What behaviors can be predicted by adding, multiplying, dividing, and composing linear functions?
PART 1 - ADDITION OF LINEAR FUNCTIONS
What can you predict about the line h(x) + g(x)?
h(x) = 2x + 1
g(x) = x + 1
Will it be linear? If so, what slope will it have? What will the x and y intercepts be? Will it intersect either of h(x) or g(x)? If so, what are the intersection points? Can you guess any answers to these questions? HOW could you determine answers to any of these questions?
First let's examine the result of h(x) + g(x) = (2x+1)+(x+1) = 3x+2.
Now that you have an explicit formula for the sum, can you answer any of the questions?
How about looking at its graph:
h(x) = 2x + 1
g(x) = x + 1
h(x) + g(x) = f(x) = 3x + 2
Let h(x) + g(x) = f(x). Is f(x) linear? How could you have predicted this without seeing the graph?
What is the of slope f(x)? How could you have predicted this without seeing the graph?
What are the x and y intercepts of f(x)?
Where does f(x) intersect h(x) and/or g(x)?
How could we have predicted these answers without graphing? Any ideas? Before you look below, try guessing the answers!
1) f(x) is linear. f(x) is in two variables, each with an exponent of 0 or 1, and the graph is a line.
2) The slope of f(x) is 3. The sum of the coefficients of the x terms is 3.
3) The x and y intercepts are (-2/3,0) and (0,2). These are found by examining the explicit formula for f(x).
4) f(x) = 3x+2 intersects h(x) = 2x+1 at point (-1,-1). This point is found by equating f(x) and g(x) and solving for x. The intersection point of f(x) and g(x) = x+1 is (-1/2,1/2) and may be found in the same fashion.
Now you predict the answers to the same questions for the addition of the following functions:
q(x) = 5x-4 and t(x) = -2x+3
Please email your questions and answers to me. Be sure to indicate that you are investigating PART 1 - ADDITION OF LINEAR FUNCTIONS
PART 2 - MULTIPLYING LINEAR FUNCTIONS
How could you answer the same four questions if you were to multiply two linear functions instead of add them? To investigate, let's use the functions q(x) and t(x).
How can you predict whether the function [q(x)]*[t(x)] will be linear? To begin investigating, let's get an explicit formula for the product of the two functions. We'll call it p(x). Let q(x) = -2x+2 and let t(x) = 3x-3. Now p(x) = (-2x+2)(3x-3). Using FOIL, we see that
1) Is p(x) a linear function? Why or why not? Answer this question without using information from the graph.
2) What can you tell me about the slope of p(x)? Make some guesses and tell me why you think your guess is right. What questions do you have about the slope of p(x)?
3) What, if any exist, are the x and y intercepts of p(x)?
4) What, if any exist, are the points of intersection of q(x) and p(x) and of t(x) and p(x)? (Hint: remember your quadratic factoring skills!)
Please email your questions and answers to me. Be sure to indicate that you are investigating PART 2 - MULTIPLICATION OF LINEAR FUNCTIONS.
PART 3 - DIVIDING LINEAR FUNCTIONS
Let r(x) = 2x-9 and let w(x) = 3x-3. First, graph r(x) and w(x) on your grapher (or plot on graph paper.)
By looking at these graphs, what would you expect the graph of [r(x)]/[w(x)] = k(x) to look like? What special problem must you address when dividing by a variable?
Use your calculator to find the value of k(2), k(1.5), k(1.05)? What is the value of k(0), k(.5), k(.95)? Finally, what is k(1)? What do these answers suggest about the graph of k(x)?
r(x) = 2x-9
and w(x) = 3x-3 so k(x) = 
.
1) Is k(x) linear? Why or why not?
2) What can you tell me about the slope of k(x)? Make some guesses and tell me why you think your guess is right. What questions do you have about the slope of k(x)?
3) What, if any exist, are the x and y intercepts of k(x)?
3a) Is there a vertical line that the graph of k(x) does not cross? If so what is the equation of that line and why is this so?
3b) Is there a horizontal line that the graph of k(x) does not cross? If so what is the equation of that line and why is this so?
4) What, if any exist, are the points of intersection of r(x) and k(x) and of w(x) and k(x)?
Please email your questions and answers to me. Be sure to indicate that you are investigating PART 3 - DIVISION OF LINEAR FUNCTIONS.
PART 4 - COMPOSITION OF LINEAR FUNCTIONS
Let f(x) = 9+2x and let d(x) = -3x-10. What will the graph of a composition of these functions look like?
What is an explicit formula for f[d(x)]? What is an explicit formula for d[f(x)]? What, if anything, do you expect the graphs of f[d(x)] and d[f(x)] to have in common or not in common? WHY? (You may find it easier to simplify your expressions before you answer.)
Now look at the graphs:
What are the slopes of the graphs of the compositions? What special relationship do these lines have? How could you have predicted this?
Make up two linear functions, graph them, and find and graph their compositions using your grapher. Can you conclude anything about graphs of compositions of linear functions? Explain your conclusion in an email to me.