y=aˣ（a>0, a≠1）: It passes through the point (0,1). For a>1, it is monotonically increasing; for 0<a<1, it is monotonically decreasing. The asymptote is the x-axis (y=0)。
Example ①
State the asymptote of y=2^(x+1)−3 and its value when x=0.
Solution:Asymptote y=−3。x=0: y=2−3=−1 → point (0,−1)
State the asymptote of y=2^(x+1)−3 and its value when x=0.
Solution:Asymptote y=−3。x=0: y=2−3=−1 → point (0,−1)
Example ②
Explain the relationship between the graphs of y=2ˣ and y=(1/2)ˣ.
Solution:They are symmetric about the y-axis。（(1/2)ˣ=2^(−x)）
Explain the relationship between the graphs of y=2ˣ and y=(1/2)ˣ.
Solution:They are symmetric about the y-axis。（(1/2)ˣ=2^(−x)）
Key Points
- aˣ is always positive / a>1: increasing, 0<a<1: decreasing / asymptote is y=k (the amount of vertical translation)
Practice Problems
- State the monotonicity and range of y=(1/3)ˣ.
- Explain the relationship between y=3ˣ and y=3^(x+2).
- Find y=a×bˣ given y=6 when x=1 and y=54 when x=3.
Solutions and Explanations
- Answer:Monotonically decreasing, y>0
- Answer:y=3^(x+2) is y=3ˣ shifted 2 units to the left (or multiplied by 9)
- Answer:y=2×3ˣ
Explanation:ab=6、ab³=54→b²=9→b=3、a=2。