Graph x=(y-3)^2

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

Step 1

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

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

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

Expand $(y - 3) (y - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

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

Move $- 3$ to the left of $y$ .

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

Multiply $- 3$ by $- 3$ .

$x = y^{2} - 3 y - 3 y + 9$

Subtract $3 y$ from $- 3 y$ .

$x = y^{2} - 6 y + 9$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $y^{2} - 6 y + 9$ .

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

$a = 1$

$b = - 6$

$c = 9$

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

$d = \frac{- 6}{2 \cdot 1}$

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 3}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 3}{1}$

Divide $- 3$ by $1$ .

$d = - 3$

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

$e = 9 - \frac{{(- 6)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 6$ to the power of $2$ .

$e = 9 - \frac{36}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 9 - \frac{36}{4}$

Divide $36$ by $4$ .

$e = 9 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 9 - 9$

Subtract $9$ from $9$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 3)}^{2} + 0$ .

${(y - 3)}^{2} + 0$

Set $x$ equal to the new right side.

$x = {(y - 3)}^{2} + 0$

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

$a = 1$

$h = 0$

$k = 3$

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

Opens Right

Find the vertex $(h, k)$ .

$(0, 3)$

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.

$(\frac{1}{4}, 3)$

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

$y = 3$

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.

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

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

Direction: Opens Right

Vertex: $(0, 3)$

Focus: $(\frac{1}{4}, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - \frac{1}{4}$

Direction: Opens Right

Vertex: $(0, 3)$

Focus: $(\frac{1}{4}, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - \frac{1}{4}$

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

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

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

Simplify the result.

Remove parentheses.

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

Any root of $1$ is $1$ .

$f (1) = 1 + 3$

Add $1$ and $3$ .

$f (1) = 4$

The final answer is $4$ .

$y = 4$

Convert $4$ to decimal.

$= 4$

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

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

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

Simplify the result.

Remove parentheses.

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

Simplify each term.

Any root of $1$ is $1$ .

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

Multiply $- 1$ by $1$ .

$f (1) = - 1 + 3$

Add $- 1$ and $3$ .

$f (1) = 2$

The final answer is $2$ .

$y = 2$

Convert $2$ to decimal.

$= 2$

Substitute the $x$ value $2$ into $f (x) = \sqrt{x} + 3$ . In this case, the point is $(2, 4.41421356)$ .

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

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

Simplify the result.

Remove parentheses.

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

The final answer is $\sqrt{2} + 3$ .

$y = \sqrt{2} + 3$

Convert $\sqrt{2} + 3$ to decimal.

$= 4.41421356$

Substitute the $x$ value $2$ into $f (x) = - \sqrt{x} + 3$ . In this case, the point is $(2, 1.58578643)$ .

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

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

Simplify the result.

Remove parentheses.

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

The final answer is $- \sqrt{2} + 3$ .

$y = - \sqrt{2} + 3$

Convert $- \sqrt{2} + 3$ to decimal.

$= 1.58578643$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 3 \\ 1 & 4 \\ 1 & 2 \\ 2 & 4.41 \\ 2 & 1.59 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(0, 3)$

Focus: $(\frac{1}{4}, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - \frac{1}{4}$

$\begin{array}{c} x & y \\ 0 & 3 \\ 1 & 4 \\ 1 & 2 \\ 2 & 4.41 \\ 2 & 1.59 \end{array}$

Step 5

Graph x=(y-3)^2

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