To graph the function [tex]\( f(x) = 3 \cdot (0.5)^x \)[/tex], let's follow these steps:
1. Identify Key Characteristics:
- The base function is [tex]\( (0.5)^x \)[/tex], which is an exponential decay function because the base [tex]\( 0.5 \)[/tex] is between 0 and 1.
- Multiplying by 3 will stretch the graph vertically by a factor of 3.
2. Determine Key Points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3 \cdot (0.5)^0 = 3 \cdot 1 = 3
\][/tex]
So, the point [tex]\( (0, 3) \)[/tex] is on the graph.
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3 \cdot (0.5)^1 = 3 \cdot 0.5 = 1.5
\][/tex]
So, the point [tex]\( (1, 1.5) \)[/tex] is on the graph.
- When [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 3 \cdot (0.5)^{-1} = 3 \cdot 2 = 6
\][/tex]
So, the point [tex]\( (-1, 6) \)[/tex] is on the graph.
3. Plot the Points:
- Plot the points [tex]\( (0, 3) \)[/tex], [tex]\( (1, 1.5) \)[/tex], and [tex]\( (-1, 6) \)[/tex].
4. Draw the Exponential Curve:
- As [tex]\( x \)[/tex] increases, [tex]\( (0.5)^x \)[/tex] gets closer to 0, so [tex]\( f(x) \)[/tex] will approach 0 but never touch the x-axis.
- As [tex]\( x \)[/tex] decreases (becomes more negative), [tex]\( (0.5)^x \)[/tex] increases exponentially, and thus [tex]\( f(x) \)[/tex] will increase exponentially since it is multiplied by 3.
5. Check for Asymptote:
- The horizontal asymptote of the function is the x-axis ([tex]\( y = 0 \)[/tex]), since the exponential function will approach 0 but never reach it as [tex]\( x \)[/tex] goes to positive infinity.
Now, examine the provided answer choices and determine which graph has the following characteristics:
- A curve passing through the points (0, 3), (1, 1.5), and (-1, 6).
- An exponential decay shape, approaching zero as x increases.
- A horizontal asymptote at y = 0.
Based on these characteristics, compare with the given options A, B, C, and D, and match the correct graph with the described points and behavior.
