Question
Functions & Their Graphs Online Practice
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Question
A volleyball is served from a height of 6 feet with an initial velocity of 33 feet per second. The situation is modeled using the function h=−6t2+33t+6
, where h is the height in feet and t is time in seconds. Using the function, which ordered pair represents the vertex?(1 point)
Responses
(51.375,2.75)
left parenthesis 51.375 comma 2.75 right parenthesis
(5.676,0)
left parenthesis 5.676 comma 0 right parenthesis
(0,6)
left parenthesis 0 comma 6 right parenthesis
(2.75,51.375)
left parenthesis 2.75 comma 51.375 right parenthesis
Remaining Attempts : 3
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Answers
GPT-4o mini
To find the vertex of the quadratic function given by \(h(t) = -6t^2 + 33t + 6\), we can use the formula for the t-coordinate of the vertex, which is given by:
\[
t = -\frac{b}{2a}
\]
where \(a\) is the coefficient of \(t^2\) and \(b\) is the coefficient of \(t\).
In your function:
- \(a = -6\)
- \(b = 33\)
Plugging in the values:
\[
t = -\frac{33}{2(-6)} = \frac{33}{12} = 2.75
\]
Next, we need to find the height \(h\) at \(t = 2.75\).
Substituting \(t = 2.75\) back into the function:
\[
h(2.75) = -6(2.75)^2 + 33(2.75) + 6
\]
Calculating \( (2.75)^2 \):
\[
(2.75)^2 = 7.5625
\]
Now substituting back into the equation:
\[
h(2.75) = -6(7.5625) + 33(2.75) + 6
\]
Calculate \(-6(7.5625)\):
\[
-6(7.5625) = -45.375
\]
Calculate \(33(2.75)\):
\[
33(2.75) = 90.75
\]
Now combine these values:
\[
h(2.75) = -45.375 + 90.75 + 6
\]
Calculating this gives:
\[
h(2.75) = 51.375
\]
Thus, the vertex of the function is at:
\[
(2.75, 51.375)
\]
Therefore, the correct response is:
**(2.75, 51.375)**
\[
t = -\frac{b}{2a}
\]
where \(a\) is the coefficient of \(t^2\) and \(b\) is the coefficient of \(t\).
In your function:
- \(a = -6\)
- \(b = 33\)
Plugging in the values:
\[
t = -\frac{33}{2(-6)} = \frac{33}{12} = 2.75
\]
Next, we need to find the height \(h\) at \(t = 2.75\).
Substituting \(t = 2.75\) back into the function:
\[
h(2.75) = -6(2.75)^2 + 33(2.75) + 6
\]
Calculating \( (2.75)^2 \):
\[
(2.75)^2 = 7.5625
\]
Now substituting back into the equation:
\[
h(2.75) = -6(7.5625) + 33(2.75) + 6
\]
Calculate \(-6(7.5625)\):
\[
-6(7.5625) = -45.375
\]
Calculate \(33(2.75)\):
\[
33(2.75) = 90.75
\]
Now combine these values:
\[
h(2.75) = -45.375 + 90.75 + 6
\]
Calculating this gives:
\[
h(2.75) = 51.375
\]
Thus, the vertex of the function is at:
\[
(2.75, 51.375)
\]
Therefore, the correct response is:
**(2.75, 51.375)**