Question

Suppose you were to graph a parabola given by the equation y=14x2+3x−1 . Which ordered pair would be the vertex of the graph?

Answer 1

To find the vertex of the parabola given by the equation \( y = 14x^2 + 3x - 1 \), we can use the formula for the x-coordinate of the vertex of a parabola in the form \( y = ax^2 + bx + c \):

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

In this equation, \( a = 14 \) and \( b = 3 \).

Now we can calculate the x-coordinate of the vertex:

\[
x = -\frac{3}{2 \cdot 14} = -\frac{3}{28}
\]

Next, we need to substitute this x-coordinate back into the equation to find the corresponding y-coordinate:

\[
y = 14\left(-\frac{3}{28}\right)^2 + 3\left(-\frac{3}{28}\right) - 1
\]

Calculating \( \left(-\frac{3}{28}\right)^2 \):

\[
\left(-\frac{3}{28}\right)^2 = \frac{9}{784}
\]

Now substituting that back into the equation:

\[
y = 14 \cdot \frac{9}{784} + 3 \cdot \left(-\frac{3}{28}\right) - 1
\]

\[
y = \frac{126}{784} - \frac{9}{28} - 1
\]

Converting \(-\frac{9}{28}\) to have a common denominator of 784:

\[
-\frac{9}{28} = -\frac{9 \cdot 28}{28 \cdot 28} = -\frac{252}{784}
\]

Now substituting into the equation for \(y\):

\[
y = \frac{126}{784} - \frac{252}{784} - \frac{784}{784}
\]

Combining these fractions:

\[
y = \frac{126 - 252 - 784}{784} = \frac{-910}{784}
\]

We can simplify \(\frac{-910}{784}\) if required, but for the vertex, we can leave it as it is.

Thus, the vertex of the graph is:

\[
\left(-\frac{3}{28}, -\frac{910}{784}\right)
\]

So the ordered pair representing the vertex of the parabola is:

\[
\boxed{\left(-\frac{3}{28}, -\frac{910}{784}\right)}
\]