Beginning with the function f(x) = log a^x, state what transformations were used on this to obtain the functions given below: (6 marks)
p(x) = - 5/8 loga^x
r(x) = log a(5 - x)
t(x) = 2 log a^2x
To determine the transformations applied to the function f(x) = log a^x in order to obtain the given functions, we will examine each function separately.
The general form of the logarithmic function is f(x) = log a(x), where a is the base, and x is the argument of the logarithm.
1) For the function p(x) = -5/8 log a^x:
- The coefficient -5/8 is a vertical stretch factor. It indicates that the logarithmic function has been vertically compressed by a factor of 5/8.
- The negative sign in front of the logarithm reflects a vertical reflection, causing the graph to be reflected about the x-axis.
2) For the function r(x) = log a(5 - x):
- The transformation (5 - x) inside the logarithm is a horizontal shift. It implies that the graph has been shifted 5 units to the right.
- There are no vertical stretches or reflections applied, so the graph retains its original shape.
3) For the function t(x) = 2 log a^2x:
- The coefficient 2 serves as a vertical stretch factor. It indicates that the logarithmic function has been vertically stretched by a factor of 2.
- There are no horizontal shifts or reflections applied, so the graph remains unchanged horizontally.
In summary, the transformations applied to the function f(x) = log a^x in order to obtain the given functions are as follows:
p(x) = -5/8 log a^x:
- Vertically compressed by a factor of 5/8
- Reflexion about the x-axis
r(x) = log a(5 - x):
- Shifted 5 units to the right horizontally
t(x) = 2 log a^2x:
- Vertically stretched by a factor of 2