I. Viewthe graphs belowin a Zoom 4 window, sketch each, and identify the degree (add the exponents).
A. (same as)degree?___ B. degree?____
Use these settings to view and sketch the next 2 graphs:
C. degree? ___ D. degree? ____
Observations:
- Look at the similarities of the graphs and describe the “end-behavior”. (How do the graphs start and end? Do they begin and end in the same direction, opposite directions? Do they begin and end positively or negatively?)______
______
- What do you think caused the ‘flip’ in the graphs of C and D? ______
- What do you notice about the degree of all the functions? ______
- Identify the x-intercept(s) of each graph.
A. ______B. ______C. ______D. ______
- For each x-intercept, tell whether the graph istangent to the x-axis (touches and turns) or crosses and continues through the x-axis?
A. ______
B. ______
C. ______
D. ______
II. View the graphs below in a Zoom 4 window, sketch each, and identify the degree.
A. (same as ) degree?___ B. degree?____
C. degree?____ D. degree?____
Observations:
- Look at the similarities of the graphs and describe the “end-behavior”. (How do the graphs start and end? Do they begin and end in the same direction, opposite directions? Do they begin and end positively or negatively?)______
______
- What do you think caused the ‘flip’ in the graphs of C and D? ______
- What do you notice about the degree of all the functions? ______
- Identify the x-intercept(s) of each graph.
A. ______B. ______C. ______D. ______
- For each x-intercept, tell whether the graph istangent to the x-axis (touches and turns) or crosses and continues through the x-axis?
A. ______
B. ______
C. ______
D. ______
III. Based upon your observations above, predict the behavior of the graph of . Sketch what you think the graph will look like on the coordinate plane provided. (Think about what your x-intercepts would be and the relationship between degree and “end behaviors”.) Do not use your calculator yet.
Now check your sketch by viewing the graph in the window shown.
Comment on any inaccuracies in your graph.
IV. Answer the following questions based on your observations of all graphs.
- Compare the graphs of polynomials of odd degree with those of even degree. What do you think this determines on your graph?
- Compare the factors with odd powers to those with even powers. What do you think this determines on your graph?
- Compare the equations with positive leading coefficients with those of negative leading coefficients. What do you think this determines on your graph?
- How do we determine the x-intercepts of a graph using only its equation?
- How do you think we can determine the y-intercept?
V. Try these without using your calculator:
Using what you have learned, match the equations to their appropriate graph.
Graph: ______
Graph: ______
Graph: ______
Graph: ______
A.B.
C.D.
In the space provided, sketch a graph of the equation without using your calculator. Note the x-intercepts and y-intercept on the axes. You need not worry about proper scaling on the y-axis.
5. 6.
7.
(Hint: Notice that all the other polynomials we’ve studied have been written in factored form!)
8. Now, write an equation for a polynomial whose graph has the following characteristics:
As and as
The graph is tangent at both its x-intercepts -5 and 2.
Write your equation here:______
What is the y-intercept of your graph?______
Challenge:You must be a pro by now! See if you can handle this one!
Sketch the graph of the equation