Real World Problems

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.

b. Find the maximum temperature during the year.

Answer 1

complete the square

-7t^2+14t+20 = -7(t^2-2t)+20 = -7(t-1)^2+27

Use this to solve other questions!

Answer 2

In looking at the results, they don't pass the common sense test. Please double check your function to make sure you didn't make an error typing it. But I'll proceed as you have written it:

There are a couple of ways to write the equation in vertex form. 1) Complete the square (the result will be in vertex form) or 2) Use x = -b/(2a) then plug the result in to get the vertex then, once you have the vertex, recreate the vertex form.

Method a) Completing the Square:

Step 1: Scoot the number off to the right

y = -7t² + 14t + 20

Step 2, Factor out what's in front of the t²:

y = -7(t² - 2t ) + 20

Step 3: Add in a number that will give a perfect square. That number is the square of half the number in front of the "t". Then subtract off the equivalent number at the end (next to the number coefficient). When subtracting off the equivalent number be mindful of what you have factored out and adjust accordingly:

y = -7(t² - 2t + 1) + 20 + 7

Notice that I had to add 1 to make the quadratic a perfect square but that by adding 1, because there was a "-7" factored out in front, I really added in -7. So I had to add 7 to make things equal out.

Step 4: Write the quadratic as a square and combine the numbers at the end:

y = -7(t - 1)² + 27

Method b) Using x = -b/(2a)

Step 1: Using x = -b/(2a), calculate the x-value of the vertex

In this case, b = 14 and a = -7

x = -14/(2•-7) = -14/-14 = 1

Step 2: Plug in x = 1 to find the y-value of the vertex:

y = -7(1)² + 14(1) + 20

y = -7 + 14 + 20

y = 27

Step 3: Write the Vertex Form using the Vertex:

Using the generic Vertex Form y = a(t - h)² + k where the vertex is (h, k) and a is the coefficient in front of t² we get:

y = -7(t - 1)² + 27

Question b: The max temp occurs at the vertex (1, 27) occurring in month 1 and is 27 degrees F. See what I mean, doesn't pass the sanity check (unless this is Antarctica)