Unit 2, Lesson 10
Multiplicity
Integrated Math 3
10.1
Warm-up
Notice and Wonder: Duplicate Factors
What 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).
10.2
Activity
Sketching Polynomials
- For polynomials
– : - Write the degree, all zeros, and complete the sentence about the end behavior.
- Sketch a possible graph.
-
Check your sketch using graphing technology.
Pause here for your teacher to check your work.
- Create your own polynomial for your partner to figure out.
- Create a polynomial with degree greater than 2 and less than 8, and write the equation in the space given.
- Trade papers with a partner, then fill out the information about their polynomial and complete a sketch.
- Trade papers back. Check your partner’s sketch using graphing technology.
Degree: Zeros:
End behavior: As
Degree: Zeros:
End behavior: As
Degree: Zeros:
End behavior: As
Degree: Zeros:
End behavior: As
Degree: Zeros:
End behavior: As
Degree: Zeros:
End behavior: As
Your polynomial:
Degree: Zeros:
End behavior: As
10.3
Activity
Using Knowledge of Zeros
-
Sketch a graph for a polynomial function
that has 3 different zeros and for all values of . - What is the smallest degree the polynomial could have?
- What is a possible equation for the polynomial? Use graphing technology to see if your equation matches your sketch.
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,
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
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
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.
multiplicity
The power to which a factor occurs in the factored form of a polynomial. For example, in the polynomial
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