Graph y^2=3x+7

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Trigonometry

Graph y^2=3x+7

$y^{2} = 3 x + 7$

Rewrite the equation as $3 x + 7 = y^{2}$ .

$3 x + 7 = y^{2}$

Subtract $7$ from both sides of the equation.

$3 x = y^{2} - 7$

Divide each term by $3$ and simplify.

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Divide each term in $3 x = y^{2} - 7$ by $3$ .

$\frac{3 x}{3} = \frac{y^{2}}{3} + \frac{- 7}{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 x}{3} = \frac{y^{2}}{3} + \frac{- 7}{3}$

Divide $x$ by $1$ .

$x = \frac{y^{2}}{3} + \frac{- 7}{3}$

Move the negative in front of the fraction.

$x = \frac{y^{2}}{3} - \frac{7}{3}$

Find the properties of the given parabola.

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Rewrite the equation in vertex form.

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Complete the square for $\frac{y^{2}}{3} - \frac{7}{3}$ .

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

$a = \frac{1}{3}, b = 0, c = - \frac{7}{3}$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 (\frac{1}{3})}$

Simplify the right side.

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

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Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{1}{3})}$

Cancel the common factors.

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Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{1}{3})}$

Rewrite the expression.

$d = \frac{0}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 \cdot 3$

Multiply $0$ by $3$ .

$d = 0$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Simplify each term.

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

$e = - \frac{7}{3} - \frac{0}{4 (\frac{1}{3})}$

Combine $4$ and $\frac{1}{3}$ .

$e = - \frac{7}{3} - \frac{0}{\frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{7}{3} - (0 (\frac{3}{4}))$

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

$e = - \frac{7}{3} - 0$

Multiply $- 1$ by $0$ .

$e = - \frac{7}{3} + 0$

Add $- \frac{7}{3}$ and $0$ .

$e = - \frac{7}{3}$

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

$\frac{1}{3} \cdot {(y + 0)}^{2} - \frac{7}{3}$

Set $x$ equal to the new right side.

$x = \frac{1}{3} \cdot {(y + 0)}^{2} - \frac{7}{3}$

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

$a = \frac{1}{3}$

$h = - \frac{7}{3}$

$k = 0$

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

Opens Right

Find the vertex $(h, k)$ .

$(- \frac{7}{3}, 0)$

Find $p$ , the distance from the vertex to the focus.

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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.

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

Simplify.

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Combine $4$ and $\frac{1}{3}$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

$\frac{3}{4}$

Find the focus.

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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.

$(- \frac{19}{12}, 0)$

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

$y = 0$

Find the directrix.

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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.

$x = - \frac{37}{12}$

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

Direction: Opens Right

Vertex: $(- \frac{7}{3}, 0)$

Focus: $(- \frac{19}{12}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{37}{12}$

Direction: Opens Right

Vertex: $(- \frac{7}{3}, 0)$

Focus: $(- \frac{19}{12}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{37}{12}$

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.

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Substitute the $x$ value $- 1.\overline{3}$ into $f (x) = \sqrt{3 x + 7}$ . In this case, the point is $(- 1.\overline{3}, \sqrt{3})$ .

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Replace the variable $x$ with $- 1.\overline{3}$ in the expression.

$f (- 1.\overline{3}) = \sqrt{3 (- 1.\overline{3}) + 7}$

Simplify the result.

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Multiply $3$ by $- 1.\overline{3}$ .

$f (- 1.\overline{3}) = \sqrt{- 4 + 7}$

Add $- 4$ and $7$ .

$f (- 1.\overline{3}) = \sqrt{3}$

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

$y = \sqrt{3}$

Substitute the $x$ value $- 1.\overline{3}$ into $f (x) = - \sqrt{3 x + 7}$ . In this case, the point is $(- 1.\overline{3}, - \sqrt{3})$ .

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Replace the variable $x$ with $- 1.\overline{3}$ in the expression.

$f (- 1.\overline{3}) = - \sqrt{3 (- 1.\overline{3}) + 7}$

Simplify the result.

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Multiply $3$ by $- 1.\overline{3}$ .

$f (- 1.\overline{3}) = - \sqrt{- 4 + 7}$

Add $- 4$ and $7$ .

$f (- 1.\overline{3}) = - \sqrt{3}$

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

$y = - \sqrt{3}$

Substitute the $x$ value $- 0.\overline{3}$ into $f (x) = \sqrt{3 x + 7}$ . In this case, the point is $(- 0.\overline{3}, \sqrt{6})$ .

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Replace the variable $x$ with $- 0.\overline{3}$ in the expression.

$f (- 0.\overline{3}) = \sqrt{3 (- 0.\overline{3}) + 7}$

Simplify the result.

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Multiply $3$ by $- 0.\overline{3}$ .

$f (- 0.\overline{3}) = \sqrt{- 1 + 7}$

Add $- 1$ and $7$ .

$f (- 0.\overline{3}) = \sqrt{6}$

The final answer is $\sqrt{6}$ .

$y = \sqrt{6}$

Substitute the $x$ value $- 0.\overline{3}$ into $f (x) = - \sqrt{3 x + 7}$ . In this case, the point is $(- 0.\overline{3}, - \sqrt{6})$ .

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Replace the variable $x$ with $- 0.\overline{3}$ in the expression.

$f (- 0.\overline{3}) = - \sqrt{3 (- 0.\overline{3}) + 7}$

Simplify the result.

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Multiply $3$ by $- 0.\overline{3}$ .

$f (- 0.\overline{3}) = - \sqrt{- 1 + 7}$

Add $- 1$ and $7$ .

$f (- 0.\overline{3}) = - \sqrt{6}$

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

$y = - \sqrt{6}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - \frac{7}{3} & 0 \\ - 1.333 & 1.73 \\ - 1.333 & - 1.73 \\ - 0.333 & 2.45 \\ - 0.333 & - 2.45 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- \frac{7}{3}, 0)$

Focus: $(- \frac{19}{12}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{37}{12}$

$\begin{array}{c} x & y \\ - \frac{7}{3} & 0 \\ - 1.333 & 1.73 \\ - 1.333 & - 1.73 \\ - 0.333 & 2.45 \\ - 0.333 & - 2.45 \end{array}$

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