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.
take a look at the one I did. They're all the same. The maximum (or minimum) is at the vertex.
A. To write the function in vertex form, we can complete the square. The general vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Given the function y = -3t^2 + 18t + 53, let's complete the square:
y = -3(t^2 - 6t) + 53
Next, we need to find the value that should be added inside the parentheses to complete the square. We can do this by halving the coefficient of t (-6) and squaring it:
y = -3(t^2 - 6t + 9) + 53 + 3(9)
Notice that we added 3(9) inside the parentheses to keep the expression equivalent. Simplifying further:
y = -3(t - 3)^2 + 80
So, the function in vertex form is y = -3(t - 3)^2 + 80.
B. To find the maximum temperature during the year, we look at the vertex form of the function. The vertex (-h, k) represents the highest point (maximum) of the parabola.
From the vertex form we obtained in part A, the vertex is (3, 80). Therefore, the maximum temperature during the year is 80 degrees Fahrenheit.
To write the function in vertex form, we need to complete the square by manipulating the equation.
The given equation is y = -3t^2 + 18t + 53.
A. To rewrite it in vertex form, we can follow these steps:
1. Factor out -3 from the terms involving t^2 and t to put the equation in the form y = -3(t^2 - 6t) + 53.
2. Complete the square by taking half of the coefficient of t (-6) and squaring it (-6/2)^2 = (-3)^2 = 9.
3. Add 9 to both sides of the equation to maintain its equality: y + 9 = -3(t^2 - 6t + 9).
4. The equation becomes y + 9 = -3(t - 3)^2.
5. Subtract 9 from both sides to isolate y: y = -3(t - 3)^2 - 9.
So, the function in vertex form is y = -3(t - 3)^2 - 9.
B. In the vertex form, the vertex of the parabolic graph is the maximum (or minimum) point. Therefore, the maximum temperature will occur at the vertex of the parabola, which in this case is (3, -9).
Since the temperature is represented by y, the maximum temperature during the year is -9 degrees Fahrenheit.
Holy CRAP! You reposted the same problem over and over! Did you actually try reviewing what the vertex form is?
geez!