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 three hundred twenty-four million two hundred seventeen thousand nine hundred fifty

Interesting facts

1324217950 has 128 divisors, whose sum is 3477178368
The reverse of 1324217950 is 0597124231
Previous prime number is 47

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 34
Digital Root 7

Name

Name	five hundred thirty-three million three hundred seventy-one thousand eight hundred seventy-eight

Interesting facts

533371878 has 16 divisors, whose sum is 836790864
The reverse of 533371878 is 878173335
Previous prime number is 117

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 45
Digital Root 9

Name

Name	one billion seven hundred eight million eight hundred seventy-one thousand six hundred six

Interesting facts

1708871606 has 8 divisors, whose sum is 2760484944
The reverse of 1708871606 is 6061788071
Previous prime number is 13

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 44
Digital Root 8

Evaluate arcsin(cos(pi/3))

Convert from Radians to Degrees 210

Graph -11-6x=33

Convert to Polar Coordinates (-5,3)

Determine the Type of Number square root of 159