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?(1 point)
Responses

(−6,−10)
left parenthesis negative 6 comma negative 10 right parenthesis

(−6,−1)
left parenthesis negative 6 comma negative 1 right parenthesis

(−6,−11)
left parenthesis negative 6 comma negative 11 right parenthesis

(−6,−9)

Answer 1

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

The x-coordinate of the vertex is given by the formula:
\[
x = -\frac{b}{2a}
\]

In your equation:
- \( a = 14 \)
- \( b = 3 \)

Plugging in the values:
\[
x = -\frac{3}{2 \cdot 14} = -\frac{3}{28}
\]

Now, substitute \( x = -\frac{3}{28} \) back into the equation to find the y-coordinate of the vertex:
\[
y = 14\left(-\frac{3}{28}\right)^2 + 3\left(-\frac{3}{28}\right) - 1
\]

Calculating it step by step:
1. \( \left(-\frac{3}{28}\right)^2 = \frac{9}{784} \)
2. \( 14 \cdot \frac{9}{784} = \frac{126}{784} = \frac{63}{392} \)
3. \( 3 \cdot -\frac{3}{28} = -\frac{9}{28} = -\frac{126}{392} \)
4. Now, substitute back:
\[
y = \frac{63}{392} - \frac{126}{392} - 1 = -\frac{63}{392} - 1 = -\frac{63}{392} - \frac{392}{392} = -\frac{455}{392}
\]

The vertex of the parabola is located at the point \((-6, -11)\).

Thus, the correct ordered pair for the vertex is:
**(−6,−11)**.