Question

You want to graph a parabola given the equation y=14x2+2x+2 . Which ordered pair would be the vertex of the graph?(1 point) Responses (−4, −4) (−4, −4) (−4, −2) (−4, −2) (−4, −1) (−4, −1) (−4, 0)

Answer 1

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

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

In this case, \( a = 14 \) and \( b = 2 \).

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

\[
x = -\frac{2}{2(14)} = -\frac{2}{28} = -\frac{1}{14}
\]

Now that we have the x-coordinate, we can substitute it back into the original equation to find the corresponding y-coordinate.

Calculating \( y \) when \( x = -\frac{1}{14} \):

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

Calculating \( 14\left(-\frac{1}{14}\right)^2 \):

\[
14\left(\frac{1}{196}\right) = \frac{14}{196} = \frac{1}{14}
\]

Now calculate \( 2\left(-\frac{1}{14}\right) \):

\[
-\frac{2}{14} = -\frac{1}{7}
\]

Now substitute back into the equation to find \( y \):

\[
y = \frac{1}{14} - \frac{1}{7} + 2
\]

We can convert \(-\frac{1}{7}\) to have a common denominator with \( \frac{1}{14} \):

\[
-\frac{1}{7} = -\frac{2}{14}
\]

Now, substituting gives:

\[
y = \frac{1}{14} - \frac{2}{14} + 2 = -\frac{1}{14} + 2
\]

To combine these, we convert 2 to have a denominator of 14:

\[
2 = \frac{28}{14}
\]

Thus,

\[
y = -\frac{1}{14} + \frac{28}{14} = \frac{27}{14}
\]

So, the vertex of the parabola is at
\[
\left(-\frac{1}{14}, \frac{27}{14}\right)
\]

Since none of the given options, (−4, −4), (−4, −2), (−4, −1), or (−4, 0), match our calculated vertex, you may want to double-check the provided options or verify the equation. The vertex calculation shows that none of those pairs are correct given the parabola defined by the equation \( y = 14x^2 + 2x + 2 \).