Trigonometry Examples

$y = - 3 x^{2} + 6 x + 5$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

$- 3 x^{2} + 6 x + 5$ .

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Step 1.1.1

Use the form

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

$a$ ,

$b$ , and

$c$ .

$a = - 3$

$b = 6$

$c = 5$

Step 1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.3

Find the value of

$d$ using the formula

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

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Step 1.1.3.1

Substitute the values of

$a$ and

$b$ into the formula

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

$d = \frac{6}{2 \cdot - 3}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of

$6$ and

$2$ .

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Step 1.1.3.2.1.1

Factor

$2$ out of

$6$ .

$d = \frac{2 \cdot 3}{2 \cdot - 3}$

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

Factor

$2$ out of

$2 \cdot - 3$ .

$d = \frac{2 \cdot 3}{2 (- 3)}$

Step 1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 \cdot - 3}$

Step 1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{3}{- 3}$

$d = \frac{3}{- 3}$

$d = \frac{3}{- 3}$

Step 1.1.3.2.2

Cancel the common factor of

$3$ and

$- 3$ .

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Step 1.1.3.2.2.1

Factor

$3$ out of

$3$ .

$d = \frac{3 (1)}{- 3}$

Step 1.1.3.2.2.2

Move the negative one from the denominator of

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

$d = - 1 \cdot 1$

$d = - 1 \cdot 1$

Step 1.1.3.2.3

Multiply

$- 1$ by

$1$ .

$d = - 1$

$d = - 1$

$d = - 1$

Step 1.1.4

Find the value of

$e$ using the formula

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

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Step 1.1.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

$e = 5 - \frac{6^{2}}{4 \cdot - 3}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Raise

$6$ to the power of

$2$ .

$e = 5 - \frac{36}{4 \cdot - 3}$

Step 1.1.4.2.1.2

Multiply

$4$ by

$- 3$ .

$e = 5 - \frac{36}{- 12}$

Step 1.1.4.2.1.3

Divide

$36$ by

$- 12$ .

$e = 5 - - 3$

Step 1.1.4.2.1.4

Multiply

$- 1$ by

$- 3$ .

$e = 5 + 3$

$e = 5 + 3$

Step 1.1.4.2.2

Add

$5$ and

$3$ .

$e = 8$

$e = 8$

$e = 8$

Step 1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$- 3 {(x - 1)}^{2} + 8$ .

$- 3 {(x - 1)}^{2} + 8$

$- 3 {(x - 1)}^{2} + 8$

Step 1.2

Set

$y$ equal to the new right side.

$y = - 3 {(x - 1)}^{2} + 8$

$y = - 3 {(x - 1)}^{2} + 8$

Step 2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = - 3$

$h = 1$

$k = 8$

Step 3

Since the value of

$a$ is negative, the parabola opens down.

Opens Down

Step 4

Find the vertex

$(h, k)$ .

$(1, 8)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply

$4$ by

$- 3$ .

$\frac{1}{- 12}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{12}$

$- \frac{1}{12}$

$- \frac{1}{12}$

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.

$(1, \frac{95}{12})$

$(1, \frac{95}{12})$

Step 7

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

$x = 1$

Step 8

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$