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,−1)
left parenthesis negative 6 comma negative 1 right parenthesis
(−6,−9)
left parenthesis negative 6 comma negative 9 right parenthesis
(−6,−10)
left parenthesis negative 6 comma negative 10 right parenthesis
(−6,−11)
. Which ordered pair would be the vertex of the graph?(1 point)
Responses
(−6,−1)
left parenthesis negative 6 comma negative 1 right parenthesis
(−6,−9)
left parenthesis negative 6 comma negative 9 right parenthesis
(−6,−10)
left parenthesis negative 6 comma negative 10 right parenthesis
(−6,−11)
Answers
GPT-4o mini
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, which is given by:
\[
x = -\frac{b}{2a}
\]
In this equation, \( a = 14 \) and \( b = 3 \).
Calculating the x-coordinate:
\[
x = -\frac{3}{2(14)} = -\frac{3}{28}
\]
Now, we can substitute this x-value back into the equation to find the corresponding y-coordinate.
First, we compute \( y \):
\[
y = 14\left(-\frac{3}{28}\right)^2 + 3\left(-\frac{3}{28}\right) - 1
\]
Calculating \( \left(-\frac{3}{28}\right)^2 \):
\[
\left(-\frac{3}{28}\right)^2 = \frac{9}{784}
\]
Now substitute it back into the equation:
\[
y = 14 \cdot \frac{9}{784} + 3 \cdot \left(-\frac{3}{28}\right) - 1
\]
Calculating \( 14 \cdot \frac{9}{784} \):
\[
14 \cdot \frac{9}{784} = \frac{126}{784} = \frac{9}{56}
\]
Calculating \( 3 \cdot \left(-\frac{3}{28}\right) \):
\[
3 \cdot \left(-\frac{3}{28}\right) = -\frac{9}{28}
\]
Now we need to find the common denominator to combine these fractions. The common denominator is 56.
Converting \( -\frac{9}{28} \) to a fraction with a denominator of 56:
\[
-\frac{9}{28} = -\frac{18}{56}
\]
Now, add everything together:
\[
y = \frac{9}{56} - \frac{18}{56} - 1
\]
\[
y = \frac{9 - 18}{56} - 1 = -\frac{9}{56} - 1 = -\frac{9}{56} - \frac{56}{56} = -\frac{65}{56}
\]
So, the vertex of the parabola is at \( \left(-\frac{3}{28}, -\frac{65}{56}\right) \).
Since the options given have integer values for the x-coordinate as -6, it seems there might be a misunderstanding in the problem or provided options. However, given your options, none of them match the calculated vertex of the parabola.
If you would like to confirm the correctness, substituting the multiple choice options into the equation may be helpful, but they do not seem to yield (−6, -10) as the correct answer based on the equation provided originally.
\[
x = -\frac{b}{2a}
\]
In this equation, \( a = 14 \) and \( b = 3 \).
Calculating the x-coordinate:
\[
x = -\frac{3}{2(14)} = -\frac{3}{28}
\]
Now, we can substitute this x-value back into the equation to find the corresponding y-coordinate.
First, we compute \( y \):
\[
y = 14\left(-\frac{3}{28}\right)^2 + 3\left(-\frac{3}{28}\right) - 1
\]
Calculating \( \left(-\frac{3}{28}\right)^2 \):
\[
\left(-\frac{3}{28}\right)^2 = \frac{9}{784}
\]
Now substitute it back into the equation:
\[
y = 14 \cdot \frac{9}{784} + 3 \cdot \left(-\frac{3}{28}\right) - 1
\]
Calculating \( 14 \cdot \frac{9}{784} \):
\[
14 \cdot \frac{9}{784} = \frac{126}{784} = \frac{9}{56}
\]
Calculating \( 3 \cdot \left(-\frac{3}{28}\right) \):
\[
3 \cdot \left(-\frac{3}{28}\right) = -\frac{9}{28}
\]
Now we need to find the common denominator to combine these fractions. The common denominator is 56.
Converting \( -\frac{9}{28} \) to a fraction with a denominator of 56:
\[
-\frac{9}{28} = -\frac{18}{56}
\]
Now, add everything together:
\[
y = \frac{9}{56} - \frac{18}{56} - 1
\]
\[
y = \frac{9 - 18}{56} - 1 = -\frac{9}{56} - 1 = -\frac{9}{56} - \frac{56}{56} = -\frac{65}{56}
\]
So, the vertex of the parabola is at \( \left(-\frac{3}{28}, -\frac{65}{56}\right) \).
Since the options given have integer values for the x-coordinate as -6, it seems there might be a misunderstanding in the problem or provided options. However, given your options, none of them match the calculated vertex of the parabola.
If you would like to confirm the correctness, substituting the multiple choice options into the equation may be helpful, but they do not seem to yield (−6, -10) as the correct answer based on the equation provided originally.