Section 2-3:  Graphing Polynomials
 
Graphing cubic functions (Mark's Math Applets)

Try the quiz at the bottom of the page!
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 Cubic Functions
In general, cubic functions are shaped like a "sideways S".
 
                        Graph of f(x) = ax3 + bx2 + cx + d
 
     a > 0          a < 0

To graph a polynomial function, first find the zeros.
If f(x) = ( x + 1)(x - 2)(x - 1),
the zeros would be at x = -1, x = 2, x =1
Now do a sign analysis of f(x) by testing a value in each interval formed by the zeros above.  Pick any number in each interval.  See the chart:
 
             x               y           sign
            -2 (-2+1)(-2-2)(-2-1)             -
             0 (0+1)(0-2)(0-1)             +
            1.5 (1.5+1)(1.5-2)(1.5-1)             -
              3 (3+1)(3-2)(3-1)            +
Where the graph is negative means it is below the x-axis, where it is positive - above.  Now sketch the graph from the information.

The graph of a quartic looks like a "W-shape" or "M-shape"
Graph of f(x) = ax4 + bx3 +cx2 + dx + e
.......................
                                  a > 0                                                                  a < 0

Effects of different factors
1)  Single factors  -- on one side positive, the other side negative

2)  Squared factors -- tangent to the x-axis at the point x = c

3)  Cubed factors --  flatten out around the point (c,0)

..............................
                                Single factors                              Tangent at x = -2 (double root)
Triple root at x = -1 (flattens out)

On to maximums and minimums
 


 
 
Current quizaroo #  2a
 
 
 
1)
Find an equation of the above graph
a)  x2(x - 2)3(x + 1)
b)  x(x - 2)(x + 1)
c)  x(x + 2)(x - 1)
d)  x2(x + 2)3(x - 1)
e)  x(x + 2)3(x - 1)
 
2)  Find the value of f(2i) for the function  f(x) = (x +1)(x - 1)
          a)  -5
b)  3
c)  -3
d)  0
e)  4i - 1
 
 
3)  Use synthetic substitution to find f(-2) for the function f(x) = 2x3 - 3x2 + 4x - 5
a)  0
b)  53
c)  -41
d)  41
e)  -53
 
4)  Find the quotient and remainder when x4 + x3 + 2x2 - 3x + 5 is divided by x - 2
a)  x3 - x2 + 4x - 11 + 27/(x - 2)
b)  x3 + 3x2 + 8x + 13 + 31/(x - 2)
c)  x3 - 2x2 + x - 15 + 2/(x - 2)
d)  x3 - x2 + 2x - 13 + 7/(x - 2)
e)  x3 + 3x2 + 5x - 1 + 15/(x - 2)
 
5)  Find the other roots if x4 + 2x3 -3x2 - 8x - 4 has roots x =2 and x = -1
a) x = -2 and x = 1
b) x = 2 and x = 1
c) x = 2 and x = -1
d) x = -2 double root
e) x = -2 and x = -1
 
 
 
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