Find the Directrix y+3=1/10(x+2)^2

$y + 3 = \frac{1}{10} {(x + 2)}^{2}$

Step 1

Rewrite the equation in vertex form.

Isolate $y$ to the left side of the equation.

Simplify $\frac{1}{10} {(x + 2)}^{2}$ .

Rewrite.

$y + 3 = 0 + 0 + \frac{1}{10} {(x + 2)}^{2}$

Simplify by adding zeros.

$y + 3 = \frac{1}{10} {(x + 2)}^{2}$

Combine $\frac{1}{10}$ and ${(x + 2)}^{2}$ .

$y + 3 = \frac{{(x + 2)}^{2}}{10}$

Subtract $3$ from both sides of the equation.

$y = \frac{{(x + 2)}^{2}}{10} - 3$

Reorder terms.

$y = \frac{1}{10} \cdot {(x + 2)}^{2} - 3$

Complete the square for $\frac{1}{10} \cdot {(x + 2)}^{2} - 3$ .

Simplify the expression.

Simplify each term.

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$\frac{1}{10} \cdot ((x + 2) (x + 2)) - 3$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{10} \cdot (x (x + 2) + 2 (x + 2)) - 3$

Apply the distributive property.

$\frac{1}{10} \cdot (x \cdot x + x \cdot 2 + 2 (x + 2)) - 3$

Apply the distributive property.

$\frac{1}{10} \cdot (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{1}{10} \cdot (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) - 3$

Move $2$ to the left of $x$ .

$\frac{1}{10} \cdot (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) - 3$

Multiply $2$ by $2$ .

$\frac{1}{10} \cdot (x^{2} + 2 x + 2 x + 4) - 3$

Add $2 x$ and $2 x$ .

$\frac{1}{10} \cdot (x^{2} + 4 x + 4) - 3$

Apply the distributive property.

$\frac{1}{10} x^{2} + \frac{1}{10} (4 x) + \frac{1}{10} \cdot 4 - 3$

Simplify.

Combine $\frac{1}{10}$ and $x^{2}$ .

$\frac{x^{2}}{10} + \frac{1}{10} (4 x) + \frac{1}{10} \cdot 4 - 3$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$\frac{x^{2}}{10} + \frac{1}{2 (5)} (4 x) + \frac{1}{10} \cdot 4 - 3$

Factor $2$ out of $4 x$ .

$\frac{x^{2}}{10} + \frac{1}{2 (5)} (2 (2 x)) + \frac{1}{10} \cdot 4 - 3$

Cancel the common factor.

$\frac{x^{2}}{10} + \frac{1}{2 \cdot 5} (2 (2 x)) + \frac{1}{10} \cdot 4 - 3$

Rewrite the expression.

$\frac{x^{2}}{10} + \frac{1}{5} (2 x) + \frac{1}{10} \cdot 4 - 3$

Combine $2$ and $\frac{1}{5}$ .

$\frac{x^{2}}{10} + \frac{2}{5} x + \frac{1}{10} \cdot 4 - 3$

Combine $\frac{2}{5}$ and $x$ .

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{1}{10} \cdot 4 - 3$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{1}{2 (5)} \cdot 4 - 3$

Factor $2$ out of $4$ .

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{1}{2 \cdot 5} \cdot (2 \cdot 2) - 3$

Cancel the common factor.

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{1}{2 \cdot 5} \cdot (2 \cdot 2) - 3$

Rewrite the expression.

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{1}{5} \cdot 2 - 3$

Combine $\frac{1}{5}$ and $2$ .

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{2}{5} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{2}{5} - 3 \cdot \frac{5}{5}$

Combine $- 3$ and $\frac{5}{5}$ .

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{2}{5} + \frac{- 3 \cdot 5}{5}$

Combine the numerators over the common denominator.

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{2 - 3 \cdot 5}{5}$

Simplify the numerator.

Multiply $- 3$ by $5$ .

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{2 - 15}{5}$

Subtract $15$ from $2$ .

$\frac{x^{2}}{10} + \frac{2 x}{5} + \frac{- 13}{5}$

Move the negative in front of the fraction.

$\frac{x^{2}}{10} + \frac{2 x}{5} - \frac{13}{5}$

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

$a = \frac{1}{10}$

$b = \frac{2}{5}$

$c = - \frac{13}{5}$

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

$d = \frac{\frac{2}{5}}{2 (\frac{1}{10})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{2}{5} \cdot \frac{1}{2 (\frac{1}{10})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{5} \cdot \frac{1}{2 (\frac{1}{10})}$

Rewrite the expression.

$d = \frac{1}{5} \cdot \frac{1}{\frac{1}{10}}$

Multiply $\frac{1}{5}$ by $\frac{1}{\frac{1}{10}}$ .

$d = \frac{1}{5 (\frac{1}{10})}$

Combine $5$ and $\frac{1}{10}$ .

$d = \frac{1}{\frac{5}{10}}$

Cancel the common factor of $5$ and $10$ .

Factor $5$ out of $5$ .

$d = \frac{1}{\frac{5 (1)}{10}}$

Cancel the common factors.

Factor $5$ out of $10$ .

$d = \frac{1}{\frac{5 \cdot 1}{5 \cdot 2}}$

Cancel the common factor.

$d = \frac{1}{\frac{5 \cdot 1}{5 \cdot 2}}$

Rewrite the expression.

$d = \frac{1}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

Multiply $2$ by $1$ .

$d = 2$

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

$e = - \frac{13}{5} - \frac{{(\frac{2}{5})}^{2}}{4 (\frac{1}{10})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{2}{5}$ .

$e = - \frac{13}{5} - \frac{\frac{2^{2}}{5^{2}}}{4 (\frac{1}{10})}$

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

$e = - \frac{13}{5} - \frac{\frac{4}{5^{2}}}{4 (\frac{1}{10})}$

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

$e = - \frac{13}{5} - \frac{\frac{4}{25}}{4 (\frac{1}{10})}$

Combine $4$ and $\frac{1}{10}$ .

$e = - \frac{13}{5} - \frac{\frac{4}{25}}{\frac{4}{10}}$

Cancel the common factor of $4$ and $10$ .

Factor $2$ out of $4$ .

$e = - \frac{13}{5} - \frac{\frac{4}{25}}{\frac{2 (2)}{10}}$

Cancel the common factors.

Factor $2$ out of $10$ .

$e = - \frac{13}{5} - \frac{\frac{4}{25}}{\frac{2 \cdot 2}{2 \cdot 5}}$

Cancel the common factor.

$e = - \frac{13}{5} - \frac{\frac{4}{25}}{\frac{2 \cdot 2}{2 \cdot 5}}$

Rewrite the expression.

$e = - \frac{13}{5} - \frac{\frac{4}{25}}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{13}{5} - (\frac{4}{25} \cdot \frac{5}{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$e = - \frac{13}{5} - (\frac{2 (2)}{25} \cdot \frac{5}{2})$

Cancel the common factor.

$e = - \frac{13}{5} - (\frac{2 \cdot 2}{25} \cdot \frac{5}{2})$

Rewrite the expression.

$e = - \frac{13}{5} - (\frac{2}{25} \cdot 5)$

Cancel the common factor of $5$ .

Factor $5$ out of $25$ .

$e = - \frac{13}{5} - (\frac{2}{5 (5)} \cdot 5)$

Cancel the common factor.

$e = - \frac{13}{5} - (\frac{2}{5 \cdot 5} \cdot 5)$

Rewrite the expression.

$e = - \frac{13}{5} - \frac{2}{5}$

Combine the numerators over the common denominator.

$e = \frac{- 13 - 2}{5}$

Subtract $2$ from $- 13$ .

$e = \frac{- 15}{5}$

Divide $- 15$ by $5$ .

$e = - 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{10} {(x + 2)}^{2} - 3$ .

$\frac{1}{10} {(x + 2)}^{2} - 3$

Set $y$ equal to the new right side.

$y = \frac{1}{10} \cdot {(x + 2)}^{2} - 3$

Step 2

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

$a = \frac{1}{10}$

$h = - 2$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(- 2, - 3)$

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.

$\frac{1}{4 \cdot \frac{1}{10}}$

Simplify.

Combine $4$ and $\frac{1}{10}$ .

$\frac{1}{\frac{4}{10}}$

Cancel the common factor of $4$ and $10$ .

Factor $2$ out of $4$ .

$\frac{1}{\frac{2 (2)}{10}}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{1}{\frac{2 \cdot 2}{2 \cdot 5}}$

Cancel the common factor.

$\frac{1}{\frac{2 \cdot 2}{2 \cdot 5}}$

Rewrite the expression.

$\frac{1}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{5}{2})$

Multiply $\frac{5}{2}$ by $1$ .

$\frac{5}{2}$

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.

$y = - \frac{11}{2}$

Step 6

Find the Directrix y+3=1/10(x+2)^2

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