Find the Directrix y=-1/10x^2+1/5x-3/5

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

Step 1

Rewrite the equation in vertex form.

Simplify each term.

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

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

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

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

Complete the square for $- \frac{x^{2}}{10} + \frac{x}{5} - \frac{3}{5}$ .

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

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

$b = \frac{1}{5}$

$c = - \frac{3}{5}$

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

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

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

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

Cancel the common factor of $2$ and $10$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $10$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $5$ .

Factor $5$ out of $- (1 \cdot 5)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = - 1 \cdot (1)$

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = - \frac{3}{5} - \frac{{(\frac{1}{5})}^{2}}{4 (- \frac{1}{10})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{1}{5}$ .

$e = - \frac{3}{5} - \frac{\frac{1^{2}}{5^{2}}}{4 (- \frac{1}{10})}$

One to any power is one.

$e = - \frac{3}{5} - \frac{\frac{1}{5^{2}}}{4 (- \frac{1}{10})}$

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

$e = - \frac{3}{5} - \frac{\frac{1}{25}}{4 (- \frac{1}{10})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{3}{5} - \frac{\frac{1}{25}}{- 4 (\frac{1}{10})}$

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

$e = - \frac{3}{5} - \frac{\frac{1}{25}}{\frac{- 4}{10}}$

Reduce the expression by cancelling the common factors.

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

Factor $2$ out of $- 4$ .

$e = - \frac{3}{5} - \frac{\frac{1}{25}}{\frac{2 (- 2)}{10}}$

Cancel the common factors.

Factor $2$ out of $10$ .

$e = - \frac{3}{5} - \frac{\frac{1}{25}}{\frac{2 \cdot - 2}{2 \cdot 5}}$

Cancel the common factor.

$e = - \frac{3}{5} - \frac{\frac{1}{25}}{\frac{2 \cdot - 2}{2 \cdot 5}}$

Rewrite the expression.

$e = - \frac{3}{5} - \frac{\frac{1}{25}}{\frac{- 2}{5}}$

Move the negative in front of the fraction.

$e = - \frac{3}{5} - \frac{\frac{1}{25}}{- \frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{3}{5} - (\frac{1}{25} (- \frac{5}{2}))$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$e = - \frac{3}{5} - (\frac{1}{25} \cdot \frac{- 5}{2})$

Factor $5$ out of $25$ .

$e = - \frac{3}{5} - (\frac{1}{5 (5)} \cdot \frac{- 5}{2})$

Factor $5$ out of $- 5$ .

$e = - \frac{3}{5} - (\frac{1}{5 \cdot 5} \cdot \frac{5 \cdot - 1}{2})$

Cancel the common factor.

$e = - \frac{3}{5} - (\frac{1}{5 \cdot 5} \cdot \frac{5 \cdot - 1}{2})$

Rewrite the expression.

$e = - \frac{3}{5} - (\frac{1}{5} \cdot \frac{- 1}{2})$

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

$e = - \frac{3}{5} - \frac{- 1}{5 \cdot 2}$

Multiply $5$ by $2$ .

$e = - \frac{3}{5} - \frac{- 1}{10}$

Move the negative in front of the fraction.

$e = - \frac{3}{5} - - \frac{1}{10}$

Multiply $- - \frac{1}{10}$ .

Multiply $- 1$ by $- 1$ .

$e = - \frac{3}{5} + 1 (\frac{1}{10})$

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

$e = - \frac{3}{5} + \frac{1}{10}$

To write $- \frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - \frac{3}{5} \cdot \frac{2}{2} + \frac{1}{10}$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{5}$ by $\frac{2}{2}$ .

$e = - \frac{3 \cdot 2}{5 \cdot 2} + \frac{1}{10}$

Multiply $5$ by $2$ .

$e = - \frac{3 \cdot 2}{10} + \frac{1}{10}$

Combine the numerators over the common denominator.

$e = \frac{- 3 \cdot 2 + 1}{10}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$e = \frac{- 6 + 1}{10}$

Add $- 6$ and $1$ .

$e = \frac{- 5}{10}$

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

Factor $5$ out of $- 5$ .

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

Cancel the common factors.

Factor $5$ out of $10$ .

$e = \frac{5 \cdot - 1}{5 \cdot 2}$

Cancel the common factor.

$e = \frac{5 \cdot - 1}{5 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 1$

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

Step 3

Find the vertex $(h, k)$ .

$(1, - \frac{1}{2})$

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.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{10})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{10})}$

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.

$y = 2$

Step 6

Find the Directrix y=-1/10x^2+1/5x-3/5

Related Questions

Support LCMGCF.com by disabling your adblocker.