Trigonometry Examples

$y^{2} - 4 y - x + 3 = 0$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

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

$- 4 y - x + 3 = - y^{2}$

Step 1.2

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

$- x + 3 = - y^{2} + 4 y$

Step 1.3

Subtract $3$ from both sides of the equation.

$- x = - y^{2} + 4 y - 3$

Step 2

Divide each term in $- x = - y^{2} + 4 y - 3$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- x = - y^{2} + 4 y - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- y^{2}}{- 1} + \frac{4 y}{- 1} + \frac{- 3}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- y^{2}}{- 1} + \frac{4 y}{- 1} + \frac{- 3}{- 1}$

Step 2.2.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{- 1} + \frac{4 y}{- 1} + \frac{- 3}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{1} + \frac{4 y}{- 1} + \frac{- 3}{- 1}$

Step 2.3.1.2

Divide $y^{2}$ by $1$ .

$x = y^{2} + \frac{4 y}{- 1} + \frac{- 3}{- 1}$

Step 2.3.1.3

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

$x = y^{2} - 1 \cdot (4 y) + \frac{- 3}{- 1}$

Step 2.3.1.4

Rewrite $- 1 \cdot (4 y)$ as $- (4 y)$ .

$x = y^{2} - (4 y) + \frac{- 3}{- 1}$

Step 2.3.1.5

Multiply $4$ by $- 1$ .

$x = y^{2} - 4 y + \frac{- 3}{- 1}$

Step 2.3.1.6

Divide $- 3$ by $- 1$ .

$x = y^{2} - 4 y + 3$

Step 3

Find the properties of the given parabola.

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Step 3.1

Rewrite the equation in vertex form.

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Step 3.1.1

Complete the square for $y^{2} - 4 y + 3$ .

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Step 3.1.1.1

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

$a = 1$

$b = - 4$

$c = 3$

Step 3.1.1.2

Consider the vertex form of a parabola.

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

Step 3.1.1.3

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

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Step 3.1.1.3.1

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

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

Step 3.1.1.3.2

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

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Step 3.1.1.3.2.1

Factor $2$ out of $- 4$ .

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

Step 3.1.1.3.2.2

Cancel the common factors.

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Step 3.1.1.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 3.1.1.3.2.2.2

Cancel the common factor.

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

Step 3.1.1.3.2.2.3

Rewrite the expression.

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

Step 3.1.1.3.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 3.1.1.4

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

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Step 3.1.1.4.1

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

$e = 3 - \frac{{(- 4)}^{2}}{4 \cdot 1}$

Step 3.1.1.4.2

Simplify the right side.

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Step 3.1.1.4.2.1

Simplify each term.

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Step 3.1.1.4.2.1.1

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

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Step 3.1.1.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 3 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}$

Step 3.1.1.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 3 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}$

Step 3.1.1.4.2.1.1.3

Raise $- 1$ to the power of $2$ .

$e = 3 - \frac{1 \cdot 4^{2}}{4 \cdot 1}$

Step 3.1.1.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 3 - \frac{4^{2}}{4 \cdot 1}$

Step 3.1.1.4.2.1.1.5

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

$e = 3 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 3.1.1.4.2.1.1.6

Cancel the common factors.

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Step 3.1.1.4.2.1.1.6.1

Factor $4$ out of $4 \cdot 1$ .

$e = 3 - \frac{4 \cdot 4}{4 (1)}$

Step 3.1.1.4.2.1.1.6.2

Cancel the common factor.

$e = 3 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 3.1.1.4.2.1.1.6.3

Rewrite the expression.

$e = 3 - \frac{4}{1}$

Step 3.1.1.4.2.1.1.6.4

Divide $4$ by $1$ .

$e = 3 - 1 \cdot 4$

Step 3.1.1.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 3 - 4$

Step 3.1.1.4.2.2

Subtract $4$ from $3$ .

$e = - 1$

Step 3.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 2)}^{2} - 1$ .

${(y - 2)}^{2} - 1$

Step 3.1.2

Set $x$ equal to the new right side.

$x = {(y - 2)}^{2} - 1$

Step 3.2

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

$a = 1$

$h = - 1$

$k = 2$

Step 3.3

Since the value of $a$ is positive, the parabola opens right.

Opens Right

Step 3.4

Find the vertex $(h, k)$ .

$(- 1, 2)$

Step 3.5

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

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Step 3.5.1

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

$\frac{1}{4 a}$

Step 3.5.2

Substitute the value of $a$ into the formula.

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

Step 3.5.3

Cancel the common factor of $1$ .

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Step 3.5.3.1

Cancel the common factor.

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

Step 3.5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 3.6

Find the focus.

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Step 3.6.1

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 3.6.2

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

$(- \frac{3}{4}, 2)$

Step 3.7

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

$y = 2$

Step 3.8

Find the directrix.

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Step 3.8.1

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 3.8.2

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

$x = - \frac{5}{4}$

Step 3.9

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

Direction: Opens Right

Vertex: $(- 1, 2)$

Focus: $(- \frac{3}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{5}{4}$

Direction: Opens Right

Vertex: $(- 1, 2)$

Focus: $(- \frac{3}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{5}{4}$

Step 4

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 4.1

Substitute the $x$ value $0$ into $f (x) = 2 + \sqrt{- 1 (- x - 1)}$ . In this case, the point is $(0, 3)$ .

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Step 4.1.1

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

$f (0) = 2 + \sqrt{- 1 (- (0) - 1)}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Multiply $- 1$ by $0$ .

$f (0) = 2 + \sqrt{- 1 (0 - 1)}$

Step 4.1.2.1.2

Subtract $1$ from $0$ .

$f (0) = 2 + \sqrt{- 1 \cdot - 1}$

Step 4.1.2.1.3

Multiply $- 1$ by $- 1$ .

$f (0) = 2 + \sqrt{1}$

Step 4.1.2.1.4

Any root of $1$ is $1$ .

$f (0) = 2 + 1$

Step 4.1.2.2

Add $2$ and $1$ .

$f (0) = 3$

Step 4.1.2.3

The final answer is $3$ .

$y = 3$

Step 4.1.3

Convert $3$ to decimal.

$= 3$

Step 4.2

Substitute the $x$ value $0$ into $f (x) = 2 - \sqrt{- 1 (- x - 1)}$ . In this case, the point is $(0, 1)$ .

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Step 4.2.1

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

$f (0) = 2 - \sqrt{- 1 (- (0) - 1)}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $- 1$ by $0$ .

$f (0) = 2 - \sqrt{- 1 (0 - 1)}$

Step 4.2.2.1.2

Subtract $1$ from $0$ .

$f (0) = 2 - \sqrt{- 1 \cdot - 1}$

Step 4.2.2.1.3

Multiply $- 1$ by $- 1$ .

$f (0) = 2 - \sqrt{1}$

Step 4.2.2.1.4

Any root of $1$ is $1$ .

$f (0) = 2 - 1 \cdot 1$

Step 4.2.2.1.5

Multiply $- 1$ by $1$ .

$f (0) = 2 - 1$

Step 4.2.2.2

Subtract $1$ from $2$ .

$f (0) = 1$

Step 4.2.2.3

The final answer is $1$ .

$y = 1$

Step 4.2.3

Convert $1$ to decimal.

$= 1$

Step 4.3

Substitute the $x$ value $1$ into $f (x) = 2 + \sqrt{- 1 (- x - 1)}$ . In this case, the point is $(1, 3.41421356)$ .

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Step 4.3.1

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

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

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Multiply $- 1$ by $1$ .

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

Step 4.3.2.1.2

Subtract $1$ from $- 1$ .

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

Step 4.3.2.1.3

Multiply $- 1$ by $- 2$ .

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

Step 4.3.2.2

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

$y = 2 + \sqrt{2}$

Step 4.3.3

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

$= 3.41421356$

Step 4.4

Substitute the $x$ value $1$ into $f (x) = 2 - \sqrt{- 1 (- x - 1)}$ . In this case, the point is $(1, 0.58578643)$ .

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Step 4.4.1

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

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

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Multiply $- 1$ by $1$ .

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

Step 4.4.2.1.2

Subtract $1$ from $- 1$ .

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

Step 4.4.2.1.3

Multiply $- 1$ by $- 2$ .

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

Step 4.4.2.2

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

$y = 2 - \sqrt{2}$

Step 4.4.3

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

$= 0.58578643$

Step 4.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 2 \\ 0 & 3 \\ 0 & 1 \\ 1 & 3.41 \\ 1 & 0.59 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 1, 2)$

Focus: $(- \frac{3}{4}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{5}{4}$

$\begin{array}{c} x & y \\ - 1 & 2 \\ 0 & 3 \\ 0 & 1 \\ 1 & 3.41 \\ 1 & 0.59 \end{array}$

Step 6