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Answer:

Step-by-step explanation:

We are provided with the information that

Function f is decreasing as it moves from quadrant 2 to quadrant 1, tending towards y=0

It intersects the y-axis at the point (0,6) and goes through (1,2).

Function g(x) also approaches y=0 in quadrant 2, but increases in quadrant 1.

It passes through (-1,2) and crosses the y-axis at (0,6).

Reflection across y-axis:

The transformation rule is identified as

Applying this rule, we derive

Then, by substituting

for x=-1

for x=0

Consequently, holds true.

The behavior of the spring can be described using either a sine or cosine function. The spring's maximum displacement is 6 inches, occurring at t=0, which we will define as the positive peak. Therefore, we can express the function as:
6sin(at+B). The spring's period is 4 minutes, which means the time factor in the equation must complete a cycle (2π) in 4 minutes. This gives us the equation 4min*a=2π, leading to a=π/2. Generally, a=2π/T where a is the coefficient and T is the period. For B, since sin(π/2)=1, we determine B=π/2 because at t=0, the equation becomes 6sin(B)=6. Therefore, we substitute to form f(t)=6sin(πt/2+π/2)=6cos(πt/2)
due to trigonometric relations.