Unit 2, Lesson 10
Multiplicity
Integrated Math 3
10.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the idea that the number of times a linear factor occurs in an equation is reflected in the shape of the graph near the intercept related to that factor, which will be useful when students sketch graphs of polynomial functions in a following activity. While students may notice and wonder many things about these images, the connections between factors, zeros, and the look of the graph near the related horizontal intercept are the important discussion points.
Launch
Arrange students in groups of 2. Display the images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Graph of a polynomial function on xy-plane. Horizontal axis, scale -3 to 9, by 1’s Vertical axis, scale -20 to 10, by 10’s. The function begins from the upper left through (-1 comma 16) decreasing through a y-intercept at (0 comma 9) to a local minimum at (3 comma 0), then increases through (7 comma 16) and remains increasing.
Graph of a polynomial function on xy-plane. Horizontal axis, scale -3 to 9, by 1’s Vertical axis, scale -20 to 10, by 10’s. The function starts near (-2 comma -20) increasing through a root at (-1 comma 0) and a y-intercept at (0 comma 9) to a local maximum near (point 333 comma 9 point 5). The function then decreases to a root and local minimum at (3 comma 0), then increases through (4 comma 5) and remains increasing.
Graph of a polynomial function with one root at x = 3, on xy-plane. The x axis is from -3 to 9, by 1’s. The x axis is from -20 to 10, by 10’s.The function starts near (point 2 comma -20) increasing to (3 comma 0) and then rapidly increasing through (5 comma 8).
$y=(x-6)(x-3)^2$
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the different look of the horizontal intercepts of the graphs does not come up during the conversation, ask students to discuss this idea and why the parts of each graph near look the way they do in the different graphs.
Tell students that this is an example of multiplicity. The multiplicity of a factor is the number of times the factor occurs when a polynomial is written in factored form. Three of the equations all have , and so the graph of the polynomial near looks quadratic, and we say that the factor has a multiplicity of 2. The other equation has , so the graph looks cubic near , and we say that the factor has a multiplicity of 3. In the last equation and graph, the factor has a multiplicity of 1, so the graph looks linear near .
Lesson Synthesis
The purpose of this discussion is for students to practice writing and sketching a graph of a polynomial based on its equation in factored form. Begin by displaying the polynomial equation . Discuss with students:
- “What is the degree of the polynomial?” (3)
- “What are the zeros of the polynomial?” (2, -1, -3)
- “What will the graph look like near ?” (The graph will look quadratic near because the factor has a multiplicity of 2.)
- “What will the graph look like near ?” (The graph will look cubic near because the factor has a multiplicity of 3.)
- “What will the graph look like near ?” (The graph will look linear near because the factor has a multiplicity of 1.)
- “What is one number you could change in your equation that would result in a graph with different end behavior?” (If the leading coefficient was the opposite sign, the end behavior of the new graph would be the opposite of the old graph. Or, increasing the multiplicity of one of the factors from 1 to 2 and changing the overall degree of the polynomial would also change the end behavior.)
Student Lesson Summary
We can combine what we know about factors, degree, end behavior, the sign of the leading coefficient, and multiplicity to sketch polynomials written in factored form. Multiplicity, or the power to which a factor occurs in the factored form of a polynomial, tells us the number of times that factor is repeated and affects the shape of the graph near the location of each zero on the horizontal axis.
For example, has three factors with no duplicates. We say that each factor, , , and , has a multiplicity of 1. This results in a graph that looks a bit like a linear function near , , and when we zoom in on each of those places.
Graph of a polynomial function on xy-plane. Horizontal axis, scale -5 to 4, by 1’s Vertical axis, scale -25 to 20, by 5’s. The function starts near (-3 point 7 comma -25) increasing through a root at (-3 comma 0) to a local maximum near (-1 point 4 comma 20 point 8). The function then decreases through a y-intercept at (0 comma 12) through a root at (1 comma 0), to a local minimum at (2 point 7 comma -12 point 6). The function then increases through a root at (4 comma 0 and remains increasing.
near Graph of a polynomial function on x-axis, zoomed view near x=3. Horizontal axis, scale -3.5 to 2.75 by .25’s. Function is increasing, looks linear with a slope of 2 and has an x-intercept at (3 comma 0).
near Graph of a polynomial function on x-axis, zoomed view near x=1. Horizontal axis, scale 0.5 to 1.25 by .25’s. Function is decreasing, looks roughly linear with a slope of -1 and has an x-intercept at (1 comma 0).
near Graph of a polynomial function on x-axis, zoomed view near x=4. Horizontal axis, scale 3.5 to 4.25 by .25’s. Function is increasing, looks roughly linear with a slope of 2 and has an x-intercept at (4 comma 0).
For , there are still three factors, but two of them are . This results in a graph that looks a bit like a quadratic near and a bit like a linear function near . We say that the factor has a multiplicity of 2 while the factor has a multiplicity of 1.
Graph of a polynomial function on xy-plane. Horizontal axis, scale -3 to 9, by 1’s Vertical axis, scale -20 to 10, by 10’s. The function starts near (-2 comma -20) increasing through a root at (-1 comma 0) and a y-intercept at (0 comma 9) to a local maximum near (point 333 comma 9 point 5). The function then decreases to a root and local minimum at (3 comma 0), then increases through (4 comma 5) and remains increasing.
near Graph of a polynomial function on x-axis, zoomed view near x=-3. Horizontal axis, scale -3.5 to -2.75 by .25’s. Function looks quadratic, is increasing to a local maximum and root at (-3 comma 0) and then decreases symmetrically.
near Graph of a polynomial function on x-axis, zoomed view near x=4. Horizontal axis, scale 3.5 to 4.25 by .25’s. Function is increasing, looks linear with a slope of 4 and has an x-intercept at (4 comma 0).
Now consider what the graph of would look like. The factors help us identify that the function has zeros at -3 and 4. We also know that since has a multiplicity of 1 and has a multiplicity of 3, the graph looks a bit like a linear polynomial crossing the -axis at -3 and a bit like a cubic polynomial crossing the -axis at 4. Since this is a 4th-degree polynomial with a positive leading coefficient, we know that as gets larger and larger in either the negative or positive direction, gets larger and larger in the positive direction.
Graph of a polynomial function on xy-plane. Horizontal axis, scale -10 to 8, by 2’s Vertical axis, scale -300 to 50, by 50’s. The function starts by decreasing through a root at (-3 comma 0) to a local minimum near (-1 comma -253). The function then increases through a y-intercept at (0 comma -192) to a root at (4 comma 0), then begins increasing.