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,−9)left parenthesis negative 6 comma negative 9 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 vertex \( x \) coordinate, which is given by
\[
x = -\frac{b}{2a}
\]
where \( a \) and \( b \) are the coefficients from the quadratic equation \( ax^2 + bx + c \). In this case, \( a = 14 \) and \( b = 3 \).
Substituting the values:
\[
x = -\frac{3}{2 \cdot 14} = -\frac{3}{28}
\]
Now we 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 \( \left(-\frac{3}{28}\right)^2 \):
\[
\left(-\frac{3}{28}\right)^2 = \frac{9}{784}
\]
Now multiply by 14:
\[
14 \cdot \frac{9}{784} = \frac{126}{784} = \frac{63}{392} = \frac{9}{56} \text{ (after simplifying) }
\]
Next, calculate \( 3 \cdot \left(-\frac{3}{28}\right) \):
\[
3 \cdot \left(-\frac{3}{28}\right) = -\frac{9}{28}
\]
Now we can combine the terms for \( y \):
\[
y = \frac{9}{56} - \frac{9}{28} - 1
\]
Convert \( -\frac{9}{28} \) to have a common denominator of 56:
\[
-\frac{9}{28} = -\frac{18}{56}
\]
So now adding those two fractions:
\[
y = \frac{9}{56} - \frac{18}{56} - 1 = -\frac{9}{56} - 1
\]
Convert -1 to have a common denominator of 56:
\[
-1 = -\frac{56}{56}
\]
So now adding these together gives:
\[
y = -\frac{9}{56} - \frac{56}{56} = -\frac{65}{56}
\]
Thus, the vertex of the parabola is:
\[
\left(-\frac{3}{28}, -\frac{65}{56}\right)
\]
However, since none of the provided options seems to match our calculations directly, we might want to re-check the problem. The original question presents options that suggest a possible calculation error on my part or a different vertex method.
You might also prefer to check the discriminant or analyzing directly the coordinates given one by one to see which ordered pair corresponds closest after analyzing the computations. However, let’s consider from provided options, none of them fit is based on standard quadratic formula calculations.
More likely, you should double-check the provided pairs:
- \((-6, -10)\)
- \((-6, -1)\)
- \((-6, -9)\)
- \((-6, -11)\)
In terms of the original polynomial/check differences against the standard vertex check (computed via changes in \( k = -\frac{b}{2a} \)). However, at face value none align, verification is recommended.
In direct calculation method, with proper substitution through vertex simplifications may be necessary to align with outcomes or justification as no matches arise.
We can conclude with strong verification.
\[
x = -\frac{b}{2a}
\]
where \( a \) and \( b \) are the coefficients from the quadratic equation \( ax^2 + bx + c \). In this case, \( a = 14 \) and \( b = 3 \).
Substituting the values:
\[
x = -\frac{3}{2 \cdot 14} = -\frac{3}{28}
\]
Now we 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 \( \left(-\frac{3}{28}\right)^2 \):
\[
\left(-\frac{3}{28}\right)^2 = \frac{9}{784}
\]
Now multiply by 14:
\[
14 \cdot \frac{9}{784} = \frac{126}{784} = \frac{63}{392} = \frac{9}{56} \text{ (after simplifying) }
\]
Next, calculate \( 3 \cdot \left(-\frac{3}{28}\right) \):
\[
3 \cdot \left(-\frac{3}{28}\right) = -\frac{9}{28}
\]
Now we can combine the terms for \( y \):
\[
y = \frac{9}{56} - \frac{9}{28} - 1
\]
Convert \( -\frac{9}{28} \) to have a common denominator of 56:
\[
-\frac{9}{28} = -\frac{18}{56}
\]
So now adding those two fractions:
\[
y = \frac{9}{56} - \frac{18}{56} - 1 = -\frac{9}{56} - 1
\]
Convert -1 to have a common denominator of 56:
\[
-1 = -\frac{56}{56}
\]
So now adding these together gives:
\[
y = -\frac{9}{56} - \frac{56}{56} = -\frac{65}{56}
\]
Thus, the vertex of the parabola is:
\[
\left(-\frac{3}{28}, -\frac{65}{56}\right)
\]
However, since none of the provided options seems to match our calculations directly, we might want to re-check the problem. The original question presents options that suggest a possible calculation error on my part or a different vertex method.
You might also prefer to check the discriminant or analyzing directly the coordinates given one by one to see which ordered pair corresponds closest after analyzing the computations. However, let’s consider from provided options, none of them fit is based on standard quadratic formula calculations.
More likely, you should double-check the provided pairs:
- \((-6, -10)\)
- \((-6, -1)\)
- \((-6, -9)\)
- \((-6, -11)\)
In terms of the original polynomial/check differences against the standard vertex check (computed via changes in \( k = -\frac{b}{2a} \)). However, at face value none align, verification is recommended.
In direct calculation method, with proper substitution through vertex simplifications may be necessary to align with outcomes or justification as no matches arise.
We can conclude with strong verification.