Algebra Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Move $7$ .

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

Step 1.2.2

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

$a = - 1$

$b = 4$

$c = 7$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Factor $2$ out of $4$ .

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

Step 1.2.4.2.1.2

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

$d = - 1 \cdot 2$

Step 1.2.4.2.2

Multiply $- 1$ by $2$ .

$d = - 2$

Step 1.2.5

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

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Step 1.2.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 7 - \frac{4^{2}}{4 \cdot - 1}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

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

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Step 1.2.5.2.1.1.1

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

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

Step 1.2.5.2.1.1.2

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

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

Step 1.2.5.2.1.2

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

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Step 1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 7 - - 4$

Step 1.2.5.2.1.2.2

Multiply $- 1$ by $- 4$ .

$e = 7 + 4$

Step 1.2.5.2.2

Add $7$ and $4$ .

$e = 11$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 2)}^{2} + 11$ .

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = - 1$

$h = 2$

$k = 11$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(2, 11)$

Step 5

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

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

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

Step 7

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

$x = 2$

Step 8