Calculus Examples

${(x - 4)}^{2} + (y + 3) = 25$

Step 1

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

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Step 1.1

Subtract ${(x - 4)}^{2}$ from both sides of the equation.

$y + 3 = 25 - {(x - 4)}^{2}$

Step 1.2

Subtract $3$ from both sides of the equation.

$y = 25 - {(x - 4)}^{2} - 3$

Step 1.3

Subtract $3$ from $25$ .

$y = - {(x - 4)}^{2} + 22$

Step 2

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

$a = - 1$

$h = 4$

$k = 22$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(4, 22)$

Step 5

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

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Step 5.1

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

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

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

$(4, \frac{87}{4})$

Step 7

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

$x = 4$

Step 8

Find the directrix.

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Step 8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 8.2

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

$y = \frac{89}{4}$

Step 9

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

Direction: Opens Down

Vertex: $(4, 22)$

Focus: $(4, \frac{87}{4})$

Axis of Symmetry: $x = 4$

Directrix: $y = \frac{89}{4}$

Step 10