Compare the functions below: f(x) = −3 sin(x − π) + 2 g(x) x y 0 8 1 3 2 0 3 −1 4 0 5 3 6 8 h(x) = (x + 7)2 − 1 which function has the smallest minimum?

To answer "which function has the smallest minimum," we'll first find the minimum of each one separately.

[1] f(x) = -3 sin(x - pi) + 2. No matter how crazy the inside of a sin function looks, the value of sin itself is always between 1 and -1. So, the minimum value for f(x) is -3*1 + 2 = -1.

[2] g(x). By looking at the table, we see that the minimum value is -1, which occurs when x = 3.

[3] h(x) = (x+7)^2 - 1. Notice that (x+7) is being squared, so the smallest that piece could be is 0 (you can never get a negative number out of (x+7)^2...). So, the minimum value of h(x) is 0 - 1 = -1.

At the end of the day, all three functions have the same minimum value! This can be confirmed on a graph. So, "which function has the smallest minimum value?" all of them!

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