Find the Directrix 12y=x^2-6x+45

$12 y = x^{2} - 6 x + 45$

Step 1

Rewrite the equation in vertex form.

Divide each term in $12 y = x^{2} - 6 x + 45$ by $12$ and simplify.

Divide each term in $12 y = x^{2} - 6 x + 45$ by $12$ .

$\frac{12 y}{12} = \frac{x^{2}}{12} + \frac{- 6 x}{12} + \frac{45}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 y}{12} = \frac{x^{2}}{12} + \frac{- 6 x}{12} + \frac{45}{12}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{12} + \frac{- 6 x}{12} + \frac{45}{12}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $12$ .

Factor $6$ out of $- 6 x$ .

$y = \frac{x^{2}}{12} + \frac{6 (- x)}{12} + \frac{45}{12}$

Cancel the common factors.

Factor $6$ out of $12$ .

$y = \frac{x^{2}}{12} + \frac{6 (- x)}{6 (2)} + \frac{45}{12}$

Cancel the common factor.

$y = \frac{x^{2}}{12} + \frac{6 (- x)}{6 \cdot 2} + \frac{45}{12}$

Rewrite the expression.

$y = \frac{x^{2}}{12} + \frac{- x}{2} + \frac{45}{12}$

Move the negative in front of the fraction.

$y = \frac{x^{2}}{12} - \frac{x}{2} + \frac{45}{12}$

Cancel the common factor of $45$ and $12$ .

Factor $3$ out of $45$ .

$y = \frac{x^{2}}{12} - \frac{x}{2} + \frac{3 (15)}{12}$

Cancel the common factors.

Factor $3$ out of $12$ .

$y = \frac{x^{2}}{12} - \frac{x}{2} + \frac{3 \cdot 15}{3 \cdot 4}$

Cancel the common factor.

$y = \frac{x^{2}}{12} - \frac{x}{2} + \frac{3 \cdot 15}{3 \cdot 4}$

Rewrite the expression.

$y = \frac{x^{2}}{12} - \frac{x}{2} + \frac{15}{4}$

Complete the square for $\frac{x^{2}}{12} - \frac{x}{2} + \frac{15}{4}$ .

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

$a = \frac{1}{12}$

$b = - \frac{1}{2}$

$c = \frac{15}{4}$

Consider the vertex form of a parabola.

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

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Combine $2$ and $\frac{1}{12}$ .

$d = - \frac{1}{2} \cdot \frac{1}{\frac{2}{12}}$

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

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $6$ by $1$ .

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Factor $2$ out of $6$ .

$d = \frac{- 1}{2} \cdot (2 (3))$

Cancel the common factor.

$d = \frac{- 1}{2} \cdot (2 \cdot 3)$

Rewrite the expression.

$d = - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$d = - 3$

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

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{15}{4} - \frac{{(- \frac{1}{2})}^{2}}{4 (\frac{1}{12})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{1}{2}$ .

$e = \frac{15}{4} - \frac{{(- 1)}^{2} {(\frac{1}{2})}^{2}}{4 (\frac{1}{12})}$

Raise $- 1$ to the power of $2$ .

$e = \frac{15}{4} - \frac{1 {(\frac{1}{2})}^{2}}{4 (\frac{1}{12})}$

Apply the product rule to $\frac{1}{2}$ .

$e = \frac{15}{4} - \frac{1 \frac{1^{2}}{2^{2}}}{4 (\frac{1}{12})}$

One to any power is one.

$e = \frac{15}{4} - \frac{1 \frac{1}{2^{2}}}{4 (\frac{1}{12})}$

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

$e = \frac{15}{4} - \frac{1 (\frac{1}{4})}{4 (\frac{1}{12})}$

Multiply $\frac{1}{4}$ by $1$ .

$e = \frac{15}{4} - \frac{\frac{1}{4}}{4 (\frac{1}{12})}$

Combine $4$ and $\frac{1}{12}$ .

$e = \frac{15}{4} - \frac{\frac{1}{4}}{\frac{4}{12}}$

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4$ .

$e = \frac{15}{4} - \frac{\frac{1}{4}}{\frac{4 (1)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

$e = \frac{15}{4} - \frac{\frac{1}{4}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Cancel the common factor.

$e = \frac{15}{4} - \frac{\frac{1}{4}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Rewrite the expression.

$e = \frac{15}{4} - \frac{\frac{1}{4}}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{15}{4} - (\frac{1}{4} \cdot 3)$

Combine $\frac{1}{4}$ and $3$ .

$e = \frac{15}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$e = \frac{15 - 3}{4}$

Subtract $3$ from $15$ .

$e = \frac{12}{4}$

Divide $12$ by $4$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{12} {(x - 3)}^{2} + 3$ .

$\frac{1}{12} {(x - 3)}^{2} + 3$

Set $y$ equal to the new right side.

$y = \frac{1}{12} \cdot {(x - 3)}^{2} + 3$

Step 2

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

$a = \frac{1}{12}$

$h = 3$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(3, 3)$

Step 4

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

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.

$\frac{1}{4 \cdot \frac{1}{12}}$

Simplify.

Combine $4$ and $\frac{1}{12}$ .

$\frac{1}{\frac{4}{12}}$

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 3$

Multiply $3$ by $1$ .

$3$

Step 5

Find the directrix.

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.

$y = 0$

Step 6

Find the Directrix 12y=x^2-6x+45

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