Write the function in vertex form. Find the maximum temperature during the year.

Question

The temperature y (in degrees Fahrenheit) after t months can be modeled by the function y=−7t^2+14t +20, where 1≤t≤12.

a.  Write the function in vertex form.

y=

b.  Find the maximum temperature during the year.

The maximum temperature is 

? degrees Fahrenheit.

Answer 1

(a) The vertex form of a parabola is y = a(x - h)² + k, where (h, k) is the coordinate of the vertex. The original problem gives the parabola in the form of y = ax² + bx + c, where a = -7, b = 14, and c = 20. Recall that to find h, you simply do -b / 2a, so h = -14 / (2 * -7) = 1. To find k, we substitute the value of h into the original equation and solve, so k = -7(1)^2 + 14(1) + 20 = 27.

Putting our values of a, h, and k into the vertex form equation, we get y = -7(x - 1)² + 27.

(b) The vertex (h, k) of a parabola is the maximum is a is negative, and the minimum if a is positive. Since a = -7, the vertex is the maximum point of the parabola. The maximum temperature is the same as k in our vertex form, which is 27 degrees Fahrenheit.