Question

Find the vertex of the parabola: y = -1/5 x^2 - 14/5 x - 3. Simplify both coordinates and write them as proper fractions, improper fractions, or integers. should I enter x = (answer) or just (answer)

Answer 1

To find the vertex of the parabola given by the equation \( y = -\frac{1}{5}x^2 - \frac{14}{5}x - 3 \), we can use the vertex formula for a quadratic function in standard form \( y = ax^2 + bx + c \). The x-coordinate of the vertex is given by the formula:

\[
x = -\frac{b}{2a}
\]

In this equation:
- \( a = -\frac{1}{5} \)
- \( b = -\frac{14}{5} \)

Let's calculate the x-coordinate of the vertex:

1. Substitute \( a \) and \( b \) into the formula:

\[
x = -\frac{-\frac{14}{5}}{2 \cdot -\frac{1}{5}} = \frac{\frac{14}{5}}{-\frac{2}{5}}
\]

2. Simplifying the right side:

\[
x = \frac{14}{5} \cdot -\frac{5}{2} = -\frac{14}{2} = -7
\]

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting \( x = -7 \) back into the original equation:

\[
y = -\frac{1}{5}(-7)^2 - \frac{14}{5}(-7) - 3
\]

Calculating \( y \):

1. Compute \(-7^2\):

\[
y = -\frac{1}{5}(49) - \frac{14}{5}(-7) - 3
\]

2. Simplifying each term:

\[
y = -\frac{49}{5} + \frac{98}{5} - 3
\]

3. Combine the fractions:

\[
y = \frac{98}{5} - \frac{49}{5} - \frac{15}{5} = \frac{98 - 49 - 15}{5} = \frac{34}{5}
\]

Thus, the vertex of the parabola is:

\[
\left(-7, \frac{34}{5}\right)
\]

If you need to enter this in a specific format, you can write it as:
- \( x = -7 \) (as "x = (answer)" format)
- The coordinates are \( (-7, \frac{34}{5}) \) overall.