


UNIT 4 : POLYNOMIAL
AND RATIONAL FUNCTIONS
LESSON 1:
POLYNOMIAL FUNCTIONS INTRODUCTION
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Example 1:
f(x) = x^{3} 4x^{2} + x + 6 is a cubic polynomial function because the largest exponent of the variable is 3. Make a table of values or use
a graphing calculator to draw its graph [see below].
Note: The function has 3 zeros (x intercepts), x = 1, 2, 3 and two
turning points A and B. Turning point A is a local maximum point as
the function changes from increasing to decreasing at A. Turning point B
is a local minimum point as the function changes from decreasing to increasing
at B. The function has a local maximum value of y = 6.1 at point A and a
local minimum value of y = 0.9 at point B.
Example 2:
Use your graphing calculator to graph and study the cubic functions below.
a) f(x) = x^{3} x^{2} + 5x 3 b) f(x) = (x 3)^{3} + 1
c) f(x) = x^{3} + 3x^{2} + 3x + 2
Note: From
the examples above we can make the following observations.
If k < 0, the function rises to the left and falls to the right (ex. 2a). This characteristic is true for all polynomial functions of odd degree (1, 3, 5, ).
Example 3:
Use your graphing calculator to graph and study the quartic functions below.
a) f(x) = x^{4} + x^{3} 5x^{2} 3x b) y = x^{4} + x^{3} 2x^{2} 3x
c) y = 0.5x^{4} x^{3} + 2x^{2} 5 d) y = x^{4} + x^{3} 2x^{2} 3x + 3
Note: From
the examples above we can make the following observations.
In (a,b,d) the leading coefficient is positive and the function rises to the left and right
If k < 0, the function falls to the left and right (ex. 3c). This characteristic is true for all polynomial functions of even degree (2, 4, 6, ).
Polynomial Functions in Factored Form f(x) = k(x a)(x b)(x c) . . . etc.
x 
y 
2 
6 
0 
6 
x 
y 
1 
16 
2 
16 