Algebra Examples

$4 y^{2} + x = 56 y - 194$

Step 1

Subtract $4 y^{2}$ from both sides of the equation.

$x = 56 y - 194 - 4 y^{2}$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Simplify $56 y - 194 - 4 y^{2}$ .

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Step 2.1.1.1

Move $- 194$ .

$x = 56 y - 4 y^{2} - 194$

Step 2.1.1.2

Reorder $56 y$ and $- 4 y^{2}$ .

$x = - 4 y^{2} + 56 y - 194$

Step 2.1.2

Complete the square for $- 4 y^{2} + 56 y - 194$ .

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Step 2.1.2.1

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

$a = - 4$

$b = 56$

$c = - 194$

Step 2.1.2.2

Consider the vertex form of a parabola.

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

Step 2.1.2.3

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

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Step 2.1.2.3.1

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

$d = \frac{56}{2 \cdot - 4}$

Step 2.1.2.3.2

Simplify the right side.

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Step 2.1.2.3.2.1

Cancel the common factor of $56$ and $2$ .

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Step 2.1.2.3.2.1.1

Factor $2$ out of $56$ .

$d = \frac{2 \cdot 28}{2 \cdot - 4}$

Step 2.1.2.3.2.1.2

Cancel the common factors.

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Step 2.1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot 28}{2 (- 4)}$

Step 2.1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 28}{2 \cdot - 4}$

Step 2.1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{28}{- 4}$

Step 2.1.2.3.2.2

Cancel the common factor of $28$ and $- 4$ .

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Step 2.1.2.3.2.2.1

Factor $4$ out of $28$ .

$d = \frac{4 (7)}{- 4}$

Step 2.1.2.3.2.2.2

Move the negative one from the denominator of $\frac{7}{- 1}$ .

$d = - 1 \cdot 7$

Step 2.1.2.3.2.3

Multiply $- 1$ by $7$ .

$d = - 7$

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

$e = - 194 - \frac{56^{2}}{4 \cdot - 4}$

Step 2.1.2.4.2

Simplify the right side.

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Step 2.1.2.4.2.1

Simplify each term.

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Step 2.1.2.4.2.1.1

Raise $56$ to the power of $2$ .

$e = - 194 - \frac{3136}{4 \cdot - 4}$

Step 2.1.2.4.2.1.2

Multiply $4$ by $- 4$ .

$e = - 194 - \frac{3136}{- 16}$

Step 2.1.2.4.2.1.3

Divide $3136$ by $- 16$ .

$e = - 194 - - 196$

Step 2.1.2.4.2.1.4

Multiply $- 1$ by $- 196$ .

$e = - 194 + 196$

Step 2.1.2.4.2.2

Add $- 194$ and $196$ .

$e = 2$

Step 2.1.2.5

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

$- 4 {(y - 7)}^{2} + 2$

Step 2.1.3

Set $x$ equal to the new right side.

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

Step 2.2

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

$a = - 4$

$h = 2$

$k = 7$

Step 2.3

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

Opens Left

Step 2.4

Find the vertex $(h, k)$ .

$(2, 7)$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Simplify.

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Step 2.5.3.1

Multiply $4$ by $- 4$ .

$\frac{1}{- 16}$

Step 2.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{16}$

Step 2.6

Find the focus.

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Step 2.6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 2.6.2

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

$(\frac{31}{16}, 7)$

Step 2.7

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

$y = 7$

Step 2.8

Find the directrix.

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Step 2.8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 2.8.2

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

$x = \frac{33}{16}$

Step 2.9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Left

Vertex: $(2, 7)$

Focus: $(\frac{31}{16}, 7)$

Axis of Symmetry: $y = 7$

Directrix: $x = \frac{33}{16}$

Direction: Opens Left

Vertex: $(2, 7)$

Focus: $(\frac{31}{16}, 7)$

Axis of Symmetry: $y = 7$

Directrix: $x = \frac{33}{16}$

Step 3

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 3.1

Substitute the $x$ value $0$ into $f (x) = \frac{14 + \sqrt{- x + 2}}{2}$ . In this case, the point is $(0, 7.70710678)$ .

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Step 3.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{14 + \sqrt{- (0) + 2}}{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify the numerator.

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Step 3.1.2.1.1

Multiply $- 1$ by $0$ .

$f (0) = \frac{14 + \sqrt{0 + 2}}{2}$

Step 3.1.2.1.2

Add $0$ and $2$ .

$f (0) = \frac{14 + \sqrt{2}}{2}$

Step 3.1.2.2

The final answer is $\frac{14 + \sqrt{2}}{2}$ .

$y = \frac{14 + \sqrt{2}}{2}$

Step 3.1.3

Convert $\frac{14 + \sqrt{2}}{2}$ to decimal.

$= 7.70710678$

Step 3.2

Substitute the $x$ value $0$ into $f (x) = \frac{14 - \sqrt{- x + 2}}{2}$ . In this case, the point is $(0, 6.29289321)$ .

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Step 3.2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{14 - \sqrt{- (0) + 2}}{2}$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify the numerator.

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Step 3.2.2.1.1

Multiply $- 1$ by $0$ .

$f (0) = \frac{14 - \sqrt{0 + 2}}{2}$

Step 3.2.2.1.2

Add $0$ and $2$ .

$f (0) = \frac{14 - \sqrt{2}}{2}$

Step 3.2.2.2

The final answer is $\frac{14 - \sqrt{2}}{2}$ .

$y = \frac{14 - \sqrt{2}}{2}$

Step 3.2.3

Convert $\frac{14 - \sqrt{2}}{2}$ to decimal.

$= 6.29289321$

Step 3.3

Substitute the $x$ value $1$ into $f (x) = \frac{14 + \sqrt{- x + 2}}{2}$ . In this case, the point is $(1, 7.5)$ .

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Step 3.3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{14 + \sqrt{- (1) + 2}}{2}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify the numerator.

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Step 3.3.2.1.1

Multiply $- 1$ by $1$ .

$f (1) = \frac{14 + \sqrt{- 1 + 2}}{2}$

Step 3.3.2.1.2

Add $- 1$ and $2$ .

$f (1) = \frac{14 + \sqrt{1}}{2}$

Step 3.3.2.1.3

Any root of $1$ is $1$ .

$f (1) = \frac{14 + 1}{2}$

Step 3.3.2.1.4

Add $14$ and $1$ .

$f (1) = \frac{15}{2}$

Step 3.3.2.2

The final answer is $\frac{15}{2}$ .

$y = \frac{15}{2}$

Step 3.3.3

Convert $\frac{15}{2}$ to decimal.

$= 7.5$

Step 3.4

Substitute the $x$ value $1$ into $f (x) = \frac{14 - \sqrt{- x + 2}}{2}$ . In this case, the point is $(1, 6.5)$ .

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Step 3.4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{14 - \sqrt{- (1) + 2}}{2}$

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Simplify the numerator.

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Step 3.4.2.1.1

Multiply $- 1$ by $1$ .

$f (1) = \frac{14 - \sqrt{- 1 + 2}}{2}$

Step 3.4.2.1.2

Add $- 1$ and $2$ .

$f (1) = \frac{14 - \sqrt{1}}{2}$

Step 3.4.2.1.3

Any root of $1$ is $1$ .

$f (1) = \frac{14 - 1 \cdot 1}{2}$

Step 3.4.2.1.4

Multiply $- 1$ by $1$ .

$f (1) = \frac{14 - 1}{2}$

Step 3.4.2.1.5

Subtract $1$ from $14$ .

$f (1) = \frac{13}{2}$

Step 3.4.2.2

The final answer is $\frac{13}{2}$ .

$y = \frac{13}{2}$

Step 3.4.3

Convert $\frac{13}{2}$ to decimal.

$= 6.5$

Step 3.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 7.71 \\ 0 & 6.29 \\ 1 & 7.5 \\ 1 & 6.5 \\ 2 & 7 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(2, 7)$

Focus: $(\frac{31}{16}, 7)$

Axis of Symmetry: $y = 7$

Directrix: $x = \frac{33}{16}$

$\begin{array}{c} x & y \\ 0 & 7.71 \\ 0 & 6.29 \\ 1 & 7.5 \\ 1 & 6.5 \\ 2 & 7 \end{array}$

Step 5