Find the Axis of Symmetry y-4x=7-x^2

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Algebra

$y - 4 x = 7 - x^{2}$

Rewrite the equation in vertex form.

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Add $4 x$ to both sides of the equation.

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

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

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Move $7$ .

$- x^{2} + 4 x + 7$

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

$a = - 1, b = 4, c = 7$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Simplify the right side.

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Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4$ .

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

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Simplify each term.

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Cancel the common factor of ${(4)}^{2}$ and $4$ .

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Factor $4$ out of ${(4)}^{2}$ .

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

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$e = 7 - (- 1 \cdot 4)$

Multiply $- 1$ by $4$ .

$e = 7 + 4$

Multiply $- 1$ by $- 4$ .

$e = 7 + 4$

Add $7$ and $4$ .

$e = 11$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $a {(x + d)}^{2} + e$ .

$- {(x - 2)}^{2} + 11$

Set $y$ equal to the new right side.

$y = - {(x - 2)}^{2} + 11$

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

$a = - 1$

$h = 2$

$k = 11$

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

Opens Down

Find the vertex $(h, k)$ .

$(2, 11)$

Find $p$ , the distance from the vertex to the focus.

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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.

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

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

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

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

Move the negative in front of the fraction.

$- \frac{1}{4}$

Find the focus.

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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.

$(2, \frac{43}{4})$

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

$x = 2$

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