewrite the equation as $x - 4 = 2 y^{2}$ .

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

Step 2

Add $4$ to both sides of the equation.

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

Step 3

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

$a = 2$

$b = 0$

$c = 4$

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

$d = \frac{0}{2 \cdot 2}$

Simplify the right side.

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

Factor $2$ out of $0$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 2}$

Rewrite the expression.

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

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

Factor $2$ out of $0$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Rewrite the expression.

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

Divide $0$ by $1$ .

$d = 0$

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $2$ .

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

Divide $0$ by $8$ .

$e = 4 - 0$

Multiply $- 1$ by $0$ .

$e = 4 + 0$

Add $4$ and $0$ .

$e = 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(y + 0)}^{2} + 4$ .

$2 {(y + 0)}^{2} + 4$

Set $x$ equal to the new right side.

$x = 2 {(y + 0)}^{2} + 4$

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

$a = 2$

$h = 4$

$k = 0$

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

Opens Right

Find the vertex $(h, k)$ .

$(4, 0)$

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.

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

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.

$(\frac{33}{8}, 0)$

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.

$x = \frac{31}{8}$

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

Direction: Opens Right

Vertex: $(4, 0)$

Focus: $(\frac{33}{8}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{31}{8}$

Direction: Opens Right

Vertex: $(4, 0)$

Focus: $(\frac{33}{8}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{31}{8}$

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 $5$ into $f (x) = \frac{\sqrt{2 (x - 4)}}{2}$ . In this case, the point is $(5, 0.70710678)$ .

Replace the variable $x$ with $5$ in the expression.

$f (5) = \frac{\sqrt{2 ((5) - 4)}}{2}$

Simplify the result.

Simplify the numerator.

Subtract $4$ from $5$ .

$f (5) = \frac{\sqrt{2 \cdot 1}}{2}$

Multiply $2$ by $1$ .

$f (5) = \frac{\sqrt{2}}{2}$

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

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

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

$= 0.70710678$

Substitute the $x$ value $5$ into $f (x) = - \frac{\sqrt{2 (x - 4)}}{2}$ . In this case, the point is $(5, - 0.70710678)$ .

Replace the variable $x$ with $5$ in the expression.

$f (5) = - \frac{\sqrt{2 ((5) - 4)}}{2}$

Simplify the result.

Simplify the numerator.

Subtract $4$ from $5$ .

$f (5) = - \frac{\sqrt{2 \cdot 1}}{2}$

Multiply $2$ by $1$ .

$f (5) = - \frac{\sqrt{2}}{2}$

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

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

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

$= - 0.70710678$

Substitute the $x$ value $6$ into $f (x) = \frac{\sqrt{2 (x - 4)}}{2}$ . In this case, the point is $(6, 1)$ .

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{\sqrt{2 ((6) - 4)}}{2}$

Simplify the result.

Simplify the numerator.

Subtract $4$ from $6$ .

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

Multiply $2$ by $2$ .

$f (6) = \frac{\sqrt{4}}{2}$

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

$f (6) = \frac{\sqrt{2^{2}}}{2}$

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

$f (6) = \frac{2}{2}$

Divide $2$ by $2$ .

$f (6) = 1$

The final answer is $1$ .

$y = 1$

Convert $1$ to decimal.

$= 1$

Substitute the $x$ value $6$ into $f (x) = - \frac{\sqrt{2 (x - 4)}}{2}$ . In this case, the point is $(6, - 1)$ .

Replace the variable $x$ with $6$ in the expression.

$f (6) = - \frac{\sqrt{2 ((6) - 4)}}{2}$

Simplify the result.

Simplify the numerator.

Subtract $4$ from $6$ .

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

Multiply $2$ by $2$ .

$f (6) = - \frac{\sqrt{4}}{2}$

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

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

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

$f (6) = - \frac{2}{2}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2$ .

Cancel the common factor.

$f (6) = - \frac{2}{2}$

Rewrite the expression.

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

Multiply $- 1$ by $1$ .

$f (6) = - 1$

The final answer is $- 1$ .

$y = - 1$

Convert $- 1$ to decimal.

$= - 1$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 4 & 0 \\ 5 & 0.71 \\ 5 & - 0.71 \\ 6 & 1 \\ 6 & - 1 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(4, 0)$

Focus: $(\frac{33}{8}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{31}{8}$

$\begin{array}{c} x & y \\ 4 & 0 \\ 5 & 0.71 \\ 5 & - 0.71 \\ 6 & 1 \\ 6 & - 1 \end{array}$

Step 6

Graph 2y^2=x-4

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