Transforming Linear Functions

Question

What value of k transforms the graph of f(x)=0.5x+3 into graph g? Describe the transformation.

1. g(x)=f(x)+k:
2. g(x)=f(x-k):
3. g(x)=kf(x):

Answer 1

When doing these types of problems it is best to plug f(x) into g(x) to better visualize what is happening.

If you plug in for #1 you will get g(x)=(0.5x+3)+k this is equal to g(x)=0.5x+(3+k).

In this case, where k is a constant, you are adding it to 3, resulting in another constant (3+k). Since you are adding k, rather than subtracting, the graph will be shifted up, k units. this will make the y-intercept of g(x) 3+k.

In #2 you will first have to plug in (x-k) into f(x). This means that anywhere you see x in f(x) you will replace it with x-k. This yields f(x-k)=0.5(x-k)+3. This means that g(x)=0.5(x-k)+3. In this case, k is directly affected x unlike the previous example where it is separate from the x term. This indicates a horizontal shift. In this case, subtracting k will shift the graph k units to the right where as adding k will shift k units to the right.

In #3 if you plug in f(x) you get g(x)=k(0.5x+3)=(0.5k)x+3k, where (0.5k is the slope of the line and 3k in the y-intercept.The resulting function, g(x) will have a different slope and y-intercept, yet still have the same x-intercept as f(x).

I cannot see the graph g(x) so I can't answer the first part of your question, but I hope this helps!