Graph x=(y-5)^2

$x = {(y - 5)}^{2}$

Step 1

Simplify ${(y - 5)}^{2}$ .

Rewrite ${(y - 5)}^{2}$ as $(y - 5) (y - 5)$ .

$x = (y - 5) (y - 5)$

Expand $(y - 5) (y - 5)$ using the FOIL Method.

Apply the distributive property.

$x = y (y - 5) - 5 (y - 5)$

Apply the distributive property.

$x = y \cdot y + y \cdot - 5 - 5 (y - 5)$

Apply the distributive property.

$x = y \cdot y + y \cdot - 5 - 5 y - 5 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$x = y^{2} + y \cdot - 5 - 5 y - 5 \cdot - 5$

Move $- 5$ to the left of $y$ .

$x = y^{2} - 5 \cdot y - 5 y - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$x = y^{2} - 5 y - 5 y + 25$

Subtract $5 y$ from $- 5 y$ .

$x = y^{2} - 10 y + 25$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $y^{2} - 10 y + 25$ .

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

$a = 1$

$b = - 10$

$c = 25$

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

$d = \frac{- 10}{2 \cdot 1}$

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 5}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 5}{1}$

Divide $- 5$ by $1$ .

$d = - 5$

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

$e = 25 - \frac{{(- 10)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 10$ to the power of $2$ .

$e = 25 - \frac{100}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 25 - \frac{100}{4}$

Divide $100$ by $4$ .

$e = 25 - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$e = 25 - 25$

Subtract $25$ from $25$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 5)}^{2} + 0$ .

${(y - 5)}^{2} + 0$

Set $x$ equal to the new right side.

$x = {(y - 5)}^{2} + 0$

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

$a = 1$

$h = 0$

$k = 5$

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

Opens Right

Find the vertex $(h, k)$ .

$(0, 5)$

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.

$(\frac{1}{4}, 5)$

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

$y = 5$

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: $(0, 5)$

Focus: $(\frac{1}{4}, 5)$

Axis of Symmetry: $y = 5$

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

Direction: Opens Right

Vertex: $(0, 5)$

Focus: $(\frac{1}{4}, 5)$

Axis of Symmetry: $y = 5$

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 $f (x) = \sqrt{x} + 5$ . In this case, the point is $(1, 6)$ .

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

$f (1) = \sqrt{1} + 5$

Simplify the result.

Remove parentheses.

$f (1) = \sqrt{1} + 5$

Any root of $1$ is $1$ .

$f (1) = 1 + 5$

Add $1$ and $5$ .

$f (1) = 6$

The final answer is $6$ .

$y = 6$

Convert $6$ to decimal.

$= 6$

Substitute the $x$ value $1$ into $f (x) = - \sqrt{x} + 5$ . In this case, the point is $(1, 4)$ .

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

$f (1) = - \sqrt{1} + 5$

Simplify the result.

Remove parentheses.

$f (1) = - \sqrt{1} + 5$

Simplify each term.

Any root of $1$ is $1$ .

$f (1) = - 1 \cdot 1 + 5$

Multiply $- 1$ by $1$ .

$f (1) = - 1 + 5$

Add $- 1$ and $5$ .

$f (1) = 4$

The final answer is $4$ .

$y = 4$

Convert $4$ to decimal.

$= 4$

Substitute the $x$ value $2$ into $f (x) = \sqrt{x} + 5$ . In this case, the point is $(2, 6.41421356)$ .

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

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

Simplify the result.

Remove parentheses.

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

The final answer is $\sqrt{2} + 5$ .

$y = \sqrt{2} + 5$

Convert $\sqrt{2} + 5$ to decimal.

$= 6.41421356$

Substitute the $x$ value $2$ into $f (x) = - \sqrt{x} + 5$ . In this case, the point is $(2, 3.58578643)$ .

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

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

Simplify the result.

Remove parentheses.

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

The final answer is $- \sqrt{2} + 5$ .

$y = - \sqrt{2} + 5$

Convert $- \sqrt{2} + 5$ to decimal.

$= 3.58578643$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 5 \\ 1 & 6 \\ 1 & 4 \\ 2 & 6.41 \\ 2 & 3.59 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(0, 5)$

Focus: $(\frac{1}{4}, 5)$

Axis of Symmetry: $y = 5$

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

$\begin{array}{c} x & y \\ 0 & 5 \\ 1 & 6 \\ 1 & 4 \\ 2 & 6.41 \\ 2 & 3.59 \end{array}$

Step 5

Graph x=(y-5)^2

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