Calculus Examples

x = 4 - y^{2}

$x = 4 - y^{2}$

Step 1

Reorder

4

$4$ and

- y^{2}

$- y^{2}$ .

x = - y^{2} + 4

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

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

- y^{2} + 4

$- y^{2} + 4$ .

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Step 2.1.1.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = - 1

$a = - 1$

b = 0

$b = 0$

c = 4

$c = 4$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

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

Step 2.1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 2.1.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{0}{2 \cdot - 1}

$d = \frac{0}{2 \cdot - 1}$

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

0

$0$ and

2

$2$ .

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Step 2.1.1.3.2.1.1

Factor

2

$2$ out of

0

$0$ .

d = \frac{2 (0)}{2 \cdot - 1}

$d = \frac{2 (0)}{2 \cdot - 1}$

Step 2.1.1.3.2.1.2

Move the negative one from the denominator of

\frac{0}{- 1}

$\frac{0}{- 1}$ .

d = - 1 \cdot 0

$d = - 1 \cdot 0$

d = - 1 \cdot 0

$d = - 1 \cdot 0$

Step 2.1.1.3.2.2

Rewrite

- 1 \cdot 0

$- 1 \cdot 0$ as

- 0

$- 0$ .

d = - 0

$d = - 0$

Step 2.1.1.3.2.3

Multiply

- 1

$- 1$ by

0

$0$ .

d = 0

$d = 0$

d = 0

$d = 0$

d = 0

$d = 0$

Step 2.1.1.4

Find the value of

e

$e$ using the formula

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

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

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Step 2.1.1.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

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

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

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

Raising

0

$0$ to any positive power yields

0

$0$ .

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

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

Step 2.1.1.4.2.1.2

Multiply

4

$4$ by

- 1

$- 1$ .

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

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

Step 2.1.1.4.2.1.3

Divide

0

$0$ by

- 4

$- 4$ .

e = 4 - 0

$e = 4 - 0$

Step 2.1.1.4.2.1.4

Multiply

- 1

$- 1$ by

0

$0$ .

e = 4 + 0

$e = 4 + 0$

e = 4 + 0

$e = 4 + 0$

Step 2.1.1.4.2.2

Add

4

$4$ and

0

$0$ .

e = 4

$e = 4$

e = 4

$e = 4$

e = 4

$e = 4$

Step 2.1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

- {(y + 0)}^{2} + 4

$- {(y + 0)}^{2} + 4$ .

- {(y + 0)}^{2} + 4

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

- {(y + 0)}^{2} + 4

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

Step 2.1.2

Set

x

$x$ equal to the new right side.

x = - {(y + 0)}^{2} + 4

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

x = - {(y + 0)}^{2} + 4

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

Step 2.2

Use the vertex form,

x = a {(y - k)}^{2} + h

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

a

$a$ ,

h

$h$ , and

k

$k$ .

a = - 1

$a = - 1$

h = 4

$h = 4$

k = 0

$k = 0$

Step 2.3

Since the value of

a

$a$ is negative, the parabola opens left.

Opens Left

Step 2.4

Find the vertex

(h, k)

$(h, k)$ .

(4, 0)

$(4, 0)$

Step 2.5

Find

p

$p$ , 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}

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of

a

$a$ into the formula.

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

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

Step 2.5.3

Cancel the common factor of

1

$1$ and

- 1

$- 1$ .

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Step 2.5.3.1

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

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

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

Step 2.5.3.2

Move the negative in front of the fraction.

- \frac{1}{4}

$- \frac{1}{4}$

- \frac{1}{4}

$- \frac{1}{4}$

- \frac{1}{4}

$- \frac{1}{4}$

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

$p$ to the x-coordinate

h

$h$ if the parabola opens left or right.

(h + p, k)

$(h + p, k)$

Step 2.6.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(\frac{15}{4}, 0)

$(\frac{15}{4}, 0)$

(\frac{15}{4}, 0)

$(\frac{15}{4}, 0)$

Step 2.7

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

y = 0

$y = 0$

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

$p$ from the x-coordinate

h

$h$ of the vertex if the parabola opens left or right.

x = h - p

$x = h - p$

Step 2.8.2

Substitute the known values of

p

$p$ and

h

$h$ into the formula and simplify.

x = \frac{17}{4}

$x = \frac{17}{4}$

x = \frac{17}{4}

$x = \frac{17}{4}$

Step 2.9

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

Direction: Opens Left

Vertex:

(4, 0)

$(4, 0)$

Focus:

(\frac{15}{4}, 0)

$(\frac{15}{4}, 0)$

Axis of Symmetry:

y = 0

$y = 0$

Directrix:

x = \frac{17}{4}

$x = \frac{17}{4}$

Direction: Opens Left

Vertex:

(4, 0)

$(4, 0)$

Focus:

(\frac{15}{4}, 0)

$(\frac{15}{4}, 0)$

Axis of Symmetry:

y = 0

$y = 0$

Directrix:

x = \frac{17}{4}

$x = \frac{17}{4}$

Step 3

Select a few

x

$x$ values, and plug them into the equation to find the corresponding

y

$y$ values. The

x

$x$ values should be selected around the vertex.

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Step 3.1

Substitute the

x

$x$ value

2

$2$ into

f (x) = \sqrt{- x + 4}

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

(2, 1.41421356)

$(2, 1.41421356)$ .

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Step 3.1.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

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

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 3.1.2.2

Add

- 2

$- 2$ and

4

$4$ .

f (2) = \sqrt{2}

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

Step 3.1.2.3

The final answer is

\sqrt{2}

$\sqrt{2}$ .

y = \sqrt{2}

$y = \sqrt{2}$

y = \sqrt{2}

$y = \sqrt{2}$

Step 3.1.3

Convert

\sqrt{2}

$\sqrt{2}$ to decimal.

= 1.41421356

$= 1.41421356$

= 1.41421356

$= 1.41421356$

Step 3.2

Substitute the

x

$x$ value

2

$2$ into

f (x) = - \sqrt{- x + 4}

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

(2, - 1.41421356)

$(2, - 1.41421356)$ .

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Step 3.2.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

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

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

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 3.2.2.2

Add

- 2

$- 2$ and

4

$4$ .

f (2) = - \sqrt{2}

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

Step 3.2.2.3

The final answer is

- \sqrt{2}

$- \sqrt{2}$ .

y = - \sqrt{2}

$y = - \sqrt{2}$

y = - \sqrt{2}

$y = - \sqrt{2}$

Step 3.2.3

Convert

- \sqrt{2}

$- \sqrt{2}$ to decimal.

= - 1.41421356

$= - 1.41421356$

= - 1.41421356

$= - 1.41421356$

Step 3.3

Substitute the

x

$x$ value

3

$3$ into

f (x) = \sqrt{- x + 4}

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

(3, 1)

$(3, 1)$ .

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Step 3.3.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = \sqrt{- (3) + 4}

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

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Multiply

- 1

$- 1$ by

3

$3$ .

f (3) = \sqrt{- 3 + 4}

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

Step 3.3.2.2

Add

- 3

$- 3$ and

4

$4$ .

f (3) = \sqrt{1}

$f (3) = \sqrt{1}$

Step 3.3.2.3

Any root of

1

$1$ is

1

$1$ .

f (3) = 1

$f (3) = 1$

Step 3.3.2.4

The final answer is

1

$1$ .

y = 1

$y = 1$

y = 1

$y = 1$

Step 3.3.3

Convert

1

$1$ to decimal.

= 1

$= 1$

= 1

$= 1$

Step 3.4

Substitute the

x

$x$ value

3

$3$ into

f (x) = - \sqrt{- x + 4}

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

(3, - 1)

$(3, - 1)$ .

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Step 3.4.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = - \sqrt{- (3) + 4}

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

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Multiply

- 1

$- 1$ by

3

$3$ .

f (3) = - \sqrt{- 3 + 4}

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

Step 3.4.2.2

Add

- 3

$- 3$ and

4

$4$ .

f (3) = - \sqrt{1}

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

Step 3.4.2.3

Any root of

1

$1$ is

1

$1$ .

f (3) = - 1 \cdot 1

$f (3) = - 1 \cdot 1$

Step 3.4.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

f (3) = - 1

$f (3) = - 1$

Step 3.4.2.5

The final answer is

- 1

$- 1$ .

y = - 1

$y = - 1$

y = - 1

$y = - 1$

Step 3.4.3

Convert

- 1

$- 1$ to decimal.

= - 1

$= - 1$

= - 1

$= - 1$

Step 3.5

Graph the parabola using its properties and the selected points.

\begin{array}{c} x & y \\ 2 & 1.41 \\ 2 & - 1.41 \\ 3 & 1 \\ 3 & - 1 \\ 4 & 0 \end{array}

$\begin{array}{c} x & y \\ 2 & 1.41 \\ 2 & - 1.41 \\ 3 & 1 \\ 3 & - 1 \\ 4 & 0 \end{array}$

\begin{array}{c} x & y \\ 2 & 1.41 \\ 2 & - 1.41 \\ 3 & 1 \\ 3 & - 1 \\ 4 & 0 \end{array}

$\begin{array}{c} x & y \\ 2 & 1.41 \\ 2 & - 1.41 \\ 3 & 1 \\ 3 & - 1 \\ 4 & 0 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex:

(4, 0)

$(4, 0)$

Focus:

(\frac{15}{4}, 0)

$(\frac{15}{4}, 0)$

Axis of Symmetry:

y = 0

$y = 0$

Directrix:

x = \frac{17}{4}

$x = \frac{17}{4}$

\begin{array}{c} x & y \\ 2 & 1.41 \\ 2 & - 1.41 \\ 3 & 1 \\ 3 & - 1 \\ 4 & 0 \end{array}

$\begin{array}{c} x & y \\ 2 & 1.41 \\ 2 & - 1.41 \\ 3 & 1 \\ 3 & - 1 \\ 4 & 0 \end{array}$

Step 5