Graph x=-y^2+2y+3

$x = - y^{2} + 2 y + 3$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- y^{2} + 2 y + 3$ .

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

$a = - 1$

$b = 2$

$c = 3$

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

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

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = 3 - \frac{2^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

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

$e = 3 - \frac{4}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = 3 - \frac{4}{- 4}$

Divide $4$ by $- 4$ .

$e = 3 - - 1$

Multiply $- 1$ by $- 1$ .

$e = 3 + 1$

Add $3$ and $1$ .

$e = 4$

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

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

Set $x$ equal to the new right side.

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

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

$a = - 1$

$h = 4$

$k = 1$

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

Opens Left

Find the vertex $(h, k)$ .

$(4, 1)$

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.

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

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

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

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

Move the negative in front of the fraction.

$- \frac{1}{4}$

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.

$(\frac{15}{4}, 1)$

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.

$x = \frac{17}{4}$

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

Direction: Opens Left

Vertex: $(4, 1)$

Focus: $(\frac{15}{4}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = \frac{17}{4}$

Direction: Opens Left

Vertex: $(4, 1)$

Focus: $(\frac{15}{4}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = \frac{17}{4}$

Step 2

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 $2$ into $f (x) = 1 + \sqrt{4 - x}$ . In this case, the point is $(2, 2.41421356)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $4$ .

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

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

$y = 1 + \sqrt{2}$

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

$= 2.41421356$

Substitute the $x$ value $2$ into $f (x) = 1 - \sqrt{4 - x}$ . In this case, the point is $(2, - 0.41421356)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $4$ .

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

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

$y = 1 - \sqrt{2}$

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

$= - 0.41421356$

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

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $4$ .

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

Any root of $1$ is $1$ .

$f (3) = 1 + 1$

Add $1$ and $1$ .

$f (3) = 2$

The final answer is $2$ .

$y = 2$

Convert $2$ to decimal.

$= 2$

Substitute the $x$ value $3$ into $f (x) = 1 - \sqrt{4 - x}$ . In this case, the point is $(3, 0)$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $4$ .

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

Any root of $1$ is $1$ .

$f (3) = 1 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (3) = 1 - 1$

Subtract $1$ from $1$ .

$f (3) = 0$

The final answer is $0$ .

$y = 0$

Convert $0$ to decimal.

$= 0$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 2.41 \\ 2 & - 0.41 \\ 3 & 2 \\ 3 & 0 \\ 4 & 1 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(4, 1)$

Focus: $(\frac{15}{4}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = \frac{17}{4}$

$\begin{array}{c} x & y \\ 2 & 2.41 \\ 2 & - 0.41 \\ 3 & 2 \\ 3 & 0 \\ 4 & 1 \end{array}$

Step 4

Graph x=-y^2+2y+3

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