To determine the behavior of the graph of the function [tex]\( y = 12(0.75)^x \)[/tex], let's analyze it step-by-step.
1. Recognize the Form of the Function:
The given function is an exponential decay function. It can be identified by its form [tex]\( y = ab^x \)[/tex] where [tex]\( a = 12 \)[/tex] and [tex]\( b = 0.75 \)[/tex]. Here [tex]\( b \)[/tex] is a positive number less than 1.
2. Understanding Exponential Decay:
In exponential decay, the base [tex]\( b \)[/tex] (which is 0.75 in this case) determines the rate at which [tex]\( y \)[/tex] decreases as [tex]\( x \)[/tex] increases. Since [tex]\( b \)[/tex] is between 0 and 1, we can conclude that as [tex]\( x \)[/tex] increases, the value of [tex]\( y \)[/tex] decreases.
3. Rate of Decrease:
Even though [tex]\( y \)[/tex] is decreasing, the rate at which [tex]\( y \)[/tex] decreases is also decreasing. This means that [tex]\( y \)[/tex] decreases quickly at first but the rate of decrease slows down as [tex]\( x \)[/tex] gets larger.
4. Statement Analysis:
We need to evaluate the provided statement: "As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases at an increasing rate."
- This statement implies that as [tex]\( x \)[/tex] gets larger, [tex]\( y \)[/tex] should be increasing and moreover, the speed of this increase should be accelerating.
- However, based on our previous analysis, we know that [tex]\( y \)[/tex] actually decreases as [tex]\( x \)[/tex] increases, and the rate of decrease slows down over time.
Conclusion:
The statement "As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases at an increasing rate" is incorrect for the graph of [tex]\( y = 12(0.75)^x \)[/tex]. The true behavior is that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases at a decreasing rate.
Thus, the correct assessment of the function [tex]\( y = 12(0.75)^x \)[/tex] is that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases at a decreasing rate. Therefore, the given statement is not true for the described function.
