Lesson 4
Standards
MM3A1b Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros.
MM3A1d Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.
Essential Question
What information can I obtain using the degree, leading coefficient, multiplicity of real zeros, and real zeros for the graph of a polynomial function?
Learning Objectives
Learning Activity
Given the list of polynomial functions (see below), please do each of the following for each function:
a) sketch graph of each using your graphing calculator or Geogebra
b) state the function’s degree
c) state the number of turns in the graph
Students will then answer the summary questions attached below.
Polynomial Functions
For each function, sketch graph, state the degree, and give the number of turns in each graph.
a) x^2 + 2x - 5
b) -x^2 - 5
c) 2x^3 - 7x + 1
d) -x^3 + 5x + 2
e) x^4 - 2x^2 + 3
f) x^5 - 5x^3 + 4x
Summary Questions
1. What is the relationship between the degree and the number of turns?
2. What is similar about the even degree functions? (a, b, e)
3. What is similar about the odd degree functions? (c, d, f)
4. What is the relationship between the number of zeros and the degree?
6E's
Engage: Students are engaged during the activity because they are using their graphing calculator or Geogebra.
Explore: The students are exploring by using the graphing calculator to find sketch the graph of each polynomial function and then writing what they notice about each graph.
Explain: Students are required to explain what they see when they graph the polynomial function and what their thoughts are about the connection between the degree, number of turns, and number of zeros.
Elaborate: Students are asked to elaborate on what they noticed about their graphs of each polynomial function by answering the summary questions.
Evaluate: A rubric is provided that will check for student understanding of the activity.
Extend: Students will extend their learning by answering the following problem for homework. We will go over it tomorrow in class.
Homework Problem: The average heights B and G (in inches) for boys and girls ages 2 to 20 can be modeled by the functions B = -0.001t^4 + 0.04t^3 - 0.57t^2 + 5.7t + 25 and G = 0.000007t^4 - 0.00276t^3 - 0.012t^2 + 3.1t +27 where t is the age in years.
a) Sketch the graphs of the two models in the same coordinate plane.
b) According to the models, what is the difference in average height between 16-year-old boy and girls?
c) During an annual physical, a doctor measures a 14-year-old to be 60 inches tall. Is the 14-year-old more likely to be male or female? Explain.
MM3A1b Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros.
MM3A1d Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.
Essential Question
What information can I obtain using the degree, leading coefficient, multiplicity of real zeros, and real zeros for the graph of a polynomial function?
Learning Objectives
- Students will be able to graph a polynomial function by using the degree, leading coefficient, multiplicity of real zeros, the real zeros, y-intercept, and domain and range.
Learning Activity
Given the list of polynomial functions (see below), please do each of the following for each function:
a) sketch graph of each using your graphing calculator or Geogebra
b) state the function’s degree
c) state the number of turns in the graph
Students will then answer the summary questions attached below.
Polynomial Functions
For each function, sketch graph, state the degree, and give the number of turns in each graph.
a) x^2 + 2x - 5
b) -x^2 - 5
c) 2x^3 - 7x + 1
d) -x^3 + 5x + 2
e) x^4 - 2x^2 + 3
f) x^5 - 5x^3 + 4x
Summary Questions
1. What is the relationship between the degree and the number of turns?
2. What is similar about the even degree functions? (a, b, e)
3. What is similar about the odd degree functions? (c, d, f)
4. What is the relationship between the number of zeros and the degree?
6E's
Engage: Students are engaged during the activity because they are using their graphing calculator or Geogebra.
Explore: The students are exploring by using the graphing calculator to find sketch the graph of each polynomial function and then writing what they notice about each graph.
Explain: Students are required to explain what they see when they graph the polynomial function and what their thoughts are about the connection between the degree, number of turns, and number of zeros.
Elaborate: Students are asked to elaborate on what they noticed about their graphs of each polynomial function by answering the summary questions.
Evaluate: A rubric is provided that will check for student understanding of the activity.
Extend: Students will extend their learning by answering the following problem for homework. We will go over it tomorrow in class.
Homework Problem: The average heights B and G (in inches) for boys and girls ages 2 to 20 can be modeled by the functions B = -0.001t^4 + 0.04t^3 - 0.57t^2 + 5.7t + 25 and G = 0.000007t^4 - 0.00276t^3 - 0.012t^2 + 3.1t +27 where t is the age in years.
a) Sketch the graphs of the two models in the same coordinate plane.
b) According to the models, what is the difference in average height between 16-year-old boy and girls?
c) During an annual physical, a doctor measures a 14-year-old to be 60 inches tall. Is the 14-year-old more likely to be male or female? Explain.
sampleassignment.pdf | |
File Size: | 530 kb |
File Type: |
Polynomial Functions Rubric
Polynomial Functions Summary Questions |
Unacceptable
0 pts. Graphs are not complete, degree is not stated, and number of turns is not stated.
Summary questions were not attempted. |
Fair
1 pt. Graphs are complete but may have some mistakes (incorrect zeros, ending behavior, y-intercept, or number of turns). Degree and number of turns is stated, but may be incorrect.
Summary questions were answered, but answers may not be correct. |
Good
2 pts. Graphs are complete and correct, meaning correct ending behavior, zeros, y-intercept, and number of turns. Degree and number of turns is stated and correct.
Summary questions were answered and are all correct. |