Graph x=1/2y^2-3

$x = \frac{1}{2} y^{2} - 3$

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

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

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $\frac{y^{2}}{2} - 3$ .

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

$a = \frac{1}{2}$

$b = 0$

$c = - 3$

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

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

Simplify the right side.

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

Factor $2$ out of $0$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = 0 \cdot 2$

Multiply $0$ by $2$ .

$d = 0$

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

$e = - 3 - \frac{0^{2}}{4 (\frac{1}{2})}$

Simplify the right side.

Simplify each term.

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

$e = - 3 - \frac{0}{4 (\frac{1}{2})}$

Combine $4$ and $\frac{1}{2}$ .

$e = - 3 - \frac{0}{\frac{4}{2}}$

Divide $4$ by $2$ .

$e = - 3 - \frac{0}{2}$

Divide $0$ by $2$ .

$e = - 3 - 0$

Multiply $- 1$ by $0$ .

$e = - 3 + 0$

Add $- 3$ and $0$ .

$e = - 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(y + 0)}^{2} - 3$ .

$\frac{1}{2} {(y + 0)}^{2} - 3$

Set $x$ equal to the new right side.

$x = \frac{1}{2} \cdot {(y + 0)}^{2} - 3$

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

$a = \frac{1}{2}$

$h = - 3$

$k = 0$

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

Opens Right

Find the vertex $(h, k)$ .

$(- 3, 0)$

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.

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

Simplify.

Combine $4$ and $\frac{1}{2}$ .

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

Divide $4$ by $2$ .

$\frac{1}{2}$

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.

$(- \frac{5}{2}, 0)$

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.

$x = - \frac{7}{2}$

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

Direction: Opens Right

Vertex: $(- 3, 0)$

Focus: $(- \frac{5}{2}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{7}{2}$

Direction: Opens Right

Vertex: $(- 3, 0)$

Focus: $(- \frac{5}{2}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{7}{2}$

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 $f (x) = \sqrt{2 (x + 3)}$ . In this case, the point is $(- 2, 1.41421356)$ .

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

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

Simplify the result.

Add $- 2$ and $3$ .

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

Multiply $2$ by $1$ .

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

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

$y = \sqrt{2}$

Convert $\sqrt{2}$ to decimal.

$= 1.41421356$

Substitute the $x$ value $- 2$ into $f (x) = - \sqrt{2 (x + 3)}$ . In this case, the point is $(- 2, - 1.41421356)$ .

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

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

Simplify the result.

Add $- 2$ and $3$ .

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

Multiply $2$ by $1$ .

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

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

$y = - \sqrt{2}$

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

$= - 1.41421356$

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

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

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

Simplify the result.

Add $- 1$ and $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

$f (- 1) = 2$

The final answer is $2$ .

$y = 2$

Convert $2$ to decimal.

$= 2$

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

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

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

Simplify the result.

Add $- 1$ and $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

Multiply $- 1$ by $2$ .

$f (- 1) = - 2$

The final answer is $- 2$ .

$y = - 2$

Convert $- 2$ to decimal.

$= - 2$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & 1.41 \\ - 2 & - 1.41 \\ - 1 & 2 \\ - 1 & - 2 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 3, 0)$

Focus: $(- \frac{5}{2}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{7}{2}$

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & 1.41 \\ - 2 & - 1.41 \\ - 1 & 2 \\ - 1 & - 2 \end{array}$

Graph x=1/2y^2-3

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