Graph y^2-2y=8x-1

$y^{2} - 2 y = 8 x - 1$

Step 1

Rewrite the equation as $8 x - 1 = y^{2} - 2 y$ .

$8 x - 1 = y^{2} - 2 y$

Step 2

Add $1$ to both sides of the equation.

$8 x = y^{2} - 2 y + 1$

Step 3

Divide each term in $8 x = y^{2} - 2 y + 1$ by $8$ and simplify.

Divide each term in $8 x = y^{2} - 2 y + 1$ by $8$ .

$\frac{8 x}{8} = \frac{y^{2}}{8} + \frac{- 2 y}{8} + \frac{1}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{y^{2}}{8} + \frac{- 2 y}{8} + \frac{1}{8}$

Divide $x$ by $1$ .

$x = \frac{y^{2}}{8} + \frac{- 2 y}{8} + \frac{1}{8}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 2$ and $8$ .

Factor $2$ out of $- 2 y$ .

$x = \frac{y^{2}}{8} + \frac{2 (- y)}{8} + \frac{1}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$x = \frac{y^{2}}{8} + \frac{2 (- y)}{2 (4)} + \frac{1}{8}$

Cancel the common factor.

$x = \frac{y^{2}}{8} + \frac{2 (- y)}{2 \cdot 4} + \frac{1}{8}$

Rewrite the expression.

$x = \frac{y^{2}}{8} + \frac{- y}{4} + \frac{1}{8}$

Move the negative in front of the fraction.

$x = \frac{y^{2}}{8} - \frac{y}{4} + \frac{1}{8}$

Step 4

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $\frac{y^{2}}{8} - \frac{y}{4} + \frac{1}{8}$ .

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

$a = \frac{1}{8}$

$b = - \frac{1}{4}$

$c = \frac{1}{8}$

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

$d = \frac{- \frac{1}{4}}{2 (\frac{1}{8})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{4} \cdot \frac{1}{2 (\frac{1}{8})}$

Combine $2$ and $\frac{1}{8}$ .

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2}{8}}$

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2$ .

$d = - \frac{1}{4} \cdot \frac{1}{\frac{2 (1)}{8}}$

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

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

Factor $4$ out of $1 \cdot 4$ .

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

Cancel the common factor.

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

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

$e = \frac{1}{8} - \frac{{(- \frac{1}{4})}^{2}}{4 (\frac{1}{8})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{1}{4}$ .

$e = \frac{1}{8} - \frac{{(- 1)}^{2} {(\frac{1}{4})}^{2}}{4 (\frac{1}{8})}$

Raise $- 1$ to the power of $2$ .

$e = \frac{1}{8} - \frac{1 {(\frac{1}{4})}^{2}}{4 (\frac{1}{8})}$

Apply the product rule to $\frac{1}{4}$ .

$e = \frac{1}{8} - \frac{1 \frac{1^{2}}{4^{2}}}{4 (\frac{1}{8})}$

One to any power is one.

$e = \frac{1}{8} - \frac{1 \frac{1}{4^{2}}}{4 (\frac{1}{8})}$

Raise $4$ to the power of $2$ .

$e = \frac{1}{8} - \frac{1 (\frac{1}{16})}{4 (\frac{1}{8})}$

Multiply $\frac{1}{16}$ by $1$ .

$e = \frac{1}{8} - \frac{\frac{1}{16}}{4 (\frac{1}{8})}$

Combine $4$ and $\frac{1}{8}$ .

$e = \frac{1}{8} - \frac{\frac{1}{16}}{\frac{4}{8}}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$e = \frac{1}{8} - \frac{\frac{1}{16}}{\frac{4 (1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$e = \frac{1}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Cancel the common factor.

$e = \frac{1}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot 1}{4 \cdot 2}}$

Rewrite the expression.

$e = \frac{1}{8} - \frac{\frac{1}{16}}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{8} - (\frac{1}{16} \cdot 2)$

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$e = \frac{1}{8} - (\frac{1}{2 (8)} \cdot 2)$

Cancel the common factor.

$e = \frac{1}{8} - (\frac{1}{2 \cdot 8} \cdot 2)$

Rewrite the expression.

$e = \frac{1}{8} - \frac{1}{8}$

Combine the numerators over the common denominator.

$e = \frac{1 - 1}{8}$

Subtract $1$ from $1$ .

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

Divide $0$ by $8$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(y - 1)}^{2} + 0$ .

$\frac{1}{8} {(y - 1)}^{2} + 0$

Set $x$ equal to the new right side.

$x = \frac{1}{8} \cdot {(y - 1)}^{2} + 0$

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

$a = \frac{1}{8}$

$h = 0$

$k = 1$

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

Opens Right

Find the vertex $(h, k)$ .

$(0, 1)$

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.

$\frac{1}{4 \cdot \frac{1}{8}}$

Simplify.

Combine $4$ and $\frac{1}{8}$ .

$\frac{1}{\frac{4}{8}}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Multiply $2$ by $1$ .

$2$

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.

$(2, 1)$

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.

$x = - 2$

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

Direction: Opens Right

Vertex: $(0, 1)$

Focus: $(2, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = - 2$

Direction: Opens Right

Vertex: $(0, 1)$

Focus: $(2, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = - 2$

Step 5

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 $f (x) = 1 + 2 \sqrt{2 x}$ . In this case, the point is $(1, 3.82842712)$ .

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

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

Simplify the result.

Multiply $2$ by $1$ .

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

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

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

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

Simplify the result.

Multiply $2$ by $1$ .

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

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

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

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

$= - 1.82842712$

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

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

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

Simplify the result.

Simplify each term.

Multiply $2$ by $2$ .

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

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

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

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

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

Multiply $2$ by $2$ .

$f (2) = 1 + 4$

Add $1$ and $4$ .

$f (2) = 5$

The final answer is $5$ .

$y = 5$

Convert $5$ to decimal.

$= 5$

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

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

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

Simplify the result.

Simplify each term.

Multiply $2$ by $2$ .

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

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

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

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

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

Multiply $- 2$ by $2$ .

$f (2) = 1 - 4$

Subtract $4$ from $1$ .

$f (2) = - 3$

The final answer is $- 3$ .

$y = - 3$

Convert $- 3$ to decimal.

$= - 3$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 1 \\ 1 & 3.83 \\ 1 & - 1.83 \\ 2 & 5 \\ 2 & - 3 \end{array}$

Step 6

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(0, 1)$

Focus: $(2, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = - 2$

$\begin{array}{c} x & y \\ 0 & 1 \\ 1 & 3.83 \\ 1 & - 1.83 \\ 2 & 5 \\ 2 & - 3 \end{array}$

Step 7

Graph y^2-2y=8x-1

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