Find the Axis of Symmetry f(x)=-1/4x^2+7

$f (x) = - \frac{1}{4} x^{2} + 7$

Step 1

Write $f (x) = - \frac{1}{4} x^{2} + 7$ as an equation.

$y = - \frac{1}{4} x^{2} + 7$

Step 2

Rewrite the equation in vertex form.

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

$y = - \frac{x^{2}}{4} + 7$

Complete the square for $- \frac{x^{2}}{4} + 7$ .

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

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

$b = 0$

$c = 7$

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

$d = \frac{0}{2 (- \frac{1}{4})}$

Simplify the right side.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{4})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{4})}$

Rewrite the expression.

$d = \frac{0}{- \frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 4)$

Multiply $0 (- 1 \cdot 4)$ .

Multiply $- 1$ by $4$ .

$d = 0 \cdot - 4$

Multiply $0$ by $- 4$ .

$d = 0$

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

$e = 7 - \frac{0^{2}}{4 (- \frac{1}{4})}$

Simplify the right side.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$e = 7 - \frac{0}{4 (- \frac{1}{4})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 7 - \frac{0}{- 4 (\frac{1}{4})}$

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

$e = 7 - \frac{0}{\frac{- 4}{4}}$

Divide $- 4$ by $4$ .

$e = 7 - \frac{0}{- 1}$

Divide $0$ by $- 1$ .

$e = 7 - 0$

Multiply $- 1$ by $0$ .

$e = 7 + 0$

Add $7$ and $0$ .

$e = 7$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(x + 0)}^{2} + 7$ .

$- \frac{1}{4} {(x + 0)}^{2} + 7$

Set $y$ equal to the new right side.

$y = - \frac{1}{4} \cdot {(x + 0)}^{2} + 7$

Step 3

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

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

$h = 0$

$k = 7$

Step 4

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(0, 7)$

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.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{4})}$

Move the negative in front of the fraction.

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

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

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

Divide $4$ by $4$ .

$- \frac{1}{1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$- \frac{1}{1}$

Rewrite the expression.

$- 1 \cdot 1$

Multiply $- 1$ 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.

$(0, 6)$

Step 8

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

$x = 0$

Step 9

Find the Axis of Symmetry f(x)=-1/4x^2+7

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