To analyze the differences between the exponential function [tex]\( y = 3^x \)[/tex] and [tex]\( y = 3^x - 1 \)[/tex], let's go through the multiple-choice options and figure out which one correctly describes the given functions.
1. Option A:
- Statement: "The graph rises to the right and [tex]\( y = 3^x - 1 \)[/tex] rises to the left."
- Analysis: An exponential function [tex]\( y = 3^x \)[/tex] increases as [tex]\( x \)[/tex] increases, which means it rises to the right. The modified function [tex]\( y = 3^x - 1 \)[/tex] behaves similarly because it is simply the original function shifted downward by 1 unit. Therefore, [tex]\( y = 3^x - 1 \)[/tex] also rises to the right, not to the left.
- Conclusion: This statement is incorrect.
2. Option B:
- Statement: "The graph of [tex]\( y = 3^x - 1 \)[/tex] is shifted one unit to the right of the function graphed."
- Analysis: The function [tex]\( y = 3^x - 1 \)[/tex] represents a vertical shift, not a horizontal one. Shifting a graph downward by 1 unit means that for every [tex]\( x \)[/tex], the value of [tex]\( y \)[/tex] is decreased by 1, but its shape and position along the [tex]\( x \)[/tex]-axis remain the same relative to the base function [tex]\( y = 3^x \)[/tex].
- Conclusion: This statement is incorrect.
3. Option C:
- Statement: "The horizontal asymptote of the graph is [tex]\( y = 0 \)[/tex] and the horizontal asymptote of [tex]\( y = 3^x - 1 \)[/tex] is"
- Analysis: For an exponential function [tex]\( y = 3^x \)[/tex], the horizontal asymptote is [tex]\( y = 0 \)[/tex] since as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 3^x \)[/tex] approaches 0. For the modified function [tex]\( y = 3^x - 1 \)[/tex], the entire graph is shifted down by 1 unit, thus the horizontal asymptote also shifts down by 1 unit: [tex]\( y = -1 \)[/tex]. However, the statement seems to be incomplete.
- Conclusion: This statement is factually correct regarding the horizontal asymptotes but is incomplete.
4. Option D:
- Statement: "The horizontal asymptote of the graph is [tex]\( y = 0 \)[/tex] and the horizontal asymptote of [tex]\( y = 3^x - 1 \)[/tex] is [tex]\( y = -1 \)[/tex]."
- Analysis: For [tex]\( y = 3^x \)[/tex], the horizontal asymptote is indeed [tex]\( y = 0 \)[/tex]. The function [tex]\( y = 3^x - 1 \)[/tex] has a horizontal asymptote shifted down by 1 unit, making it [tex]\( y = -1 \)[/tex].
- Conclusion: This statement is complete and factually correct.
Given this detailed analysis, we can conclude that the correct answer is:
Option D: The horizontal asymptote of the graph is [tex]\( y = 0 \)[/tex] and the horizontal asymptote of [tex]\( y = 3^x - 1 \)[/tex] is [tex]\( y = -1 \)[/tex].
