Question

A music club was formed in the month of January last year. The function m(t) = t2 - 4t + 25 models the number of club members for the first 10 months, where t is the number of months since the club's inception. Complete the square, and graph this function. What is the minimum number of members the club had during the year? A. 29 members B. 24 members C. 22 members D. 21 members

Answer 1

D

Explanation:

We have to find the vertex of m(t) = t² - 4t + 25

a = 1

b = 4

c = 25

To do that we first need to move the 25 to the other side by subtracting 25 from both sides

m(t) = t² - 4t + 25

- 25 - 25

m(t) - 25 = t² - 4t

To continue completing the square, we have to add 4 to both sides since it is the square of half of b.

m(t) - 25 + 4 = t² - 4t + 4

m(t) - 21 = t² - 4t + 4

Now, we already found the minimum amount of members, but let's finish the square completion to find the vertex.

m(t) - 21 = t² - 4t + 4

Factor the right side

m(t) - 21 = (t-2)(t-2)

m(t) - 21 = (t-2)²

Finish it by adding 21 to both sides

m(t) - 21 = (t-2)²

+ 21 +21

m(t) = (t-2)² + 21

With this vertex form equation, we can tell that the vertex is (2,21)

Meaning 21 is the lowest number for m(t) for this equation.