Calculus Examples

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

Step 1

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

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

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 $- y^{2} + 2 y$ .

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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 $c$ .

$a = - 1$

$b = 2$

$c = 0$

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 $d = \frac{b}{2 a}$ .

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

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 $2$ .

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Step 2.1.1.3.2.1.1

Cancel the common factor.

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

Step 2.1.1.3.2.1.2

Rewrite the expression.

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

Step 2.1.1.3.2.1.3

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

$d = - 1 \cdot 1$

Step 2.1.1.3.2.2

Multiply $- 1$ by $1$ .

$d = - 1$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{2^{2}}{4 \cdot - 1}$

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

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

$e = 0 - \frac{4}{4 \cdot - 1}$

Step 2.1.1.4.2.1.2

Multiply $4$ by $- 1$ .

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

Step 2.1.1.4.2.1.3

Divide $4$ by $- 4$ .

$e = 0 - - 1$

Step 2.1.1.4.2.1.4

Multiply $- 1$ by $- 1$ .

$e = 0 + 1$

Step 2.1.1.4.2.2

Add $0$ and $1$ .

$e = 1$

Step 2.1.1.5

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

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

Step 2.1.2

Set $x$ equal to the new right side.

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

Step 2.2

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

$a = - 1$

$h = 1$

$k = 1$

Step 2.3

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

Opens Left

Step 2.4

Find the vertex $(h, k)$ .

$(1, 1)$

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.

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

Step 2.5.3

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

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Step 2.5.3.1

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

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

Step 2.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

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.

$(\frac{3}{4}, 1)$

Step 2.7

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

$y = 1$

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.

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

Step 2.9

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

Direction: Opens Left

Vertex: $(1, 1)$

Focus: $(\frac{3}{4}, 1)$

Axis of Symmetry: $y = 1$

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

Direction: Opens Left

Vertex: $(1, 1)$

Focus: $(\frac{3}{4}, 1)$

Axis of Symmetry: $y = 1$

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

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 $- 1$ into $f (x) = 1 + \sqrt{1 - x}$ . In this case, the point is $(- 1, 2.41421356)$ .

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Step 3.1.1

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

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.2.1.2

Add $1$ and $1$ .

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

Step 3.1.2.2

The final answer is $1 + \sqrt{2}$ .

$y = 1 + \sqrt{2}$

Step 3.1.3

Convert $1 + \sqrt{2}$ to decimal.

$= 2.41421356$

Step 3.2

Substitute the $x$ value $- 1$ into $f (x) = 1 - \sqrt{1 - x}$ . In this case, the point is $(- 1, - 0.41421356)$ .

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Step 3.2.1

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

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

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2.1.2

Add $1$ and $1$ .

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

Step 3.2.2.2

The final answer is $1 - \sqrt{2}$ .

$y = 1 - \sqrt{2}$

Step 3.2.3

Convert $1 - \sqrt{2}$ to decimal.

$= - 0.41421356$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Multiply $- 1$ by $0$ .

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

Step 3.3.2.1.2

Add $1$ and $0$ .

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

Step 3.3.2.1.3

Any root of $1$ is $1$ .

$f (0) = 1 + 1$

Step 3.3.2.2

Add $1$ and $1$ .

$f (0) = 2$

Step 3.3.2.3

The final answer is $2$ .

$y = 2$

Step 3.3.3

Convert $2$ to decimal.

$= 2$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Multiply $- 1$ by $0$ .

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

Step 3.4.2.1.2

Add $1$ and $0$ .

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

Step 3.4.2.1.3

Any root of $1$ is $1$ .

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

Step 3.4.2.1.4

Multiply $- 1$ by $1$ .

$f (0) = 1 - 1$

Step 3.4.2.2

Subtract $1$ from $1$ .

$f (0) = 0$

Step 3.4.2.3

The final answer is $0$ .

$y = 0$

Step 3.4.3

Convert $0$ to decimal.

$= 0$

Step 3.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & 2.41 \\ - 1 & - 0.41 \\ 0 & 2 \\ 0 & 0 \\ 1 & 1 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(1, 1)$

Focus: $(\frac{3}{4}, 1)$

Axis of Symmetry: $y = 1$

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

$\begin{array}{c} x & y \\ - 1 & 2.41 \\ - 1 & - 0.41 \\ 0 & 2 \\ 0 & 0 \\ 1 & 1 \end{array}$

Step 5