Find the Axis of Symmetry f(x)=1/4x^2-2x+2

$f (x) = \frac{1}{4} x^{2} - 2 x + 2$

Step 1

Write $f (x) = \frac{1}{4} x^{2} - 2 x + 2$ as an equation.

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

Step 2

Rewrite the equation in vertex form.

Combine $\frac{1}{4}$ and $x^{2}$ .

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

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

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

$a = \frac{1}{4}$

$b = - 2$

$c = 2$

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

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

Simplify the right side.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$d = - 4$

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

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

Simplify the right side.

Simplify each term.

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

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

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

$e = 2 - \frac{4}{\frac{4}{4}}$

Divide $4$ by $4$ .

$e = 2 - \frac{4}{1}$

Divide $4$ by $1$ .

$e = 2 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 2 - 4$

Subtract $4$ from $2$ .

$e = - 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(x - 4)}^{2} - 2$ .

$\frac{1}{4} {(x - 4)}^{2} - 2$

Set $y$ equal to the new right side.

$y = \frac{1}{4} \cdot {(x - 4)}^{2} - 2$

Step 3

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

$a = \frac{1}{4}$

$h = 4$

$k = - 2$

Step 4

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

Opens Up

Step 5

Find the vertex $(h, k)$ .

$(4, - 2)$

Step 6

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.

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

Simplify.

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

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

Simplify by dividing numbers.

Divide $4$ by $4$ .

$\frac{1}{1}$

Divide $1$ by $1$ .

$1$

Step 7

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.

$(4, - 1)$

Step 8

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

$x = 4$

Step 9

Find the Axis of Symmetry f(x)=1/4x^2-2x+2

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