Find the Directrix y=-1/2x^2+14/5x+17/53

$y = - \frac{1}{2} x^{2} + \frac{14}{5} x + \frac{17}{53}$

Step 1

Rewrite the equation in vertex form.

Simplify each term.

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

$y = - \frac{x^{2}}{2} + \frac{14}{5} x + \frac{17}{53}$

Combine $\frac{14}{5}$ and $x$ .

$y = - \frac{x^{2}}{2} + \frac{14 x}{5} + \frac{17}{53}$

Complete the square for $- \frac{x^{2}}{2} + \frac{14 x}{5} + \frac{17}{53}$ .

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

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

$b = \frac{14}{5}$

$c = \frac{17}{53}$

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

$d = \frac{\frac{14}{5}}{2 (- \frac{1}{2})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{14}{5} \cdot \frac{1}{2 (- \frac{1}{2})}$

Cancel the common factor of $2$ .

Factor $2$ out of $14$ .

$d = \frac{2 (7)}{5} \cdot \frac{1}{2 (- \frac{1}{2})}$

Cancel the common factor.

$d = \frac{2 \cdot 7}{5} \cdot \frac{1}{2 (- \frac{1}{2})}$

Rewrite the expression.

$d = \frac{7}{5} \cdot \frac{1}{- \frac{1}{2}}$

Multiply $\frac{7}{5}$ by $\frac{1}{- \frac{1}{2}}$ .

$d = \frac{7}{5 (- \frac{1}{2})}$

Multiply $- 1$ by $5$ .

$d = \frac{7}{- 5 (\frac{1}{2})}$

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

$d = \frac{7}{\frac{- 5}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 7 (\frac{2}{- 5})$

Move the negative in front of the fraction.

$d = 7 (- \frac{2}{5})$

Multiply $7 (- \frac{2}{5})$ .

Multiply $- 1$ by $7$ .

$d = - 7 (\frac{2}{5})$

Combine $- 7$ and $\frac{2}{5}$ .

$d = \frac{- 7 \cdot 2}{5}$

Multiply $- 7$ by $2$ .

$d = \frac{- 14}{5}$

Move the negative in front of the fraction.

$d = - \frac{14}{5}$

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

$e = \frac{17}{53} - \frac{{(\frac{14}{5})}^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{14}{5}$ .

$e = \frac{17}{53} - \frac{\frac{14^{2}}{5^{2}}}{4 (- \frac{1}{2})}$

Raise $14$ to the power of $2$ .

$e = \frac{17}{53} - \frac{\frac{196}{5^{2}}}{4 (- \frac{1}{2})}$

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

$e = \frac{17}{53} - \frac{\frac{196}{25}}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{17}{53} - \frac{\frac{196}{25}}{- 4 (\frac{1}{2})}$

Combine $- 4$ and $\frac{1}{2}$ .

$e = \frac{17}{53} - \frac{\frac{196}{25}}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = \frac{17}{53} - \frac{\frac{196}{25}}{- 2}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{17}{53} - (\frac{196}{25} \cdot \frac{1}{- 2})$

Cancel the common factor of $2$ .

Factor $2$ out of $196$ .

$e = \frac{17}{53} - (\frac{2 (98)}{25} \cdot \frac{1}{- 2})$

Factor $2$ out of $- 2$ .

$e = \frac{17}{53} - (\frac{2 \cdot 98}{25} \cdot \frac{1}{2 \cdot - 1})$

Cancel the common factor.

$e = \frac{17}{53} - (\frac{2 \cdot 98}{25} \cdot \frac{1}{2 \cdot - 1})$

Rewrite the expression.

$e = \frac{17}{53} - (\frac{98}{25} \cdot \frac{1}{- 1})$

Multiply $\frac{98}{25}$ by $\frac{1}{- 1}$ .

$e = \frac{17}{53} - \frac{98}{25 \cdot - 1}$

Multiply $25$ by $- 1$ .

$e = \frac{17}{53} - \frac{98}{- 25}$

Move the negative in front of the fraction.

$e = \frac{17}{53} - - \frac{98}{25}$

Multiply $- - \frac{98}{25}$ .

Multiply $- 1$ by $- 1$ .

$e = \frac{17}{53} + 1 (\frac{98}{25})$

Multiply $\frac{98}{25}$ by $1$ .

$e = \frac{17}{53} + \frac{98}{25}$

To write $\frac{17}{53}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$e = \frac{17}{53} \cdot \frac{25}{25} + \frac{98}{25}$

To write $\frac{98}{25}$ as a fraction with a common denominator, multiply by $\frac{53}{53}$ .

$e = \frac{17}{53} \cdot \frac{25}{25} + \frac{98}{25} \cdot \frac{53}{53}$

Write each expression with a common denominator of $1325$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{17}{53}$ by $\frac{25}{25}$ .

$e = \frac{17 \cdot 25}{53 \cdot 25} + \frac{98}{25} \cdot \frac{53}{53}$

Multiply $53$ by $25$ .

$e = \frac{17 \cdot 25}{1325} + \frac{98}{25} \cdot \frac{53}{53}$

Multiply $\frac{98}{25}$ by $\frac{53}{53}$ .

$e = \frac{17 \cdot 25}{1325} + \frac{98 \cdot 53}{25 \cdot 53}$

Multiply $25$ by $53$ .

$e = \frac{17 \cdot 25}{1325} + \frac{98 \cdot 53}{1325}$

Combine the numerators over the common denominator.

$e = \frac{17 \cdot 25 + 98 \cdot 53}{1325}$

Simplify the numerator.

Multiply $17$ by $25$ .

$e = \frac{425 + 98 \cdot 53}{1325}$

Multiply $98$ by $53$ .

$e = \frac{425 + 5194}{1325}$

Add $425$ and $5194$ .

$e = \frac{5619}{1325}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - \frac{14}{5})}^{2} + \frac{5619}{1325}$ .

$- \frac{1}{2} {(x - \frac{14}{5})}^{2} + \frac{5619}{1325}$

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x - \frac{14}{5})}^{2} + \frac{5619}{1325}$

Step 2

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

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

$h = \frac{14}{5}$

$k = \frac{5619}{1325}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{14}{5}, \frac{5619}{1325})$

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.

$\frac{1}{4 \cdot (- \frac{1}{2})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{2})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Combine $4$ and $\frac{1}{2}$ .

$- \frac{1}{\frac{4}{2}}$

Divide $4$ by $2$ .

$- \frac{1}{2}$

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.

$y = \frac{12563}{2650}$

Step 6

Find the Directrix y=-1/2x^2+14/5x+17/53

Related Questions

Support LCMGCF.com by disabling your adblocker.