Graph x=-1/2y^2-4y-7

$x = - \frac{1}{2} y^{2} - 4 y - 7$

Step 1

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

$x = - \frac{y^{2}}{2} - 4 y - 7$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- \frac{y^{2}}{2} - 4 y - 7$ .

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

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

$b = - 4$

$c = - 7$

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

$e = - 7 - \frac{{(- 4)}^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

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

Rewrite $- 4$ as $- 1 (4)$ .

$e = - 7 - \frac{{(- 1 (4))}^{2}}{4 (- \frac{1}{2})}$

Apply the product rule to $- 1 (4)$ .

$e = - 7 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 (- \frac{1}{2})}$

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

$e = - 7 - \frac{1 \cdot 4^{2}}{4 (- \frac{1}{2})}$

Multiply $4^{2}$ by $1$ .

$e = - 7 - \frac{4^{2}}{4 (- \frac{1}{2})}$

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

$e = - 7 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Cancel the common factors.

Cancel the common factor.

$e = - 7 - \frac{4 \cdot 4}{4 (- \frac{1}{2})}$

Rewrite the expression.

$e = - 7 - \frac{4}{- \frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - 7 - (4 (- 1 \cdot 2))$

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

Multiply $- 1$ by $2$ .

$e = - 7 - (4 \cdot - 2)$

Multiply $4$ by $- 2$ .

$e = - 7 - - 8$

Multiply $- 1$ by $- 8$ .

$e = - 7 + 8$

Add $- 7$ 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.

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

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.

$x = \frac{3}{2}$

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

Direction: Opens Left

Vertex: $(1, - 4)$

Focus: $(\frac{1}{2}, - 4)$

Axis of Symmetry: $y = - 4$

Directrix: $x = \frac{3}{2}$

Direction: Opens Left

Vertex: $(1, - 4)$

Focus: $(\frac{1}{2}, - 4)$

Axis of Symmetry: $y = - 4$

Directrix: $x = \frac{3}{2}$

Step 3

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

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

Multiply $- 1$ by $2$ .

$f (- 1) = - 4 - 2$

Subtract $2$ from $- 4$ .

$f (- 1) = - 6$

The final answer is $- 6$ .

$y = - 6$

Convert $- 6$ to decimal.

$= - 6$

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

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

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

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

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

$f (- 1) = - 4 + 2$

Add $- 4$ and $2$ .

$f (- 1) = - 2$

The final answer is $- 2$ .

$y = - 2$

Convert $- 2$ to decimal.

$= - 2$

Substitute the $x$ value $0$ into $f (x) = - 4 - \sqrt{2 (- x + 1)}$ . In this case, the point is $(0, - 5.41421356)$ .

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

$f (0) = - 4 - \sqrt{2 (- (0) + 1)}$

Simplify the result.

Simplify each term.

Multiply $- 1$ by $0$ .

$f (0) = - 4 - \sqrt{2 (0 + 1)}$

Add $0$ and $1$ .

$f (0) = - 4 - \sqrt{2 \cdot 1}$

Multiply $2$ by $1$ .

$f (0) = - 4 - \sqrt{2}$

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

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

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

$= - 5.41421356$

Substitute the $x$ value $0$ into $f (x) = - 4 + \sqrt{2 (- x + 1)}$ . In this case, the point is $(0, - 2.58578643)$ .

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

$f (0) = - 4 + \sqrt{2 (- (0) + 1)}$

Simplify the result.

Simplify each term.

Multiply $- 1$ by $0$ .

$f (0) = - 4 + \sqrt{2 (0 + 1)}$

Add $0$ and $1$ .

$f (0) = - 4 + \sqrt{2 \cdot 1}$

Multiply $2$ by $1$ .

$f (0) = - 4 + \sqrt{2}$

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

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

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

$= - 2.58578643$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 6 \\ - 1 & - 2 \\ 0 & - 5.41 \\ 0 & - 2.59 \\ 1 & - 4 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(1, - 4)$

Focus: $(\frac{1}{2}, - 4)$

Axis of Symmetry: $y = - 4$

Directrix: $x = \frac{3}{2}$

$\begin{array}{c} x & y \\ - 1 & - 6 \\ - 1 & - 2 \\ 0 & - 5.41 \\ 0 & - 2.59 \\ 1 & - 4 \end{array}$

Step 5

Graph x=-1/2y^2-4y-7

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