Graph x=5(y-1)^2

$x = 5 {(y - 1)}^{2}$

Step 1

Simplify $5 {(y - 1)}^{2}$ .

Rewrite ${(y - 1)}^{2}$ as $(y - 1) (y - 1)$ .

$x = 5 ((y - 1) (y - 1))$

Expand $(y - 1) (y - 1)$ using the FOIL Method.

Apply the distributive property.

$x = 5 (y (y - 1) - 1 (y - 1))$

Apply the distributive property.

$x = 5 (y \cdot y + y \cdot - 1 - 1 (y - 1))$

Apply the distributive property.

$x = 5 (y \cdot y + y \cdot - 1 - 1 y - 1 \cdot - 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$x = 5 (y^{2} + y \cdot - 1 - 1 y - 1 \cdot - 1)$

Move $- 1$ to the left of $y$ .

$x = 5 (y^{2} - 1 \cdot y - 1 y - 1 \cdot - 1)$

Rewrite $- 1 y$ as $- y$ .

$x = 5 (y^{2} - y - 1 y - 1 \cdot - 1)$

Rewrite $- 1 y$ as $- y$ .

$x = 5 (y^{2} - y - y - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

$x = 5 (y^{2} - y - y + 1)$

Subtract $y$ from $- y$ .

$x = 5 (y^{2} - 2 y + 1)$

Apply the distributive property.

$x = 5 y^{2} + 5 (- 2 y) + 5 \cdot 1$

Simplify.

Multiply $- 2$ by $5$ .

$x = 5 y^{2} - 10 y + 5 \cdot 1$

Multiply $5$ by $1$ .

$x = 5 y^{2} - 10 y + 5$

Step 2

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $5 y^{2} - 10 y + 5$ .

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

$a = 5$

$b = - 10$

$c = 5$

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

$d = \frac{- 10}{2 \cdot 5}$

Simplify the right side.

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

Factor $2$ out of $- 10$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 5$ and $5$ .

Factor $5$ out of $- 5$ .

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

Cancel the common factors.

Factor $5$ out of $5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ 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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 5 - \frac{{(- 10)}^{2}}{4 \cdot 5}$

Simplify the right side.

Simplify each term.

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

$e = 5 - \frac{100}{4 \cdot 5}$

Multiply $4$ by $5$ .

$e = 5 - \frac{100}{20}$

Divide $100$ by $20$ .

$e = 5 - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$e = 5 - 5$

Subtract $5$ from $5$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(y - 1)}^{2} + 0$ .

$5 {(y - 1)}^{2} + 0$

Set $x$ equal to the new right side.

$x = 5 {(y - 1)}^{2} + 0$

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

$a = 5$

$h = 0$

$k = 1$

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

Opens Right

Find the vertex $(h, k)$ .

$(0, 1)$

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.

$\frac{1}{4 \cdot 5}$

Multiply $4$ by $5$ .

$\frac{1}{20}$

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.

$(\frac{1}{20}, 1)$

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.

$x = - \frac{1}{20}$

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

Direction: Opens Right

Vertex: $(0, 1)$

Focus: $(\frac{1}{20}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = - \frac{1}{20}$

Direction: Opens Right

Vertex: $(0, 1)$

Focus: $(\frac{1}{20}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = - \frac{1}{20}$

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 $f (x) = \frac{\sqrt{5 x}}{5} + 1$ . In this case, the point is $(1, 1.44721359)$ .

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

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

Simplify the result.

Multiply $5$ by $1$ .

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

The final answer is $\frac{\sqrt{5}}{5} + 1$ .

$y = \frac{\sqrt{5}}{5} + 1$

Convert $\frac{\sqrt{5}}{5} + 1$ to decimal.

$= 1.44721359$

Substitute the $x$ value $1$ into $f (x) = - \frac{\sqrt{5 x}}{5} + 1$ . In this case, the point is $(1, 0.5527864)$ .

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

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

Simplify the result.

Multiply $5$ by $1$ .

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

The final answer is $- \frac{\sqrt{5}}{5} + 1$ .

$y = - \frac{\sqrt{5}}{5} + 1$

Convert $- \frac{\sqrt{5}}{5} + 1$ to decimal.

$= 0.5527864$

Substitute the $x$ value $2$ into $f (x) = \frac{\sqrt{5 x}}{5} + 1$ . In this case, the point is $(2, 1.63245553)$ .

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

$f (2) = \frac{\sqrt{5 (2)}}{5} + 1$

Simplify the result.

Multiply $5$ by $2$ .

$f (2) = \frac{\sqrt{10}}{5} + 1$

The final answer is $\frac{\sqrt{10}}{5} + 1$ .

$y = \frac{\sqrt{10}}{5} + 1$

Convert $\frac{\sqrt{10}}{5} + 1$ to decimal.

$= 1.63245553$

Substitute the $x$ value $2$ into $f (x) = - \frac{\sqrt{5 x}}{5} + 1$ . In this case, the point is $(2, 0.36754446)$ .

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

$f (2) = - \frac{\sqrt{5 (2)}}{5} + 1$

Simplify the result.

Multiply $5$ by $2$ .

$f (2) = - \frac{\sqrt{10}}{5} + 1$

The final answer is $- \frac{\sqrt{10}}{5} + 1$ .

$y = - \frac{\sqrt{10}}{5} + 1$

Convert $- \frac{\sqrt{10}}{5} + 1$ to decimal.

$= 0.36754446$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 1 \\ 1 & 1.45 \\ 1 & 0.55 \\ 2 & 1.63 \\ 2 & 0.37 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(0, 1)$

Focus: $(\frac{1}{20}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = - \frac{1}{20}$

$\begin{array}{c} x & y \\ 0 & 1 \\ 1 & 1.45 \\ 1 & 0.55 \\ 2 & 1.63 \\ 2 & 0.37 \end{array}$

Step 5

Graph x=5(y-1)^2

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