Graph -1/4*(y+2)^2=(x-1)

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

Step 1

Rewrite the equation as $x - 1 = - \frac{1}{4} \cdot {(y + 2)}^{2}$ .

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

Step 2

Simplify $- \frac{1}{4} \cdot {(y + 2)}^{2}$ .

Rewrite.

$x - 1 = 0 + 0 - \frac{1}{4} \cdot {(y + 2)}^{2}$

Rewrite ${(y + 2)}^{2}$ as $(y + 2) (y + 2)$ .

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

Expand $(y + 2) (y + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

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

Move $2$ to the left of $y$ .

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

Multiply $2$ by $2$ .

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

Add $2 y$ and $2 y$ .

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

Apply the distributive property.

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

Simplify.

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

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

Cancel the common factor of $4$ .

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

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

Factor $4$ out of $4 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Rewrite $- 1 y$ as $- y$ .

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

Step 3

Move all terms not containing $x$ to the right side of the equation.

Add $1$ to both sides of the equation.

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

Combine the opposite terms in $- \frac{y^{2}}{4} - y - 1 + 1$ .

Add $- 1$ and $1$ .

$x = - \frac{y^{2}}{4} - y + 0$

Add $- \frac{y^{2}}{4} - y$ and $0$ .

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

Step 4

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

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

$b = - 1$

$c = 0$

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

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

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

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

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

Multiply $2$ by $1$ .

$d = 2$

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

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

Simplify the right side.

Simplify each term.

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

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

Simplify the denominator.

Multiply $- 1$ by $4$ .

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

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

$e = 0 - \frac{1}{\frac{- 4}{4}}$

Divide $- 4$ by $4$ .

$e = 0 - \frac{1}{- 1}$

Divide $1$ by $- 1$ .

$e = 0 - - 1$

Multiply $- 1$ by $- 1$ .

$e = 0 + 1$

Add $0$ and $1$ .

$e = 1$

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

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

Set $x$ equal to the new right side.

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

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

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

$h = 1$

$k = - 2$

Since the value of $a$ is negative, the parabola opens left.

Opens Left

Find the vertex $(h, k)$ .

$(1, - 2)$

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.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

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

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

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

Divide $4$ by $4$ .

$- \frac{1}{1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$- \frac{1}{1}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ 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.

$(0, - 2)$

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

$y = - 2$

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 Left

Vertex: $(1, - 2)$

Focus: $(0, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = 2$

Direction: Opens Left

Vertex: $(1, - 2)$

Focus: $(0, - 2)$

Axis of Symmetry: $y = - 2$

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 $f (x) = 2 \sqrt{- x + 1} - 2$ . In this case, the point is $(- 1, 0.82842712)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

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

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

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

$= 0.82842712$

Substitute the $x$ value $- 1$ into $f (x) = - 2 \sqrt{- x + 1} - 2$ . In this case, the point is $(- 1, - 4.82842712)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

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

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

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

$= - 4.82842712$

Substitute the $x$ value $0$ into $f (x) = 2 \sqrt{- x + 1} - 2$ . In this case, the point is $(0, 0)$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = 2 \sqrt{- (0) + 1} - 2$

Simplify the result.

Simplify each term.

Multiply $- 1$ by $0$ .

$f (0) = 2 \sqrt{0 + 1} - 2$

Add $0$ and $1$ .

$f (0) = 2 \sqrt{1} - 2$

Any root of $1$ is $1$ .

$f (0) = 2 \cdot 1 - 2$

Multiply $2$ by $1$ .

$f (0) = 2 - 2$

Subtract $2$ from $2$ .

$f (0) = 0$

The final answer is $0$ .

$y = 0$

Convert $0$ to decimal.

$= 0$

Substitute the $x$ value $0$ into $f (x) = - 2 \sqrt{- x + 1} - 2$ . In this case, the point is $(0, - 4)$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 2 \sqrt{- (0) + 1} - 2$

Simplify the result.

Simplify each term.

Multiply $- 1$ by $0$ .

$f (0) = - 2 \sqrt{0 + 1} - 2$

Add $0$ and $1$ .

$f (0) = - 2 \sqrt{1} - 2$

Any root of $1$ is $1$ .

$f (0) = - 2 \cdot 1 - 2$

Multiply $- 2$ by $1$ .

$f (0) = - 2 - 2$

Subtract $2$ from $- 2$ .

$f (0) = - 4$

The final answer is $- 4$ .

$y = - 4$

Convert $- 4$ to decimal.

$= - 4$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 0.83 \\ - 1 & - 4.83 \\ 0 & 0 \\ 0 & - 4 \\ 1 & - 2 \end{array}$

Step 6

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(1, - 2)$

Focus: $(0, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = 2$

$\begin{array}{c} x & y \\ - 1 & 0.83 \\ - 1 & - 4.83 \\ 0 & 0 \\ 0 & - 4 \\ 1 & - 2 \end{array}$

Step 7

Graph -1/4*(y+2)^2=(x-1)

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