Which function is shown in the graph below?
A) y=(1/2)^x+3 -1
B) y=(1/2)^x-3 +1
C) y=(1/2)^x-1 +3
D) y=(1/2)^x+1 -3
The Correct Answer and Explanation is :
To determine which function is represented by the graph, we need to analyze the key features of the provided options and the graph itself.
Function Analysis
- Form of the Function: All options are in the form (y = a^x + k), where (a) is a base and (k) is a vertical shift. The base is (1/2), indicating a decreasing exponential function.
- Vertical Shift: The vertical shift (k) affects the horizontal asymptote of the graph. If (k > 0), the asymptote will be (y = k) (above the x-axis), and if (k < 0), it will be (y = k) (below the x-axis).
Evaluating Options
- A) (y = (1/2)^x + 3 – 1 = (1/2)^x + 2): The horizontal asymptote is at (y = 2).
- B) (y = (1/2)^x – 3 + 1 = (1/2)^x – 2): The horizontal asymptote is at (y = -2).
- C) (y = (1/2)^x – 1 + 3 = (1/2)^x + 2): The horizontal asymptote is at (y = 2).
- D) (y = (1/2)^x + 1 – 3 = (1/2)^x – 2): The horizontal asymptote is at (y = -2).
Asymptote Determination
To match with the graph, identify the horizontal asymptote:
- If the graph approaches a specific line as (x) approaches infinity, it indicates the value of (k).
Conclusion
Assuming the graph’s asymptote is at (y = -2), the correct function would be either B) (y = (1/2)^x – 2) or D) (y = (1/2)^x – 2). Both options provide the same asymptote.
In examining the behavior as (x) approaches positive infinity, the graph would decrease towards this line, confirming the match. If the graph displays a shift upwards, options A and C can be eliminated. Thus, you can select option B or D, depending on the specific curve behavior near the asymptote.
To finalize which one is correct, observe the specific details in the graph about the intercepts and behavior at x-values approaching zero and positive infinity.