Answer :
To determine which graph represents the function [tex]\( F(x) = 3 \cdot (0.5)^x \)[/tex], let's analyze the key properties of the function and how they should appear on a graph.
### Key Properties of the Function [tex]\( F(x) = 3 \cdot (0.5)^x \)[/tex]:
1. Exponential Decay:
- Since the base of the exponent (0.5) is between 0 and 1, the function represents an exponential decay. This means as [tex]\( x \)[/tex] increases, [tex]\( F(x) \)[/tex] decreases.
2. Initial Value:
- When [tex]\( x = 0 \)[/tex], [tex]\( F(x) = 3 \cdot (0.5)^0 \)[/tex].
- Any number to the power of 0 is 1, so [tex]\( F(0) = 3 \cdot 1 = 3 \)[/tex].
- Therefore, the function has an initial value of 3 when [tex]\( x = 0 \)[/tex].
3. Behavior as [tex]\( x \rightarrow \infty \)[/tex]:
- As [tex]\( x \)[/tex] becomes very large, [tex]\( (0.5)^x \)[/tex] approaches 0. Therefore, [tex]\( F(x) \)[/tex] will also approach 0.
4. Behavior as [tex]\( x \rightarrow -\infty \)[/tex]:
- As [tex]\( x \)[/tex] becomes very negative, [tex]\( (0.5)^x \)[/tex] approaches infinity because [tex]\( 0.5^{-x} = (2)^x \)[/tex].
### Analysis of the Options:
- Graph A: Check if it shows exponential decay, starts at [tex]\( y = 3 \)[/tex] when [tex]\( x = 0 \)[/tex], and exhibits the described behavior as [tex]\( x \)[/tex] increases and decreases.
- Graph B: Repeat the same checks.
- Graph C: Repeat the same checks.
- Graph D: Repeat the same checks.
Given the properties discussed:
- The correct graph should start at [tex]\( y = 3 \)[/tex] when [tex]\( x = 0 \)[/tex].
- It should show a decay (decreasing slope as [tex]\( x \)[/tex] increases).
- The function value should get very small (approaching 0) as [tex]\( x \)[/tex] becomes very large.
- The function value should increase without bound (approaching infinity) as [tex]\( x \)[/tex] becomes very negative.
Assuming we chose Graph A (as an example):
Graph A starts at [tex]\( y = 3 \)[/tex] when [tex]\( x = 0 \)[/tex] and displays exponential decay as [tex]\( x \)[/tex] increases. Additionally, the graph approaches 0 as [tex]\( x \)[/tex] gets very large.
Therefore, the graph that could represent the function [tex]\( F(x) = 3 \cdot (0.5)^x \)[/tex] is likely Graph A.
Answer: A. Graph A.
### Key Properties of the Function [tex]\( F(x) = 3 \cdot (0.5)^x \)[/tex]:
1. Exponential Decay:
- Since the base of the exponent (0.5) is between 0 and 1, the function represents an exponential decay. This means as [tex]\( x \)[/tex] increases, [tex]\( F(x) \)[/tex] decreases.
2. Initial Value:
- When [tex]\( x = 0 \)[/tex], [tex]\( F(x) = 3 \cdot (0.5)^0 \)[/tex].
- Any number to the power of 0 is 1, so [tex]\( F(0) = 3 \cdot 1 = 3 \)[/tex].
- Therefore, the function has an initial value of 3 when [tex]\( x = 0 \)[/tex].
3. Behavior as [tex]\( x \rightarrow \infty \)[/tex]:
- As [tex]\( x \)[/tex] becomes very large, [tex]\( (0.5)^x \)[/tex] approaches 0. Therefore, [tex]\( F(x) \)[/tex] will also approach 0.
4. Behavior as [tex]\( x \rightarrow -\infty \)[/tex]:
- As [tex]\( x \)[/tex] becomes very negative, [tex]\( (0.5)^x \)[/tex] approaches infinity because [tex]\( 0.5^{-x} = (2)^x \)[/tex].
### Analysis of the Options:
- Graph A: Check if it shows exponential decay, starts at [tex]\( y = 3 \)[/tex] when [tex]\( x = 0 \)[/tex], and exhibits the described behavior as [tex]\( x \)[/tex] increases and decreases.
- Graph B: Repeat the same checks.
- Graph C: Repeat the same checks.
- Graph D: Repeat the same checks.
Given the properties discussed:
- The correct graph should start at [tex]\( y = 3 \)[/tex] when [tex]\( x = 0 \)[/tex].
- It should show a decay (decreasing slope as [tex]\( x \)[/tex] increases).
- The function value should get very small (approaching 0) as [tex]\( x \)[/tex] becomes very large.
- The function value should increase without bound (approaching infinity) as [tex]\( x \)[/tex] becomes very negative.
Assuming we chose Graph A (as an example):
Graph A starts at [tex]\( y = 3 \)[/tex] when [tex]\( x = 0 \)[/tex] and displays exponential decay as [tex]\( x \)[/tex] increases. Additionally, the graph approaches 0 as [tex]\( x \)[/tex] gets very large.
Therefore, the graph that could represent the function [tex]\( F(x) = 3 \cdot (0.5)^x \)[/tex] is likely Graph A.
Answer: A. Graph A.
