4. y=-(x+2)(x+1)²(x−3) A. leading term: B. behavior of the graph: (n is __________ and an_0) C. x-intercepts: the polynomial in factored form is y= D. multiplicity of roots:_ E. y-intercept:_ F. number of turning points: G. sketch:
4. y=-(x+2)(x+1)²(x−3) A. leading term: B. behavior of the graph: (n is __________ and an_0) C. x-intercepts: the polynomial in factored form is y= D. multiplicity of roots:_ E. y-intercept:_ F. number of turning points: G. sketch:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Graphing a Polynomial Function
To sketch the graph of a polynomial function we need to consider the following:
A. leading term
B. behavior of the graph
The graph of a polynomial function:
1. comes down from the extreme left and goes up to the extreme right if n is even and an>0.
2. comes up from the extreme left and goes up to the extreme right if n is odd and an>0.
3. comes up from the extreme left and goes down to the extreme right if n is even and an<0.
4. comes down from the extreme left and goes down to the extreme right if n is odd and an<0.
a>0
an <0
n is even
n is odd
~ ~
C. x-intercepts
D. multiplicity of roots
If r is a zero of odd multiplicity, the graph of P(x) crosses the x-axis at r.
if r is a zero of even multiplicity, the graph of P(x) is tangent to the x-axis at r.
E.y-intercept
F. number of turning points
Turning points are points where the graph from increasing to decreasing function value, or vice versa. The number of turning points
in the graph of a polynomial is strictly less than the degree of the polynomial. Also, we must note that;
1. Quartic Functions: have an odd number of turning points; at most 3 turning points
2. Quintic functions: have an even number of turning points, at most 4 turning points
3. The number of turning points is at most (n-1).
Transcribed Image Text:4. y=-(x+2)(x+1)²(x−3)
A. leading term:
B. behavior of the graph:
(nis
and an_0)
C. x-intercepts:
the polynomial in factored form is
y=
D. multiplicity of roots:_
E. y-intercept:_
F. number of turning points:
G. sketch:
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