the powers are in descending order. the largest power on x. should be first term. n th degree...
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The powers are in descending order.
The largest power on x. Should be first term. nth degree
vertices
How many times an x-int repeats.
LHB RHB
LHB RHB
a > 0, positive value
a < 0, negative value
LHB RHB
a > 0, positive value
a < 0, negative value
Notice the
right hand side
doesn’t change
Two concepts that we need to know about turning points.
1. The most turning points a polynomial function can have is n – 1.
2. Turning points can reduce by 2 only, until it reaches a 1 or 0.- this is based on odd multiplicity of 3 or higher.
Example. 64 35 xxxP1. Most turning points n – 1 = 5 – 1 = 42. Possible turning points, 4, 2, 0
126 xxxP1. Most turning points n – 1 = 6 – 1 = 52. Possible turning points, 5, 3, 1
-1
-1
1
1
-1
-1
1
1
y = x
-1
-1
1
1
-1
-1
1
1
y = x
y = x3 y = x5
Even multiplicity always does a “Touch & Go”. Never crosses the x-axis.
Odd multiplicity always crosses the x-axis.
2312212 322 xxxxxxxxPx * x2 = -2 * (+1)2 =
(0, -2)
(2, 0) 1(-1, 0) 2
Turning Pts = 2 or 0
1;2
01;02
012 2
xx
xx
xx
Occurs twice,Multiplicity of 2
Graph in your calculator to find relative max and min for intervals of increasing and decreasing.
242 6.12.0242.0 xxxxxxP Turning Pts = 3 or 1
+ 0
2;4;0
02;04;02.0
0242.02
2
xxx
xxx
xxx
Occurs twice,Multiplicity of 2
(-2, 0) 1
(0, 0) 2
(4, 0) 1
Graph in your calculator to find relative max and min for intervals of increasing and decreasing.
Not a constant
01243 23 xxxTurning Pts = 2 or 0 03432 xxx
3,2,2
0322
0342
x
xxx
xx
(2, 0) 1
(-3, 0) 1 (-2, 0) 1
(0, -12)
Factor by grouping
Graph in your calculator to find relative max and min for intervals of increasing and decreasing.
3x
0511 xxx
1x 1x 5x01x 01x 05x
0512 xxFOIL conjugate pair.
FOIL again.
55
05523
23
xxxxf
xxx
5x
01 23 xx
0x 1x03 x
01
012
x
x
01223 xxxFOIL.
Distribute.
345
345
2
02
xxxxf
xxx