Start by identifying the transformations taking place to go from f(x) to g(x).
See solution.
Practice makes perfect
We can start by identifying the transformations taking place to go from f(x)= log (x) to g(x)=log100 x-1. By comparing both functions, we can note that g(x) is a transformation of the form shown below.
4cm f( ax) + k [-0.5em]
f(x) = log (x) → 1.5cm [-0.5em]
3.8cm g(x) = log( 100 x) + ( -1)
In this case, the parameter a= 100 causes a horizontal shrink by a factor of 1100, shrinking the graphs towards the y-axis. On the other hand, the parameter k = -1 shifts the graph vertically down by 1 unit. Two different ways to obtain g(x) from f(x) are by applying these transformations in a different order.
We can of course find more transformations to go from f(x) to g(x). We have already shown the two more efficient ones, but there are many more ways to do it. A third example is shown below.
Transformation
Transformation Notation
Resulting Function
Original Function
f(x)
f(x)= log (x)
1. Reflection in x-axis
f(x) → - f(x)
f_2(x) = - log(x)
2. Vertical translation (up by 1 unit)
f_2(x) → f_2(x) +1
f_3(x) = - log(x) + 1
3. Horizontal shrink by a factor of 100
f_3(x) → f_3(100x)
f_4(x) = - log(100x) + 1
4. Reflection in x-axis
f_4(x) → - f_4(x)
g(x) = log(100x) - 1
This series of transformations are depicted in the graph below. Give it a try!
