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 > 0
a < 0
.......................
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