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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Name

Name	one billion six hundred twenty-seven million three hundred thirty-eight thousand three hundred thirty-five

Interesting facts

1627338335 has 16 divisors, whose sum is 1787443200
The reverse of 1627338335 is 5338337261
Previous prime number is 31

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	one billion one hundred twenty-five million thirty-six thousand forty-four

Interesting facts

1125036044 has 32 divisors, whose sum is 2762548416
The reverse of 1125036044 is 4406305211
Previous prime number is 4643

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 26
Digital Root 8

Name

Name	one billion seven hundred forty-three million one hundred seventy-four thousand four hundred ninety-two

Interesting facts

1743174492 has 16 divisors, whose sum is 3999047832
The reverse of 1743174492 is 2944713471
Previous prime number is 51

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 42
Digital Root 6

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