You have experienced transformations with both exponential functions and with quadratic functions. Does this apply to any function? Go to Desmos and try the following functions:
- \(f(x) = x^3\)
- \(f(x) = x^3 + 2\). Compare this to #1, how is it different?
- \(f(x) = (x-1)^3\). Compare this to #1, how is it different?
- \(f(x) = (x + 1)^3 -2\). Compare this again to #1, how is it different?
- \(f(x) = 2x^3\). Again compare this to #1 and describe the difference.
- Given the function \(f(x)\), Describe what happens in the graph of \(f(x) = 2 (x-1)^2 -3\).
Did you see that #2 is a vertical translation, #3 is a horizontal translation, #4 has both a vertical and a horizontal translation, #5 is a dialation (each point in the original is multiplied by 2 and #6 has a vertical translation down 3, horizontal translation right 1 and a dialation.
Can you generalize this to any function? You could go to Desmos again and try it with a quartic, a trigonometric function. What do you see?