3.8 – End Behavior (day 1)
Linear
Quartic
Quadratic
Cubic
Quintic
End Behavior
Even Functions: Positive Leading Coefficient
→
∞,
→ ∞,
→
→
Minimum/Maximum?
Domain:
Range:
Smiley ☺
End Behavior
Even Functions: Negative Leading Coefficient
→
∞,
→ ∞,
→
→
Minimum/Maximum?
Frowny
Domain:
Range:
End Behavior
Odd Functions: Positive Leading Coefficient
→
∞,
→ ∞,
→
→
Minimum/Maximum?
Snakes from 3rd to 1st
Domain:
Range:
End Behavior
Odd Functions: Negative Leading Coefficient
→
∞,
→ ∞,
→
→
Minimum/Maximum?
Domain:
Range:
Snakes from 2nd to 4th
End Behavior
Note: The degree of the polynomial = the number of zeros of the function.
Not all zeros can be seen on the x-axis because some zeros are imaginary.
Some zeros also have multiplicity, meaning that zero represents multiple roots.
If the degree of the polynomial is n, then the graph will have n – 1 “turning points”,
and will have AT MOST n real zeros.
End Behavior
Multiplicity
The point of multiplicities with respect to graphing is that any factors that occur an
even number of time (twice, four times, six times, etc) are squares, so they don't
change signs since squares are always positive.
This means that the x-intercept corresponding to an even-multiplicity zero can't cross
the x-axis, because the zero can't cause the graph to change sign from positive (above
the x-axis) to negative (below the x-axis), or vice versa.
End Behavior
Compare these fourth degree polynomials. [Circle the real zeros, determine if any have
multiplicity, are there any imaginary zeros.]
End Behavior
Compare these fifth degree polynomials. [Circle the real zeros, determine if any have
multiplicity, are there any imaginary zeros.]
End Behavior
Answer the following questions about the graphs given below:
A) Is the function even or odd?
B) What is the minimum degree of the function?
C) What can you determine about the leading coefficient?
D) How many real zeros are there? (include multiplicity)
End Behavior
Answer the following questions about the graphs given below:
A) Is the function even or odd?
B) What is the minimum degree of the function?
C) What can you determine about the leading coefficient?
D) How many real zeros are there? (include multiplicity)
End Behavior
Answer the following questions about the graphs given below:
A) Is the function even or odd?
B) What is the minimum degree of the function?
C) What can you determine about the leading coefficient?
D) How many real zeros are there? (include multiplicity)
End Behavior
Answer the following questions about the graphs given below:
A) Is the function even or odd?
B) What is the minimum degree of the function?
C) What can you determine about the leading coefficient?
D) How many real zeros are there? (include multiplicity)
End Behavior
Determine the end behavior of the following functions.
7
→
∞,
→ ∞,
8
→ ______
→ ______
Minimum/Maximum?
2
→
∞,
→ ∞,
6
5
→ ______
→ ______
Minimum/Maximum?
End Behavior
Determine the end behavior of the following functions.
4
→
8
∞,
→ ∞,
9
→ ______
→ ______
Minimum/Maximum?
5
→
∞,
→ ∞,
4
10
→ ______
→ ______
Minimum/Maximum?
19
3.8 – Homework
3.8 Worksheet