Graph y=(3x-1)^2

$y = {(3 x - 1)}^{2}$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for ${(3 x - 1)}^{2}$ .

Simplify the expression.

Rewrite ${(3 x - 1)}^{2}$ as $(3 x - 1) (3 x - 1)$ .

$(3 x - 1) (3 x - 1)$

Expand $(3 x - 1) (3 x - 1)$ using the FOIL Method.

Apply the distributive property.

$3 x (3 x - 1) - 1 (3 x - 1)$

Apply the distributive property.

$3 x (3 x) + 3 x \cdot - 1 - 1 (3 x - 1)$

Apply the distributive property.

$3 x (3 x) + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 \cdot 3 x \cdot x + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$3 \cdot 3 (x \cdot x) + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

Multiply $x$ by $x$ .

$3 \cdot 3 x^{2} + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

Multiply $3$ by $3$ .

$9 x^{2} + 3 x \cdot - 1 - 1 (3 x) - 1 \cdot - 1$

Multiply $- 1$ by $3$ .

$9 x^{2} - 3 x - 1 (3 x) - 1 \cdot - 1$

Multiply $3$ by $- 1$ .

$9 x^{2} - 3 x - 3 x - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$9 x^{2} - 3 x - 3 x + 1$

Subtract $3 x$ from $- 3 x$ .

$9 x^{2} - 6 x + 1$

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

$a = 9$

$b = - 6$

$c = 1$

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

$d = \frac{- 6}{2 \cdot 9}$

Simplify the right side.

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

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 3}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{- 3}{9}$

Cancel the common factor of $- 3$ and $9$ .

Factor $3$ out of $- 3$ .

$d = \frac{3 (- 1)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $9$ .

$e = 1 - \frac{36}{36}$

Cancel the common factor of $36$ .

Cancel the common factor.

$e = 1 - \frac{36}{36}$

Rewrite the expression.

$e = 1 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = 1 - 1$

Subtract $1$ from $1$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - \frac{1}{3})}^{2} + 0$ .

$9 {(x - \frac{1}{3})}^{2} + 0$

Set $y$ equal to the new right side.

$y = 9 {(x - \frac{1}{3})}^{2} + 0$

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

$a = 9$

$h = \frac{1}{3}$

$k = 0$

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

Opens Up

Find the vertex $(h, k)$ .

$(\frac{1}{3}, 0)$

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.

$\frac{1}{4 \cdot 9}$

Multiply $4$ by $9$ .

$\frac{1}{36}$

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{1}{3}, \frac{1}{36})$

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

$x = \frac{1}{3}$

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = - \frac{1}{36}$

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

Direction: Opens Up

Vertex: $(\frac{1}{3}, 0)$

Focus: $(\frac{1}{3}, \frac{1}{36})$

Axis of Symmetry: $x = \frac{1}{3}$

Directrix: $y = - \frac{1}{36}$

Direction: Opens Up

Vertex: $(\frac{1}{3}, 0)$

Focus: $(\frac{1}{3}, \frac{1}{36})$

Axis of Symmetry: $x = \frac{1}{3}$

Directrix: $y = - \frac{1}{36}$

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.

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

$f (- 1) = 9 {(- 1)}^{2} - 6 \cdot - 1 + 1$

Simplify the result.

Simplify each term.

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

$f (- 1) = 9 \cdot 1 - 6 \cdot - 1 + 1$

Multiply $9$ by $1$ .

$f (- 1) = 9 - 6 \cdot - 1 + 1$

Multiply $- 6$ by $- 1$ .

$f (- 1) = 9 + 6 + 1$

Simplify by adding numbers.

Add $9$ and $6$ .

$f (- 1) = 15 + 1$

Add $15$ and $1$ .

$f (- 1) = 16$

The final answer is $16$ .

$16$

The $y$ value at $x = - 1$ is $16$ .

$y = 16$

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

$f (1) = 9 {(1)}^{2} - 6 \cdot 1 + 1$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 9 \cdot 1 - 6 \cdot 1 + 1$

Multiply $9$ by $1$ .

$f (1) = 9 - 6 \cdot 1 + 1$

Multiply $- 6$ by $1$ .

$f (1) = 9 - 6 + 1$

Simplify by adding and subtracting.

Subtract $6$ from $9$ .

$f (1) = 3 + 1$

Add $3$ and $1$ .

$f (1) = 4$

The final answer is $4$ .

$4$

The $y$ value at $x = 1$ is $4$ .

$y = 4$

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

$f (2) = 9 {(2)}^{2} - 6 \cdot 2 + 1$

Simplify the result.

Simplify each term.

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

$f (2) = 9 \cdot 4 - 6 \cdot 2 + 1$

Multiply $9$ by $4$ .

$f (2) = 36 - 6 \cdot 2 + 1$

Multiply $- 6$ by $2$ .

$f (2) = 36 - 12 + 1$

Simplify by adding and subtracting.

Subtract $12$ from $36$ .

$f (2) = 24 + 1$

Add $24$ and $1$ .

$f (2) = 25$

The final answer is $25$ .

$25$

The $y$ value at $x = 2$ is $25$ .

$y = 25$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 16 \\ \frac{1}{3} & 0 \\ 1 & 4 \\ 2 & 25 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{1}{3}, 0)$

Focus: $(\frac{1}{3}, \frac{1}{36})$

Axis of Symmetry: $x = \frac{1}{3}$

Directrix: $y = - \frac{1}{36}$

$\begin{array}{c} x & y \\ - 1 & 16 \\ \frac{1}{3} & 0 \\ 1 & 4 \\ 2 & 25 \end{array}$

Step 4

Graph y=(3x-1)^2

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