The temperature y (in degrees Fahrenheit) after t months can be modeled by the function y=−t2+18t −31, where 1≤t≤12.

Question

The temperature y (in degrees Fahrenheit) after t months can be modeled by the function y=−t2+18t −31, where 1≤t≤12.

Write the function in vertex form.

Find the maximum temperature during the year.

Answer 1

Hi Zoe,

Vertex form is a very helpful way to write quadratic equations. It looks like this: y = a(x - h)² + k. It gives you the vertex coordinates (h, k), the scale of the graph (a), and it's also easy to find points to plot.

So how do we get to vertex form? The key is in (x - h)²: this is a square of a binomial, so why don't we try completing the square! Given an equation of the form y = ax² + bx + c, we complete the square as follows:

1) Make sure a is positive! We can't square a binomial and get a negative a. In our case, a is negative (-t²), so we'll have to detach that negative from the a like so: y = -(t² - 18t) - 31. Now, we'll be able to complete the square within the parentheses.

2) Find what number d times 2 equals b. This is because when we expand a squared binomial like (x + a)², we get x² + 2ax + c. So our d is 2d = -18 (notice, b became negative because we detached the negative from a and b in the first step), and d = -9.

3) Find d squared (81), and add it inside the parentheses: y = -(t - 18x + 81) - 31 + ___. There's a blank there because we can't simply put an 81 to only the right side, unless we cancel it out at the same time. You might think we'll cancel it out by subtracting 81, since we added it inside the parentheses, but be careful! Because everything is multiplied by -1 inside the parentheses, we actually subtracted 81 from the right side. That means we have to actually add 81 to the blank. Then, if we combine the -31 and +81, our equation becomes y = -(t² - 18x + 81) + 50.

4) Finally, factor the expression inside the parentheses: (t² - 18x + 81) becomes (t - 9)² (verify that's correct!) and our final equation, in vertex form, is y = -(t - 9)² + 50.

Now, finding the maximum temperature is as simple as plugging in 9 for t. Why? If t = 9, -(t - 9)² equals 0, and subsequently, y = 50. But if t is anything else, -(t - 9)² becomes some negative, and so y < 50. So the maximum is 50°F

There's plenty more to say about this problem and vertex form in general, but that's enough for your solution. Hope that helped :)

Answer 2

y = −t² + 18t − 31, where 1 ≤ t ≤ 12.

Let a = -1, b = 18, and c = -31. Use these coefficients to find the vertex.

x = -b/2a

x = -18/2(-1) = -18/(-2) = 9

y = -(9)² + 18(9) - 31 = -81 + 162 - 31 = 50

Vertex form: y = a(x - h)² + k

y = −(x - 9)² + 50

The maximum temperature during the year is 50 degrees Fahrenheit (50^oF).