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

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

Step 1

Simplify ${(y - 2)}^{2} + 3$ .

Simplify each term.

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

$x = (y - 2) (y - 2) + 3$

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

Apply the distributive property.

$x = y (y - 2) - 2 (y - 2) + 3$

Apply the distributive property.

$x = y \cdot y + y \cdot - 2 - 2 (y - 2) + 3$

Apply the distributive property.

$x = y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2 + 3$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

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

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

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

Multiply $- 2$ by $- 2$ .

$x = y^{2} - 2 y - 2 y + 4 + 3$

Subtract $2 y$ from $- 2 y$ .

$x = y^{2} - 4 y + 4 + 3$

Add $4$ and $3$ .

$x = y^{2} - 4 y + 7$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

$a = 1$

$b = - 4$

$c = 7$

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

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

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

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify the right side.

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$e = 7 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 7 - 4$

Subtract $4$ from $7$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 2)}^{2} + 3$ .

${(y - 2)}^{2} + 3$

Set $x$ equal to the new right side.

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

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

$a = 1$

$h = 3$

$k = 2$

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

Opens Right

Find the vertex $(h, k)$ .

$(3, 2)$

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.

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

Cancel the common factor of $1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{4}$

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.

$(\frac{13}{4}, 2)$

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.

$x = \frac{11}{4}$

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

Direction: Opens Right

Vertex: $(3, 2)$

Focus: $(\frac{13}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = \frac{11}{4}$

Direction: Opens Right

Vertex: $(3, 2)$

Focus: $(\frac{13}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = \frac{11}{4}$

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

Replace the variable $x$ with $4$ in the expression.

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

Simplify the result.

Simplify each term.

Subtract $3$ from $4$ .

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

Any root of $1$ is $1$ .

$f (4) = 1 + 2$

Add $1$ and $2$ .

$f (4) = 3$

The final answer is $3$ .

$y = 3$

Convert $3$ to decimal.

$= 3$

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

Replace the variable $x$ with $4$ in the expression.

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

Simplify the result.

Simplify each term.

Subtract $3$ from $4$ .

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

Any root of $1$ is $1$ .

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

Multiply $- 1$ by $1$ .

$f (4) = - 1 + 2$

Add $- 1$ and $2$ .

$f (4) = 1$

The final answer is $1$ .

$y = 1$

Convert $1$ to decimal.

$= 1$

Substitute the $x$ value $5$ into $f (x) = \sqrt{x - 3} + 2$ . In this case, the point is $(5, 3.41421356)$ .

Replace the variable $x$ with $5$ in the expression.

$f (5) = \sqrt{(5) - 3} + 2$

Simplify the result.

Subtract $3$ from $5$ .

$f (5) = \sqrt{2} + 2$

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

$y = \sqrt{2} + 2$

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

$= 3.41421356$

Substitute the $x$ value $5$ into $f (x) = - \sqrt{x - 3} + 2$ . In this case, the point is $(5, 0.58578643)$ .

Replace the variable $x$ with $5$ in the expression.

$f (5) = - \sqrt{(5) - 3} + 2$

Simplify the result.

Subtract $3$ from $5$ .

$f (5) = - \sqrt{2} + 2$

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

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

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

$= 0.58578643$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 3 & 2 \\ 4 & 3 \\ 4 & 1 \\ 5 & 3.41 \\ 5 & 0.59 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(3, 2)$

Focus: $(\frac{13}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = \frac{11}{4}$

$\begin{array}{c} x & y \\ 3 & 2 \\ 4 & 3 \\ 4 & 1 \\ 5 & 3.41 \\ 5 & 0.59 \end{array}$

Step 5

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

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