Mini-Lecture 5.1
Polynomial Functions and Models
Learning Objectives:
1.
2.
3.
4.
5.
Identify Polynomial Functions and Their Degree (p. 330)
Graph Polynomial Functions Using Transformations (p. 333)
Identify the Real Zeros of a Polynomial Function and Their Multiplicity (p. 334)
Analyze the Graph of a Polynomial Function (p. 339)
Build Cubic Models from Data (p. 342)
Examples:
1. Determine the end behavior of the graph of the given polynomial function.
f ( x)  6 x 5  3 x 3  x  5
2. Find the real zeros of the given polynomial function and give their multiplicity. State
whether the graph crosses or touches the x-axis or turns around at each zero.
f ( x)  4  x  3  x  1
3
2
3. Analyze the given polynomial function according to the Summary on page 341.
f ( x)  x3  3x 2  16 x  48
4. Graph the polynomial function. f ( x)  x3  4 x 2  4 x
Teaching Notes:
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Encourage students to find the end behavior because it helps to know, in
general, what the graph should look like before beginning to graph it.
Remind the students when using a graphing utility calculator to select a viewing
window that will show the end behavior of the graph.
Emphasize the role that the multiplicity of the x-intercepts plays.
When you begin to teach how to find the zeros of functions, advise students that
they may need to review factoring.
Emphasize the two summaries on graphing polynomial functions in the book.
Answer: 1) up on the right and down on the left; 2) zero of 3 with multiplicity 3 crosses, zero of -1 with
multiplicity 2 touches and turns; 3) behaves like y  x , x-intercepts -3, -4, 4, y-intercept -48, zeros -3, -4,
4 each with a multiplicity of one, the graph crosses at each zero, turning points (1.52, -61.88) and
(-3.52, 1.88), domain and range-the set of all real numbers, increasing on  , 3.52  , 1.52,   ,
3
decreasing on (3.52,1.52) ;
3)
4)
y
5
x
5