Answer :
To determine which graph corresponds to the function [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^x \)[/tex], we can analyze the behavior of the function and use some sample points to verify. Here's a step-by-step solution:
### Step-by-Step Analysis
1. Understand the Function:
The function given is [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^x \)[/tex].
- Base: [tex]\( \frac{1}{2} \)[/tex]
- Multiplier: 4
This indicates an exponential decay function because the base [tex]\( \frac{1}{2} \)[/tex] is between 0 and 1.
2. Identify Key Characteristics:
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] decreases (since [tex]\( \left( \frac{1}{2} \right)^x \)[/tex] gets smaller).
- As [tex]\( x \)[/tex] decreases, [tex]\( f(x) \)[/tex] increases exponentially (since [tex]\( \left( \frac{1}{2} \right)^x \)[/tex] gets larger for negative [tex]\( x \)[/tex]).
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^0 = 4 \)[/tex].
- The function will never be negative and never touch the x-axis but will approach it (asymptote at [tex]\( y = 0 \)[/tex]).
3. Calculate Sample Points:
Let's use the following sample points to plot/highlight the function's behavior:
- [tex]\( f(-10) = 4096 \)[/tex]
- [tex]\( f(-9) = 2048 \)[/tex]
- [tex]\( f(-8) = 1024 \)[/tex]
- [tex]\( f(-7) = 512 \)[/tex]
- [tex]\( f(-6) = 256 \)[/tex]
- [tex]\( f(-5) = 128 \)[/tex]
- [tex]\( f(-4) = 64 \)[/tex]
- [tex]\( f(-3) = 32 \)[/tex]
- [tex]\( f(-2) = 16 \)[/tex]
- [tex]\( f(-1) = 8 \)[/tex]
- [tex]\( f(0) = 4 \)[/tex]
- [tex]\( f(1) = 2 \)[/tex]
- [tex]\( f(2) = 1 \)[/tex]
- [tex]\( f(3) = 0.5 \)[/tex]
- [tex]\( f(4) = 0.25 \)[/tex]
- [tex]\( f(5) = 0.125 \)[/tex]
- [tex]\( f(6) = 0.0625 \)[/tex]
- [tex]\( f(7) = 0.03125 \)[/tex]
- [tex]\( f(8) = 0.015625 \)[/tex]
- [tex]\( f(9) = 0.0078125 \)[/tex]
- [tex]\( f(10) = 0.00390625 \)[/tex]
### Conclusion and Graph Characteristics
From these points, you can observe that the function:
- Starts very high for large negative [tex]\( x \)[/tex] values (e.g., at [tex]\( x = -10 \)[/tex], [tex]\( y = 4096 \)[/tex]).
- Cuts through [tex]\( (0, 4) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Decreases rapidly towards 0 for positive [tex]\( x \)[/tex] values (e.g., at [tex]\( x = 10 \)[/tex], [tex]\( y = 0.00390625 \)[/tex]).
### Identifying the Graph
The correct graph of [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^x \)[/tex] should:
- Decline from a high positive value on the left to a value approaching zero on the right.
- Pass through [tex]\( (0, 4) \)[/tex].
From the given options, look for the graph that features these properties - a sharp decline from left to right, passing through the y-value of 4 at [tex]\( x = 0 \)[/tex]. This should help you identify the correct graph representing [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^x \)[/tex].
### Step-by-Step Analysis
1. Understand the Function:
The function given is [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^x \)[/tex].
- Base: [tex]\( \frac{1}{2} \)[/tex]
- Multiplier: 4
This indicates an exponential decay function because the base [tex]\( \frac{1}{2} \)[/tex] is between 0 and 1.
2. Identify Key Characteristics:
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] decreases (since [tex]\( \left( \frac{1}{2} \right)^x \)[/tex] gets smaller).
- As [tex]\( x \)[/tex] decreases, [tex]\( f(x) \)[/tex] increases exponentially (since [tex]\( \left( \frac{1}{2} \right)^x \)[/tex] gets larger for negative [tex]\( x \)[/tex]).
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^0 = 4 \)[/tex].
- The function will never be negative and never touch the x-axis but will approach it (asymptote at [tex]\( y = 0 \)[/tex]).
3. Calculate Sample Points:
Let's use the following sample points to plot/highlight the function's behavior:
- [tex]\( f(-10) = 4096 \)[/tex]
- [tex]\( f(-9) = 2048 \)[/tex]
- [tex]\( f(-8) = 1024 \)[/tex]
- [tex]\( f(-7) = 512 \)[/tex]
- [tex]\( f(-6) = 256 \)[/tex]
- [tex]\( f(-5) = 128 \)[/tex]
- [tex]\( f(-4) = 64 \)[/tex]
- [tex]\( f(-3) = 32 \)[/tex]
- [tex]\( f(-2) = 16 \)[/tex]
- [tex]\( f(-1) = 8 \)[/tex]
- [tex]\( f(0) = 4 \)[/tex]
- [tex]\( f(1) = 2 \)[/tex]
- [tex]\( f(2) = 1 \)[/tex]
- [tex]\( f(3) = 0.5 \)[/tex]
- [tex]\( f(4) = 0.25 \)[/tex]
- [tex]\( f(5) = 0.125 \)[/tex]
- [tex]\( f(6) = 0.0625 \)[/tex]
- [tex]\( f(7) = 0.03125 \)[/tex]
- [tex]\( f(8) = 0.015625 \)[/tex]
- [tex]\( f(9) = 0.0078125 \)[/tex]
- [tex]\( f(10) = 0.00390625 \)[/tex]
### Conclusion and Graph Characteristics
From these points, you can observe that the function:
- Starts very high for large negative [tex]\( x \)[/tex] values (e.g., at [tex]\( x = -10 \)[/tex], [tex]\( y = 4096 \)[/tex]).
- Cuts through [tex]\( (0, 4) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Decreases rapidly towards 0 for positive [tex]\( x \)[/tex] values (e.g., at [tex]\( x = 10 \)[/tex], [tex]\( y = 0.00390625 \)[/tex]).
### Identifying the Graph
The correct graph of [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^x \)[/tex] should:
- Decline from a high positive value on the left to a value approaching zero on the right.
- Pass through [tex]\( (0, 4) \)[/tex].
From the given options, look for the graph that features these properties - a sharp decline from left to right, passing through the y-value of 4 at [tex]\( x = 0 \)[/tex]. This should help you identify the correct graph representing [tex]\( f(x) = 4 \left( \frac{1}{2} \right)^x \)[/tex].
