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

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

Step 1

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

Rewrite.

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

Rewrite ${(y - 4)}^{2}$ as $(y - 4) (y - 4)$ .

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

Expand $(y - 4) (y - 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

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

Move $- 4$ to the left of $y$ .

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

Multiply $- 4$ by $- 4$ .

$x + 1 = - \frac{1}{2} \cdot (y^{2} - 4 y - 4 y + 16)$

Subtract $4 y$ from $- 4 y$ .

$x + 1 = - \frac{1}{2} \cdot (y^{2} - 8 y + 16)$

Apply the distributive property.

$x + 1 = - \frac{1}{2} y^{2} - \frac{1}{2} (- 8 y) - \frac{1}{2} \cdot 16$

Simplify.

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

$x + 1 = - \frac{y^{2}}{2} - \frac{1}{2} (- 8 y) - \frac{1}{2} \cdot 16$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x + 1 = - \frac{y^{2}}{2} + \frac{- 1}{2} (- 8 y) - \frac{1}{2} \cdot 16$

Factor $2$ out of $- 8 y$ .

$x + 1 = - \frac{y^{2}}{2} + \frac{- 1}{2} (2 (- 4 y)) - \frac{1}{2} \cdot 16$

Cancel the common factor.

$x + 1 = - \frac{y^{2}}{2} + \frac{- 1}{2} (2 (- 4 y)) - \frac{1}{2} \cdot 16$

Rewrite the expression.

$x + 1 = - \frac{y^{2}}{2} - 1 (- 4 y) - \frac{1}{2} \cdot 16$

Multiply $- 4$ by $- 1$ .

$x + 1 = - \frac{y^{2}}{2} + 4 y - \frac{1}{2} \cdot 16$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x + 1 = - \frac{y^{2}}{2} + 4 y + \frac{- 1}{2} \cdot 16$

Factor $2$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $8$ .

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

Step 2

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

Subtract $1$ from both sides of the equation.

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

Subtract $1$ from $- 8$ .

$x = - \frac{y^{2}}{2} + 4 y - 9$

Step 3

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- \frac{y^{2}}{2} + 4 y - 9$ .

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

$a = - \frac{1}{2}$

$b = 4$

$c = - 9$

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

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

Simplify the right side.

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

Factor $2$ out of $4$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $2 (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$d = 2 \cdot - 2$

Multiply $2$ by $- 2$ .

$d = - 4$

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

$e = - 9 - \frac{4^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $4^{2}$ and $4$ .

Factor $4$ out of $4^{2}$ .

$e = - 9 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Cancel the common factors.

Cancel the common factor.

$e = - 9 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Rewrite the expression.

$e = - 9 - \frac{4}{- \frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - 9 - (4 (- 1 \cdot 2))$

Multiply $- (4 (- 1 \cdot 2))$ .

Multiply $- 1$ by $2$ .

$e = - 9 - (4 \cdot - 2)$

Multiply $4$ by $- 2$ .

$e = - 9 - - 8$

Multiply $- 1$ by $- 8$ .

$e = - 9 + 8$

Add $- 9$ and $8$ .

$e = - 1$

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

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

Set $x$ equal to the new right side.

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

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

$a = - \frac{1}{2}$

$h = - 1$

$k = 4$

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

Opens Left

Find the vertex $(h, k)$ .

$(- 1, 4)$

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.

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

Simplify.

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

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

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

Move the negative in front of the fraction.

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

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

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

Divide $4$ by $2$ .

$- \frac{1}{2}$

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.

$(- \frac{3}{2}, 4)$

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

$y = 4$

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.

$x = - \frac{1}{2}$

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

Direction: Opens Left

Vertex: $(- 1, 4)$

Focus: $(- \frac{3}{2}, 4)$

Axis of Symmetry: $y = 4$

Directrix: $x = - \frac{1}{2}$

Direction: Opens Left

Vertex: $(- 1, 4)$

Focus: $(- \frac{3}{2}, 4)$

Axis of Symmetry: $y = 4$

Directrix: $x = - \frac{1}{2}$

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 $- 3$ into $f (x) = \sqrt{2 (- x - 1)} + 4$ . In this case, the point is $(- 3, 6)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \sqrt{2 (- (- 3) - 1)} + 4$

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 3$ .

$f (- 3) = \sqrt{2 (3 - 1)} + 4$

Subtract $1$ from $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

$f (- 3) = 2 + 4$

Add $2$ and $4$ .

$f (- 3) = 6$

The final answer is $6$ .

$y = 6$

Convert $6$ to decimal.

$= 6$

Substitute the $x$ value $- 3$ into $f (x) = - \sqrt{2 (- x - 1)} + 4$ . In this case, the point is $(- 3, 2)$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - \sqrt{2 (- (- 3) - 1)} + 4$

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 3$ .

$f (- 3) = - \sqrt{2 (3 - 1)} + 4$

Subtract $1$ from $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

$f (- 3) = - 1 \cdot 2 + 4$

Multiply $- 1$ by $2$ .

$f (- 3) = - 2 + 4$

Add $- 2$ and $4$ .

$f (- 3) = 2$

The final answer is $2$ .

$y = 2$

Convert $2$ to decimal.

$= 2$

Substitute the $x$ value $- 2$ into $f (x) = \sqrt{2 (- x - 1)} + 4$ . In this case, the point is $(- 2, 5.41421356)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 2$ .

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

Subtract $1$ from $2$ .

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

Multiply $2$ by $1$ .

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

The final answer is $\sqrt{2} + 4$ .

$y = \sqrt{2} + 4$

Convert $\sqrt{2} + 4$ to decimal.

$= 5.41421356$

Substitute the $x$ value $- 2$ into $f (x) = - \sqrt{2 (- x - 1)} + 4$ . In this case, the point is $(- 2, 2.58578643)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 2$ .

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

Subtract $1$ from $2$ .

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

Multiply $2$ by $1$ .

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

The final answer is $- \sqrt{2} + 4$ .

$y = - \sqrt{2} + 4$

Convert $- \sqrt{2} + 4$ to decimal.

$= 2.58578643$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 6 \\ - 3 & 2 \\ - 2 & 5.41 \\ - 2 & 2.59 \\ - 1 & 4 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(- 1, 4)$

Focus: $(- \frac{3}{2}, 4)$

Axis of Symmetry: $y = 4$

Directrix: $x = - \frac{1}{2}$

$\begin{array}{c} x & y \\ - 3 & 6 \\ - 3 & 2 \\ - 2 & 5.41 \\ - 2 & 2.59 \\ - 1 & 4 \end{array}$

Step 6

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

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