Pre-Algebra Examples

$49 x = - 4 y^{2} + 196$

Step 1

Divide each term in $49 x = - 4 y^{2} + 196$ by $49$ and simplify.

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Step 1.1

Divide each term in $49 x = - 4 y^{2} + 196$ by $49$ .

$\frac{49 x}{49} = \frac{- 4 y^{2}}{49} + \frac{196}{49}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $49$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{49 x}{49} = \frac{- 4 y^{2}}{49} + \frac{196}{49}$

Step 1.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4 y^{2}}{49} + \frac{196}{49}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Move the negative in front of the fraction.

$x = - \frac{4 y^{2}}{49} + \frac{196}{49}$

Step 1.3.1.2

Divide $196$ by $49$ .

$x = - \frac{4 y^{2}}{49} + 4$

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

Complete the square for $- \frac{4 y^{2}}{49} + 4$ .

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Step 2.1.1.1

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

$a = - \frac{4}{49}$

$b = 0$

$c = 4$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$d = \frac{0}{2 (- \frac{4}{49})}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

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

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Step 2.1.1.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{4}{49})}$

Step 2.1.1.3.2.1.2

Cancel the common factors.

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Step 2.1.1.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{4}{49})}$

Step 2.1.1.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{4}{49}}$

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- \frac{49}{4})$

Step 2.1.1.3.2.3

Multiply $0 (- \frac{49}{4})$ .

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Step 2.1.1.3.2.3.1

Multiply $- 1$ by $0$ .

$d = 0 (\frac{49}{4})$

Step 2.1.1.3.2.3.2

Multiply $0$ by $\frac{49}{4}$ .

$d = 0$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

$e = 4 - \frac{0^{2}}{4 (- \frac{4}{49})}$

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

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

$e = 4 - \frac{0}{4 (- \frac{4}{49})}$

Step 2.1.1.4.2.1.2

Simplify the denominator.

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Step 2.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 4 - \frac{0}{- 4 (\frac{4}{49})}$

Step 2.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{4}{49}$ .

$e = 4 - \frac{0}{\frac{- 4 \cdot 4}{49}}$

Step 2.1.1.4.2.1.3

Multiply $- 4$ by $4$ .

$e = 4 - \frac{0}{\frac{- 16}{49}}$

Step 2.1.1.4.2.1.4

Move the negative in front of the fraction.

$e = 4 - \frac{0}{- \frac{16}{49}}$

Step 2.1.1.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = 4 - (0 (- \frac{49}{16}))$

Step 2.1.1.4.2.1.6

Multiply $0 (- \frac{49}{16})$ .

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Step 2.1.1.4.2.1.6.1

Multiply $- 1$ by $0$ .

$e = 4 - (0 (\frac{49}{16}))$

Step 2.1.1.4.2.1.6.2

Multiply $0$ by $\frac{49}{16}$ .

$e = 4 - 0$

Step 2.1.1.4.2.1.7

Multiply $- 1$ by $0$ .

$e = 4 + 0$

Step 2.1.1.4.2.2

Add $4$ and $0$ .

$e = 4$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{4}{49} {(y + 0)}^{2} + 4$ .

$- \frac{4}{49} {(y + 0)}^{2} + 4$

Step 2.1.2

Set $x$ equal to the new right side.

$x = - \frac{4}{49} \cdot {(y + 0)}^{2} + 4$

Step 2.2

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

$a = - \frac{4}{49}$

$h = 4$

$k = 0$

Step 2.3

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

Opens Left

Step 2.4

Find the vertex $(h, k)$ .

$(4, 0)$

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.

$\frac{1}{4 \cdot (- \frac{4}{49})}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

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

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Step 2.5.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{4}{49})}$

Step 2.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{4}{49})}$

Step 2.5.3.2

Combine $4$ and $\frac{4}{49}$ .

$- \frac{1}{\frac{4 \cdot 4}{49}}$

Step 2.5.3.3

Multiply $4$ by $4$ .

$- \frac{1}{\frac{16}{49}}$

Step 2.5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{49}{16}))$

Step 2.5.3.5

Multiply $\frac{49}{16}$ by $1$ .

$- \frac{49}{16}$

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.

$(\frac{15}{16}, 0)$

Step 2.7

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

$y = 0$

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.

$x = \frac{113}{16}$

Step 2.9

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

Direction: Opens Left

Vertex: $(4, 0)$

Focus: $(\frac{15}{16}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{113}{16}$

Direction: Opens Left

Vertex: $(4, 0)$

Focus: $(\frac{15}{16}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{113}{16}$

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 $2$ into $f (x) = \frac{7 \sqrt{- x + 4}}{2}$ . In this case, the point is $(2, 4.94974746)$ .

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Step 3.1.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{7 \sqrt{- (2) + 4}}{2}$

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 $2$ .

$f (2) = \frac{7 \sqrt{- 2 + 4}}{2}$

Step 3.1.2.1.2

Add $- 2$ and $4$ .

$f (2) = \frac{7 \sqrt{2}}{2}$

Step 3.1.2.2

The final answer is $\frac{7 \sqrt{2}}{2}$ .

$y = \frac{7 \sqrt{2}}{2}$

Step 3.1.3

Convert $\frac{7 \sqrt{2}}{2}$ to decimal.

$= 4.94974746$

Step 3.2

Substitute the $x$ value $2$ into $f (x) = - \frac{7 \sqrt{- x + 4}}{2}$ . In this case, the point is $(2, - 4.94974746)$ .

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Step 3.2.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = - \frac{7 \sqrt{- (2) + 4}}{2}$

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 $2$ .

$f (2) = - \frac{7 \sqrt{- 2 + 4}}{2}$

Step 3.2.2.1.2

Add $- 2$ and $4$ .

$f (2) = - \frac{7 \sqrt{2}}{2}$

Step 3.2.2.2

The final answer is $- \frac{7 \sqrt{2}}{2}$ .

$y = - \frac{7 \sqrt{2}}{2}$

Step 3.2.3

Convert $- \frac{7 \sqrt{2}}{2}$ to decimal.

$= - 4.94974746$

Step 3.3

Substitute the $x$ value $3$ into $f (x) = \frac{7 \sqrt{- x + 4}}{2}$ . In this case, the point is $(3, 3.5)$ .

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Step 3.3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{7 \sqrt{- (3) + 4}}{2}$

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 $3$ .

$f (3) = \frac{7 \sqrt{- 3 + 4}}{2}$

Step 3.3.2.1.2

Add $- 3$ and $4$ .

$f (3) = \frac{7 \sqrt{1}}{2}$

Step 3.3.2.1.3

Any root of $1$ is $1$ .

$f (3) = \frac{7 \cdot 1}{2}$

Step 3.3.2.2

Multiply $7$ by $1$ .

$f (3) = \frac{7}{2}$

Step 3.3.2.3

The final answer is $\frac{7}{2}$ .

$y = \frac{7}{2}$

Step 3.3.3

Convert $\frac{7}{2}$ to decimal.

$= 3.5$

Step 3.4

Substitute the $x$ value $3$ into $f (x) = - \frac{7 \sqrt{- x + 4}}{2}$ . In this case, the point is $(3, - 3.5)$ .

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Step 3.4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = - \frac{7 \sqrt{- (3) + 4}}{2}$

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 $3$ .

$f (3) = - \frac{7 \sqrt{- 3 + 4}}{2}$

Step 3.4.2.1.2

Add $- 3$ and $4$ .

$f (3) = - \frac{7 \sqrt{1}}{2}$

Step 3.4.2.1.3

Any root of $1$ is $1$ .

$f (3) = - \frac{7 \cdot 1}{2}$

Step 3.4.2.2

Multiply $7$ by $1$ .

$f (3) = - \frac{7}{2}$

Step 3.4.2.3

The final answer is $- \frac{7}{2}$ .

$y = - \frac{7}{2}$

Step 3.4.3

Convert $- \frac{7}{2}$ to decimal.

$= - 3.5$

Step 3.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & 4.95 \\ 2 & - 4.95 \\ 3 & 3.5 \\ 3 & - 3.5 \\ 4 & 0 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(4, 0)$

Focus: $(\frac{15}{16}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{113}{16}$

$\begin{array}{c} x & y \\ 2 & 4.95 \\ 2 & - 4.95 \\ 3 & 3.5 \\ 3 & - 3.5 \\ 4 & 0 \end{array}$

Step 5