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.[br][br][i][b]Challenge: Adding & subtracting functions[/b][/i][br] Choose a linear function for f(x). [br] Find a function g(x), which when added to f(x) gives you a constant function.[br] Can you adjust g(x) so that you can get any constant function?[br] What happens to the graph of your sum function as you vary the slope of g(x)? the y-intercept of g(x)?[br][br][i][b]Challenge: Understanding factoring[/b][/i][br] Choose a quadratic function for f(x) – [br] e.g., set f(x) = [math]x^2 + x – 2[/math] by setting the [i][b]a [/b][/i]slider to 1, the [b][i]b[/i][/b] slider to 1 and the [b][i]c[/i][/b] slider to -2[br] Choose a linear function for g(x) – [br] e.g., set g(x) = [math]– x + 1[/math] by setting the [b][i]A[/i][/b] slider to – 1 and the [b][i]B[/i][/b] slider to 1.[br] Choose an operation on the left side of the screen. [br][br] 1. What can you say about the quotient f(x) / g(x)? Is it linear? quadratic? why?[br] 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?[br] 3. Reset the linear function g(x) to [math] – x + 1[/math] 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?[br] 4. Suppose the linear function had been in the form g(x) = [math]A(x – B)[/math]. Would the same things have happened when you vary A and B? Why or why not?[br] 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?[br][br][i][b]Challenge: Understanding multiplication of functions and FOIL[/b][/i][br] Choose linear functions for f(x) and g(x) – say f(x) = [math]2x + 1[/math] and g(x) = [math]– x + 3[/math][br] 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.[br] 1. What happens when you multiply f(x) and g(x)? [br] 2. How does the graph of the product function depend on the[br] slope of f(x)? slope of g(x)?[br] y-intercept of f(x)? y-intercept of g(x),[br] x-intercept of f(x)? x-intercept of g(x)?[br] 3. Can you make a parabola with no real roots, such as [math]x^2 + 4[/math] by multiplying two linear functions? Why or why not?[br][br][i][b]Challenge: Understanding composition[/b][/i][br] Choose a linear function f(x).[br] Can you find a function g(x) such that the composed function is f(x)…[br] translated horizontally? translated vertically?[br] dilated horizontally? dilated vertically?[br] reflected in the x-axis? reflected in the y-axis?[br][br]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?[br][br][color=#ff0000]What other questions [could/would] you ask?[/color]