An Exploration of Combining Linear Functions

by Margaret Morgan

(for College Algebra Students)

This assignment will allow you to explore the results when two linear functions, f(x) and g(x) are combined in each of the following ways:

h(x) = f(x) + g(x)

h(x) = f(x) g(x)

h(x) = f(x)/g(x)

h(x) = f(g(x))

We will choose multiple versions of f(x) and g(x) and look at the resulting graphs for h(x). In particular, we will pay attention to the resulting shape of the graph of h(x), its intercepts, and any features of the graph of h(x) that can be related back f(x) and g(x).

(In a classroom situation, I would ask students to generate functions and to answer the questions listed. For the purpose of this assignment, I have chosen functions and included answers.)

Let's begin by looking at an f(x) and g(x) with positive slopes and positive y-intercept.

 f(x) = 4x + 2 f(x) = 2x + 3

Combining these two functions, we get the following resulting functions.

(Prior to displaying the graphs, I would ask students what shape they would expect the graph to have)

 h(x) = f(x) + g(x) = 4x + 2 + 2x + 3 = 6x + 5 h(x) = f(x)g(x) = (4x + 2)(2x + 3) = 8x^2 + 16x + 6 h(x) = f(x)/g(x) = (4x + 2)/(2x + 3) h(x) = f((g(x)) = 4 (2x + 3) + 2 = 8x + 14

What shape does each of the graphs have? Is the shape what you expected it to be before you saw the graph?

h(x) = f(x) + g(x) -- Line

h(x) = f(x) g(x) -- Parabola

h(x) = f(x)/g(x) -- Hyperbola

h(x) = f(g(x)) -- Line

What are the x and y intercepts of f(x) and g(x)?

f(x) = 4x + 2; x-intecept = -.5; y-intercept = 2
g(x) = 2x + 3; x-intercept = -1.5; y-intercept = 3

What are the x and y intercepts of h(x) = f(x) + g(x) = 6x + 5? How do these intercepts relate to the x and y intercepts of the individual functions?

x-intercept = -5/6; y-intercept = 5; The y-intercept is the sum of y-intercepts of the individual functions.

What are the x and y intercepts of h(x) = f(x) g(x) = 8x^2 + 16x + 6? How do these intercepts relate to the x and y intercepts of the individual functions?

x-intercept = -.5 and -1.5; y-intercept = 6; The y-intercept is the product of y-intercepts of the individual functions.The x-intercepts are the same as the original functions.

What are the x and y intercepts of h(x) = f(x)/g(x) = (4x + 2)/(2x + 3)? How do these intercepts relate to the x and y intercepts of the individual functions?

x-intercept = -.5; y-intercept = 2/3; The y-intercept is the quotient of y-intercepts of the individual functions. The x-intercept is the x-intercept of the numerator function.

What happens at the x-intercept of the denominator function? Why?

What are the x and y intercepts of h(x) = f(g(x)) = 8x + 14? How do these intercepts relate to the x and y intercepts of the individual functions?

x-intercept = -7/4; y-intercept = 14;

Now, let's look at f(x) and g(x) such that both f(x) and g(x) have a negative slope and positive y-intercept.

 f(x) = -x + 2 g(x) = -2x + 2 h(x) = f(x) + g(x) = -3x + 4 h(x) = f(x)g(x) = 2x^2 - 6x + 4 h(x) = f(x)/g(x) = (-x + 2)/(-2x + 2) h(x) = f(g(x)) = 2x

Do the relationships we found for the intercepts in the previous example still hold?

In the previous example, the slope of each line was positive. In this example, we start with lines with negative slopes, but the composition gives us a line with a positive slope. Why?

Now, let's look at f(x) and g(x) such that f(x) has a positive slope and g(x) has a negative slope.

 f(x) = 2x + 4 g(x) = -x - 2 h(x) = f(x) + g(x) = x + 2 h(x) = f(x) g(x) = -2x^2 - 8x - 8 h(x) = f(x)/g(x) = (2x + 4) / (-x -2) h(x) = f(g(x)) = -2x

Homework question: Determine whether our hypotheses regarding intercepts still hold for this example. Explain why or why not.

In this example, f(x) with positive slope and g(x) with negative slope yielded h(x) = f(x) + g(x) with positive slope. Will this always hold? If so, explain why. If not, give a counter example.

In this example, f(x) with positive slope and g(x) with negative slope yielded h(x) = f(x) g(x) opening up. Will this always hold? If so, explain why. If not, give a counter example.

In this example, f(x) with positive slope and g(x) with negative slope yielded h(x) = f(g(x)) with negative slope. Will this always hold? If so, explain why. If not, give a counter example.

The graph of h(x) = f(x)/g(x) is different than the graphs of this function in the earlier examples--why? What happens when x = -2?

Homework assignment: Play with other examples as needed; write a synopsis of today's classroom discussion.