Starting with the function , is it possible to build any possible linear function of the form using these transformations? If yes, can you prove it? If no, can you find a counterexample?
Starting with the function , is it possible to build any possible quadratic function of the form using these transformations? If yes, can you prove it? If no, can you find a counterexample?
Starting with the function , is it possible to build any possible absolute value function of the form
using these transformations? If yes, can you prove it? If no, can you find a counterexample?Can you build a constant function with this environment? Why or why not?
Think about the following questions. For the cases examined in this environment
• Vertical sliding of f(x) leads to f(x) + a. [Which way does the function slide if a > 0? a < 0?]
• Horizontal sliding of f(x) leads to f(x+a). [Which way does the function slide if a > 0? a < 0?]
• Vertical stretching & squeezing of f(x) leads to af(x). [What happens to the function if a < 0? 0 < a < 1? a > 1? ]
• Horizontal stretching & squeezing of f(x) leads to af(x). [What happens to the function if a < 0? 0< a < 1? a > 1? ]
Do you believe these statements are true for any function of one variable? If so, can you prove it? If not, can you find a counterexample?
Can you build a cubic function with this environment? Why or why not?