Functions as Objects - operations on functions
- Judah L Schwartz
Using this tool you can select two functions - each of which can be a linear, quadratic, absolute value function. You can then add, subtract, multiply, divide or compose them. Challenge: Adding & subtracting functions Choose a linear function for f(x). Find a function g(x), which when added to f(x) gives you a constant function. Can you adjust g(x) so that you can get any constant function? What happens to the graph of your sum function as you vary the slope of g(x)? the y-intercept of g(x)? Challenge: Understanding factoring Choose a quadratic function for f(x) – e.g., set f(x) = by setting the a slider to 1, the b slider to 1 and the c slider to -2 Choose a linear function for g(x) – e.g., set g(x) = by setting the A slider to – 1 and the B slider to 1. Choose an operation on the left side of the screen. 1. What can you say about the quotient f(x) / g(x)? Is it linear? quadratic? why? 2. What happens to the quotient when you vary the slope of the linear function? why? can you explain why the quotient has the shape it does? 3. Reset the linear function g(x) to and now vary the y-intercept of g(x). What happens to the quotient when you vary the y-intercept of the linear function? why? can you explain why the quotient has the shape it does? 4. Suppose the linear function had been in the form g(x) = . Would the same things have happened when you vary A and B? Why or why not? 5. What would the quotient look like if you divided g(x) by f(x) rather than f(x) by g(x)? How does this quotient vary as you vary the y-intercept of the linear function? Challenge: Understanding multiplication of functions and FOIL Choose linear functions for f(x) and g(x) – say f(x) = and g(x) = Before choosing the multiply operation, predict the general shape of the product function and where it will be positive and where it will be negative. 1. What happens when you multiply f(x) and g(x)? 2. How does the graph of the product function depend on the slope of f(x)? slope of g(x)? y-intercept of f(x)? y-intercept of g(x), x-intercept of f(x)? x-intercept of g(x)? 3. Can you make a parabola with no real roots, such as by multiplying two linear functions? Why or why not? Challenge: Understanding composition Choose a linear function f(x). Can you find a function g(x) such that the composed function is f(x)… translated horizontally? translated vertically? dilated horizontally? dilated vertically? reflected in the x-axis? reflected in the y-axis? Does it make a difference if you choose f(g(x)) or g(f(x))? If so, what is the difference? If not, why not? What other questions [could/would] you ask?