Graph 4x-2y=y^2-23

$4 x - 2 y = y^{2} - 23$

Step 1

Add $2 y$ to both sides of the equation.

$4 x = y^{2} - 23 + 2 y$

Step 2

Divide each term in $4 x = y^{2} - 23 + 2 y$ by $4$ and simplify.

Divide each term in $4 x = y^{2} - 23 + 2 y$ by $4$ .

$\frac{4 x}{4} = \frac{y^{2}}{4} + \frac{- 23}{4} + \frac{2 y}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{y^{2}}{4} + \frac{- 23}{4} + \frac{2 y}{4}$

Divide $x$ by $1$ .

$x = \frac{y^{2}}{4} + \frac{- 23}{4} + \frac{2 y}{4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = \frac{y^{2}}{4} - \frac{23}{4} + \frac{2 y}{4}$

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

Factor $2$ out of $2 y$ .

$x = \frac{y^{2}}{4} - \frac{23}{4} + \frac{2 (y)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = \frac{y^{2}}{4} - \frac{23}{4} + \frac{2 y}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{y^{2}}{4} - \frac{23}{4} + \frac{2 y}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{y^{2}}{4} - \frac{23}{4} + \frac{y}{2}$

Step 3

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Move $- \frac{23}{4}$ .

$x = \frac{y^{2}}{4} + \frac{y}{2} - \frac{23}{4}$

Complete the square for $\frac{y^{2}}{4} + \frac{y}{2} - \frac{23}{4}$ .

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

$a = \frac{1}{4}$

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

$c = - \frac{23}{4}$

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

$d = \frac{\frac{1}{2}}{2 (\frac{1}{4})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Factor $2$ out of $1 \cdot 2$ .

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

Cancel the common factor.

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

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

$e = - \frac{23}{4} - \frac{{(\frac{1}{2})}^{2}}{4 (\frac{1}{4})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{1}{2}$ .

$e = - \frac{23}{4} - \frac{\frac{1^{2}}{2^{2}}}{4 (\frac{1}{4})}$

One to any power is one.

$e = - \frac{23}{4} - \frac{\frac{1}{2^{2}}}{4 (\frac{1}{4})}$

Raise $2$ to the power of $2$ .

$e = - \frac{23}{4} - \frac{\frac{1}{4}}{4 (\frac{1}{4})}$

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

$e = - \frac{23}{4} - \frac{\frac{1}{4}}{\frac{4}{4}}$

Divide $4$ by $4$ .

$e = - \frac{23}{4} - \frac{\frac{1}{4}}{1}$

Divide $\frac{1}{4}$ by $1$ .

$e = - \frac{23}{4} - \frac{1}{4}$

Combine the numerators over the common denominator.

$e = \frac{- 23 - 1}{4}$

Subtract $1$ from $- 23$ .

$e = \frac{- 24}{4}$

Divide $- 24$ by $4$ .

$e = - 6$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(y + 1)}^{2} - 6$ .

$\frac{1}{4} {(y + 1)}^{2} - 6$

Set $x$ equal to the new right side.

$x = \frac{1}{4} \cdot {(y + 1)}^{2} - 6$

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

$a = \frac{1}{4}$

$h = - 6$

$k = - 1$

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

Opens Right

Find the vertex $(h, k)$ .

$(- 6, - 1)$

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.

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

Simplify.

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

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

Simplify by dividing numbers.

Divide $4$ by $4$ .

$\frac{1}{1}$

Divide $1$ by $1$ .

$1$

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.

$(- 5, - 1)$

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.

$x = - 7$

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

Direction: Opens Right

Vertex: $(- 6, - 1)$

Focus: $(- 5, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - 7$

Direction: Opens Right

Vertex: $(- 6, - 1)$

Focus: $(- 5, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - 7$

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 $f (x) = - 1 - 2 \sqrt{x + 6}$ . In this case, the point is $(- 5, - 3)$ .

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

$f (- 5) = - 1 - 2 \sqrt{(- 5) + 6}$

Simplify the result.

Simplify each term.

Add $- 5$ and $6$ .

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

Any root of $1$ is $1$ .

$f (- 5) = - 1 - 2 \cdot 1$

Multiply $- 2$ by $1$ .

$f (- 5) = - 1 - 2$

Subtract $2$ from $- 1$ .

$f (- 5) = - 3$

The final answer is $- 3$ .

$y = - 3$

Convert $- 3$ to decimal.

$= - 3$

Substitute the $x$ value $- 5$ into $f (x) = - 1 + 2 \sqrt{x + 6}$ . In this case, the point is $(- 5, 1)$ .

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

$f (- 5) = - 1 + 2 \sqrt{(- 5) + 6}$

Simplify the result.

Simplify each term.

Add $- 5$ and $6$ .

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

Any root of $1$ is $1$ .

$f (- 5) = - 1 + 2 \cdot 1$

Multiply $2$ by $1$ .

$f (- 5) = - 1 + 2$

Add $- 1$ and $2$ .

$f (- 5) = 1$

The final answer is $1$ .

$y = 1$

Convert $1$ to decimal.

$= 1$

Substitute the $x$ value $- 4$ into $f (x) = - 1 - 2 \sqrt{x + 6}$ . In this case, the point is $(- 4, - 3.82842712)$ .

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = - 1 - 2 \sqrt{(- 4) + 6}$

Simplify the result.

Add $- 4$ and $6$ .

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

The final answer is $- 1 - 2 \sqrt{2}$ .

$y = - 1 - 2 \sqrt{2}$

Convert $- 1 - 2 \sqrt{2}$ to decimal.

$= - 3.82842712$

Substitute the $x$ value $- 4$ into $f (x) = - 1 + 2 \sqrt{x + 6}$ . In this case, the point is $(- 4, 1.82842712)$ .

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = - 1 + 2 \sqrt{(- 4) + 6}$

Simplify the result.

Add $- 4$ and $6$ .

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

The final answer is $- 1 + 2 \sqrt{2}$ .

$y = - 1 + 2 \sqrt{2}$

Convert $- 1 + 2 \sqrt{2}$ to decimal.

$= 1.82842712$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 6 & - 1 \\ - 5 & - 3 \\ - 5 & 1 \\ - 4 & - 3.83 \\ - 4 & 1.83 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 6, - 1)$

Focus: $(- 5, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - 7$

$\begin{array}{c} x & y \\ - 6 & - 1 \\ - 5 & - 3 \\ - 5 & 1 \\ - 4 & - 3.83 \\ - 4 & 1.83 \end{array}$

Step 6

Graph 4x-2y=y^2-23

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