The temperature y (in degrees Fahrenheit) after t months can be modeled by the function y=-3t^2+18t+53 where 1<t<12.
A. Write the function in vertex form y=__
B. Find the maximum temperature during the year.
A. To write the function in vertex form, we need to complete the square. The vertex form of a quadratic function is given by:
y = a(t - h)^2 + k
where (h, k) is the vertex of the parabola. To convert the given function into vertex form, we need to factor out the coefficient of t^2, which is -3:
y = -3(t^2 - 6t) + 53
Now, we need to complete the square for the expression inside the parentheses. To do this, we take half of the coefficient of t (-6) and square it:
(-6/2)^2 = (-3)^2 = 9
Now, we can rewrite the function:
y = -3(t^2 - 6t + 9) + 53 - 3(9)
= -3(t - 3)^2 + 20
So, the function in vertex form is y = -3(t - 3)^2 + 20.
B. The maximum temperature occurs at the vertex of the parabola. In the vertex form, the vertex is given by the values (h, k). In this case, the vertex is (3, 20).
Therefore, the maximum temperature during the year is 20 degrees Fahrenheit.
To write the function in vertex form, we need to complete the square. The vertex form of a quadratic function is given by y = a(t - h)^2 + k, where (h, k) represents the vertex of the parabola.
A. To rewrite the function y = -3t^2 + 18t + 53 in vertex form, we follow these steps:
Step 1: Divide the coefficient of t^2 (in this case, -3) by 2: -3/2 = -1.5.
Step 2: Square the result: (-1.5)^2 = 2.25.
y = -3t^2 + 18t + 53
= -3(t^2 - 6t) + 53 [Notice that we factored out -3 from the first two terms]
= -3(t^2 - 6t + 9) + 53 - 3(9) [We add and subtract (b/2)^2 = 9 to complete the square]
= -3(t - 3)^2 + 26
Therefore, the function in vertex form is y = -3(t - 3)^2 + 26.
B. To find the maximum temperature during the year, we need to determine the y-coordinate of the vertex of the parabola. In this case, the vertex form of the function is y = -3(t - 3)^2 + 26.
Comparing with the general vertex form y = a(t - h)^2 + k, we can conclude that the vertex is at the point (h, k) = (3, 26).
Therefore, the maximum temperature during the year is 26 degrees Fahrenheit.