Graph y=x^2-12x+2

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Trigonometry

Graph y=x^2-12x+2

$y = x^{2} - 12 x + 2$

Find the properties of the given parabola.

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Rewrite the equation in vertex form.

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Complete the square for $x^{2} - 12 x + 2$ .

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

$a = 1, b = - 12, c = 2$

Consider the vertex form of a parabola.

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

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

$d = - \frac{12}{2 (1)}$

Simplify the right side.

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Cancel the common factor of $12$ and $2$ .

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Factor $2$ out of $12$ .

$d = - \frac{2 \cdot 6}{2 \cdot 1}$

Cancel the common factors.

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

$d = - \frac{2 \cdot 6}{2 (1)}$

Cancel the common factor.

$d = - \frac{2 \cdot 6}{2 \cdot 1}$

Rewrite the expression.

$d = - \frac{6}{1}$

Divide $6$ by $1$ .

$d = - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$d = - 6$

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

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Simplify each term.

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Raise $- 12$ to the power of $2$ .

$e = 2 - \frac{144}{4 (1)}$

Multiply $4$ by $1$ .

$e = 2 - \frac{144}{4}$

Divide $144$ by $4$ .

$e = 2 - 1 \cdot 36$

Multiply $- 1$ by $36$ .

$e = 2 - 36$

Subtract $36$ from $2$ .

$e = - 34$

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

${(x - 6)}^{2} - 34$

Set $y$ equal to the new right side.

$y = {(x - 6)}^{2} - 34$

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

$a = 1$

$h = 6$

$k = - 34$

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

Opens Up

Find the vertex $(h, k)$ .

$(6, - 34)$

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

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

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

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

Cancel the common factor of $1$ .

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Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{4}$

Find the focus.

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

$(h, k + p)$

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

$(6, - \frac{135}{4})$

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

$x = 6$

Find the directrix.

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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.

$y = k - p$

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

$y = - \frac{137}{4}$

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

Direction: Opens Up

Vertex: $(6, - 34)$

Focus: $(6, - \frac{135}{4})$

Axis of Symmetry: $x = 6$

Directrix: $y = - \frac{137}{4}$

Direction: Opens Up

Vertex: $(6, - 34)$

Focus: $(6, - \frac{135}{4})$

Axis of Symmetry: $x = 6$

Directrix: $y = - \frac{137}{4}$

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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Replace the variable $x$ with $5$ in the expression.

$f (5) = {(5)}^{2} - 12 \cdot 5 + 2$

Simplify the result.

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Simplify each term.

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Raise $5$ to the power of $2$ .

$f (5) = 25 - 12 \cdot 5 + 2$

Multiply $- 12$ by $5$ .

$f (5) = 25 - 60 + 2$

Simplify by adding and subtracting.

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Subtract $60$ from $25$ .

$f (5) = - 35 + 2$

Add $- 35$ and $2$ .

$f (5) = - 33$

The final answer is $- 33$ .

$- 33$

The $y$ value at $x = 5$ is $- 33$ .

$y = - 33$

Replace the variable $x$ with $4$ in the expression.

$f (4) = {(4)}^{2} - 12 \cdot 4 + 2$

Simplify the result.

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Simplify each term.

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Raise $4$ to the power of $2$ .

$f (4) = 16 - 12 \cdot 4 + 2$

Multiply $- 12$ by $4$ .

$f (4) = 16 - 48 + 2$

Simplify by adding and subtracting.

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Subtract $48$ from $16$ .

$f (4) = - 32 + 2$

Add $- 32$ and $2$ .

$f (4) = - 30$

The final answer is $- 30$ .

$- 30$

The $y$ value at $x = 4$ is $- 30$ .

$y = - 30$

Replace the variable $x$ with $7$ in the expression.

$f (7) = {(7)}^{2} - 12 \cdot 7 + 2$

Simplify the result.

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Simplify each term.

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Raise $7$ to the power of $2$ .

$f (7) = 49 - 12 \cdot 7 + 2$

Multiply $- 12$ by $7$ .

$f (7) = 49 - 84 + 2$

Simplify by adding and subtracting.

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Subtract $84$ from $49$ .

$f (7) = - 35 + 2$

Add $- 35$ and $2$ .

$f (7) = - 33$

The final answer is $- 33$ .

$- 33$

The $y$ value at $x = 7$ is $- 33$ .

$y = - 33$

Replace the variable $x$ with $8$ in the expression.

$f (8) = {(8)}^{2} - 12 \cdot 8 + 2$

Simplify the result.

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Simplify each term.

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Raise $8$ to the power of $2$ .

$f (8) = 64 - 12 \cdot 8 + 2$

Multiply $- 12$ by $8$ .

$f (8) = 64 - 96 + 2$

Simplify by adding and subtracting.

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Subtract $96$ from $64$ .

$f (8) = - 32 + 2$

Add $- 32$ and $2$ .

$f (8) = - 30$

The final answer is $- 30$ .

$- 30$

The $y$ value at $x = 8$ is $- 30$ .

$y = - 30$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 4 & - 30 \\ 5 & - 33 \\ 6 & - 34 \\ 7 & - 33 \\ 8 & - 30 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(6, - 34)$

Focus: $(6, - \frac{135}{4})$

Axis of Symmetry: $x = 6$

Directrix: $y = - \frac{137}{4}$

$\begin{array}{c} x & y \\ 4 & - 30 \\ 5 & - 33 \\ 6 & - 34 \\ 7 & - 33 \\ 8 & - 30 \end{array}$

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Name

Name	one billion two hundred million six hundred eighty-four thousand seven hundred eighty-five

Interesting facts

1200684785 has 8 divisors, whose sum is 1447114920
The reverse of 1200684785 is 5874860021
Previous prime number is 229

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	one billion nine hundred ninety-three million four hundred sixty-eight thousand seventy-seven

Interesting facts

1993468077 has 32 divisors, whose sum is 3975961600
The reverse of 1993468077 is 7708643991
Previous prime number is 103

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 54
Digital Root 9

Name

Name	one billion six hundred eighty million seventy-nine thousand one hundred fifty-five

Interesting facts

1680079155 has 8 divisors, whose sum is 1830214656
The reverse of 1680079155 is 5519700861
Previous prime number is 47

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 42
Digital Root 6

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