Graphing Polynomial and Rational Functions



[pic]

I. View the graphs below in a Zoom 4 window, sketch each, and identify the degree.

A. [pic](same as[pic]) degree?___ B. [pic] degree?____

[pic] (Use these settings to view and sketch the next 2 graphs:

[pic] [pic]

C. [pic]degree? ___ D. [pic]degree? ____

Observations:

• What similarities do you see in the shape of the graphs? _____________________________________________________________

• 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 is tangent 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. [pic](same as [pic]) degree?___ B. [pic] degree?____

C. [pic] degree?____ D. [pic] degree?____

Observations:

• What similarities do you see in the shape of the graphs? _____________________________________________________________

• 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 is tangent 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 [pic]. Sketch what you think the graph will look like on the coordinate plane provided. 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.

[pic] [pic]

C. D.

[pic] [pic]

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. [pic] 6. [pic]

[pic] [pic]

7. [pic]

(Hint: Notice that all the other polynomials we’ve studied have been written in factored form!)

[pic]

8. Now, write an equation for a polynomial whose graph has the following characteristics:

➢ As [pic] and as [pic]

➢ 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 [pic]

[pic]

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[pic]

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