Graph y^2=16/3x

$y^{2} = \frac{16}{3} x$

Rewrite the equation as $\frac{16}{3} x = y^{2}$ .

$\frac{16}{3} x = y^{2}$

Multiply both sides of the equation by $\frac{3}{16}$ .

$\frac{3}{16} (\frac{16}{3} x) = \frac{3}{16} y^{2}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{16} (\frac{16}{3} x)$ .

Combine $\frac{16}{3}$ and $x$ .

$\frac{3}{16} \cdot \frac{16 x}{3} = \frac{3}{16} y^{2}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{16} \cdot \frac{16 x}{3} = \frac{3}{16} y^{2}$

Rewrite the expression.

$\frac{1}{16} (16 x) = \frac{3}{16} y^{2}$

Cancel the common factor of $16$ .

Factor $16$ out of $16 x$ .

$\frac{1}{16} (16 (x)) = \frac{3}{16} y^{2}$

Cancel the common factor.

$\frac{1}{16} (16 x) = \frac{3}{16} y^{2}$

Rewrite the expression.

$x = \frac{3}{16} y^{2}$

Simplify the right side.

Combine $\frac{3}{16}$ and $y^{2}$ .

$x = \frac{3 y^{2}}{16}$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $\frac{3 y^{2}}{16}$ .

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

$a = \frac{3}{16}$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 (\frac{3}{16})}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{3}{16})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{3}{16})}$

Rewrite the expression.

$d = \frac{0}{\frac{3}{16}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (\frac{16}{3})$

Multiply $0$ by $\frac{16}{3}$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 (\frac{3}{16})}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{0}{4 (\frac{3}{16})}$

Combine $4$ and $\frac{3}{16}$ .

$e = 0 - \frac{0}{\frac{4 \cdot 3}{16}}$

Multiply $4$ by $3$ .

$e = 0 - \frac{0}{\frac{12}{16}}$

Cancel the common factor of $12$ and $16$ .

Factor $4$ out of $12$ .

$e = 0 - \frac{0}{\frac{4 (3)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $0$ by $\frac{4}{3}$ .

$e = 0 - 0$

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 $\frac{3}{16} y^{2}$ .

$\frac{3}{16} y^{2}$

Set $x$ equal to the new right side.

$x = \frac{3}{16} y^{2}$

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

$a = \frac{3}{16}$

$h = 0$

$k = 0$

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

Opens Right

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.

$\frac{1}{4 \cdot \frac{3}{16}}$

Simplify.

Combine $4$ and $\frac{3}{16}$ .

$\frac{1}{\frac{4 \cdot 3}{16}}$

Reduce the expression by cancelling the common factors.

Multiply $4$ by $3$ .

$\frac{1}{\frac{12}{16}}$

Cancel the common factor of $12$ and $16$ .

Factor $4$ out of $12$ .

$\frac{1}{\frac{4 (3)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$\frac{1}{\frac{4 \cdot 3}{4 \cdot 4}}$

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 3}{4 \cdot 4}}$

Rewrite the expression.

$\frac{1}{\frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{4}{3})$

Multiply $\frac{4}{3}$ by $1$ .

$\frac{4}{3}$

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.

$(\frac{4}{3}, 0)$

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.

$x = - \frac{4}{3}$

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

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{4}{3}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{4}{3}$

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{4}{3}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{4}{3}$

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 $1$ into $f (x) = \frac{4 \sqrt{3 x}}{3}$ . In this case, the point is $(1, 2.30940107)$ .

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

$f (1) = \frac{4 \sqrt{3 (1)}}{3}$

Simplify the result.

Multiply $3$ by $1$ .

$f (1) = \frac{4 \sqrt{3}}{3}$

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

$y = \frac{4 \sqrt{3}}{3}$

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

$= 2.30940107$

Substitute the $x$ value $1$ into $f (x) = - \frac{4 \sqrt{3 x}}{3}$ . In this case, the point is $(1, - 2.30940107)$ .

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

$f (1) = - \frac{4 \sqrt{3 (1)}}{3}$

Simplify the result.

Multiply $3$ by $1$ .

$f (1) = - \frac{4 \sqrt{3}}{3}$

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

$y = - \frac{4 \sqrt{3}}{3}$

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

$= - 2.30940107$

Substitute the $x$ value $2$ into $f (x) = \frac{4 \sqrt{3 x}}{3}$ . In this case, the point is $(2, 3.26598632)$ .

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

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

Simplify the result.

Multiply $3$ by $2$ .

$f (2) = \frac{4 \sqrt{6}}{3}$

The final answer is $\frac{4 \sqrt{6}}{3}$ .

$y = \frac{4 \sqrt{6}}{3}$

Convert $\frac{4 \sqrt{6}}{3}$ to decimal.

$= 3.26598632$

Substitute the $x$ value $2$ into $f (x) = - \frac{4 \sqrt{3 x}}{3}$ . In this case, the point is $(2, - 3.26598632)$ .

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

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

Simplify the result.

Multiply $3$ by $2$ .

$f (2) = - \frac{4 \sqrt{6}}{3}$

The final answer is $- \frac{4 \sqrt{6}}{3}$ .

$y = - \frac{4 \sqrt{6}}{3}$

Convert $- \frac{4 \sqrt{6}}{3}$ to decimal.

$= - 3.26598632$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 2.31 \\ 1 & - 2.31 \\ 2 & 3.27 \\ 2 & - 3.27 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{4}{3}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{4}{3}$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 2.31 \\ 1 & - 2.31 \\ 2 & 3.27 \\ 2 & - 3.27 \end{array}$

Graph y^2=16/3x

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