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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