Compare the functions shown below:

f (x) = 2 sin (3x + π) - 2

g (x) = (x - 3) 2 - 1

h (x)

x y

-2 3

-1 - 2

0 - 5

1 - 6

2 - 5

3 - 2

4 3

Which function has the smallest minimum y-value?

The correct answer is:

h (x)

Explanation:

Graphing f (x) and tracing the function, we find the smallest y-value to be - 4.

We can use the form of the equation for g (x). It is in vertex form, which is

g (x) = a (x-h) ²+k, where (h, k) is the vertex.

In our function, g (x) = (x-3) ²-1, we have a vertex of (3, - 1). The vertex of a quadratic function is either the maximum or minimum. Since the value of a would be 1, the graph would open upward; thus the vertex would be a minimum, and - 1 would be the minimum y-value.

We can see from the table that the smallest y-value in h (x) is - 6. This is smaller than the other two, so this is the smallest.

F (x) = 2 sin (3x + pi) - 2 has the minimum y-value of - 4

g (x) = (x - 3) ^2 - 1 has a minimum y-value of - 1

h (x) has the minimum y-value of - 6

Therefore, h (x) has the smallest minimum y-value.