Question

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?

Answer 1

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.

Answer 2

The correct answer is:

h(x)

Step-by-step 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.