Find the Axis of Symmetry y=1/3x^2-4/3x+1/3

$y = \frac{1}{3} x^{2} - \frac{4}{3} x + \frac{1}{3}$

Step 1

Rewrite the equation in vertex form.

Simplify each term.

Combine $\frac{1}{3}$ and $x^{2}$ .

$y = \frac{x^{2}}{3} - \frac{4}{3} x + \frac{1}{3}$

Combine $x$ and $\frac{4}{3}$ .

$y = \frac{x^{2}}{3} - \frac{x \cdot 4}{3} + \frac{1}{3}$

Move $4$ to the left of $x$ .

$y = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$

Complete the square for $\frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$ .

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

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

$b = - \frac{4}{3}$

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

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

$d = \frac{- \frac{4}{3}}{2 (\frac{1}{3})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{4}{3} \cdot \frac{1}{2 (\frac{1}{3})}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$d = \frac{- 4}{3} \cdot \frac{1}{2 (\frac{1}{3})}$

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{- 2}{3}$ by $\frac{1}{\frac{1}{3}}$ .

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

Combine $3$ and $\frac{1}{3}$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

$e = \frac{1}{3} - \frac{{(- \frac{4}{3})}^{2}}{4 (\frac{1}{3})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{4}{3}$ .

$e = \frac{1}{3} - \frac{{(- 1)}^{2} {(\frac{4}{3})}^{2}}{4 (\frac{1}{3})}$

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

$e = \frac{1}{3} - \frac{1 {(\frac{4}{3})}^{2}}{4 (\frac{1}{3})}$

Apply the product rule to $\frac{4}{3}$ .

$e = \frac{1}{3} - \frac{1 \frac{4^{2}}{3^{2}}}{4 (\frac{1}{3})}$

Raise $4$ to the power of $2$ .

$e = \frac{1}{3} - \frac{1 \frac{16}{3^{2}}}{4 (\frac{1}{3})}$

Raise $3$ to the power of $2$ .

$e = \frac{1}{3} - \frac{1 (\frac{16}{9})}{4 (\frac{1}{3})}$

Multiply $\frac{16}{9}$ by $1$ .

$e = \frac{1}{3} - \frac{\frac{16}{9}}{4 (\frac{1}{3})}$

Combine $4$ and $\frac{1}{3}$ .

$e = \frac{1}{3} - \frac{\frac{16}{9}}{\frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{3} - (\frac{16}{9} \cdot \frac{3}{4})$

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

$e = \frac{1}{3} - (\frac{4 (4)}{9} \cdot \frac{3}{4})$

Cancel the common factor.

$e = \frac{1}{3} - (\frac{4 \cdot 4}{9} \cdot \frac{3}{4})$

Rewrite the expression.

$e = \frac{1}{3} - (\frac{4}{9} \cdot 3)$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$e = \frac{1}{3} - (\frac{4}{3 (3)} \cdot 3)$

Cancel the common factor.

$e = \frac{1}{3} - (\frac{4}{3 \cdot 3} \cdot 3)$

Rewrite the expression.

$e = \frac{1}{3} - \frac{4}{3}$

Combine the numerators over the common denominator.

$e = \frac{1 - 4}{3}$

Subtract $4$ from $1$ .

$e = \frac{- 3}{3}$

Divide $- 3$ by $3$ .

$e = - 1$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 2$

$k = - 1$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(2, - 1)$

Step 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.

$\frac{1}{4 \cdot \frac{1}{3}}$

Simplify.

Combine $4$ and $\frac{1}{3}$ .

$\frac{1}{\frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{3}{4}$ by $1$ .

$\frac{3}{4}$

Step 6

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.

$(2, - \frac{1}{4})$

Step 7

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

$x = 2$

Step 8

Find the Axis of Symmetry y=1/3x^2-4/3x+1/3

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