Transcript 5.2 Notes: Graphing of Polynomial Functions

5.2 Notes: Graphing of Polynomial Functions
Features of Graphs of Polynomial Functions:
1. Graphs are continuous
2. At most, n – 1 turns, where n = degree of polynomial
Ex. x4 has at most ____
3 turns
y = x4 – 9x2
y = –2x3 + x
3. If the leading coefficient is positive, the graph rises to
the right. If it negative, the graph falls to the right.
y = x4 – 9x2
y = –2x3 + x
Example 1: State the maximum number of
turns and describe the right behavior in the
graph of each function:
1 3
a. f ( x)  x  1
2
n–1=3–1=
turns: ____
2
Positive coefficient
right end
↑
behavior: ____
Example 1: State the maximum number of
turns and describe the right behavior in the
graph of each function:
b. f ( x)   x  2 x  1
2
n–1=2–1=
turns: ____
1
Negative coefficient
right end
↓
behavior: ____
Example 1: State the maximum number of
turns and describe the right behavior in the
graph of each function:
c. f ( x)  ( x)( x  2)( x  1)( x  5)
x4
n–1=4–1=
turns: ____
3
Positive coefficient
right end
↑
behavior: ____
Example 1: State the maximum number of
turns and describe the right behavior in the
graph of each function:
d.
f ( x)  ( x  2)( x  1)( x  3)
x3
n–1=3–1=
turns: ____
2
Positive coefficient
right end
↑
behavior: ____
Even and Odd functions:
1. If the degree of the polynomial is even, the
graph has the same behavior to the left and
right (rise-rise or fall-fall)
y = x4 – 9x2
Even and Odd functions:
2. If the degree of the polynomial is odd, the
graph has the opposite behavior to the left
and right (rise-fall or fall-rise)
y = –2x3 + x
Example #3: Describe the left and right
behaviors of the graphs.
a. f ( x)  2 x  4 x  3
3
Negative coefficient
Odd
Right end
↓
behavior: ____
Left end
↑
behavior: ____
Example #3: Describe the left and right
behaviors of the graphs.
b. f ( x)  x  x  3x
4
3
Positive coefficient
Even
Right end
↑
behavior: ____
Left end
↑
behavior: ____
Example #3: Describe the left and right
behaviors of the graphs.
c. f ( x)  4 x  x  2 x  7
8
5
Negative coefficient
Even
Right end
↓
behavior: ____
Left end
↓
behavior: ____
Example #3: Describe the left and right
behaviors of the graphs.
d.
f ( x)   x  16
7
Negative coefficient
Odd
Right end
↓
behavior: ____
Left end
↑
behavior: ____
Intercepts:
The y-intercept of the graph of a function
x = 0.
(0, ___)
occurs when ________.
zeros of the
The x-intercepts are the _______
function. At an x-intercept, the value of the
0
y = 0 (____, 0)
function is ______.
Example #4: Sketch the graph of
f ( x)  ( x  3)( x  2)( x  1)
2
a. Max # of turns: _________
↑
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
3, –2, 1
d. x – intercepts: __________
6
e. y – intercept(s): __________
x3
(0-3)(0+2)(0-1)
6
f ( x)  ( x  3)( x  2)( x  1)
2
a. Max # of turns: _________
↑
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
3, –2, 1
d. x – intercepts: __________
e. y – intercept(s): __________
6
Example #5: Sketch the graph of
f ( x)  x( x  3)( x  3)
2
a. Max # of turns: _________
↑
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
0, 3, –1
d. x – intercepts: __________
0
e. y – intercept(s): __________
x3
(0)(0-3)(0+3)
0
f ( x)  x( x  3)( x  3)
2
a. Max # of turns: _________
↑
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
0, 3, –1
d. x – intercepts: __________
e. y – intercept(s): __________
0
Example #6: Sketch the graph of
f ( x)  ( x  2)( x  1)( x  4)( x  3)
–x4
3
a. Max # of turns: _________
-(0-2)(0+1)(0-4)(0+3)
↓
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
2, –1, 4, –3
d. x – intercepts: __________
–24
e. y – intercept(s): __________
–24
f ( x)  ( x  2)( x  1)( x  4)( x  3)
3
a. Max # of turns: _________
↓
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
2, –1, 4, –3
d. x – intercepts: __________
–24
e. y – intercept(s): __________
Example #5: Sketch the graph of
3
2
f ( x)  x  3x  4 x 12
2
a. Max # of turns: _________
↑
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
d. x – intercepts: __________
3
2 )(
(
f ( x)  x  3x  4 x 12 )
x2(x + 3) –4( x + 3)
(x2 – 4)(x + 3)
x 2
x –2
2x + –2x
(x + 2)(x – 2)(x + 3)
–2, 2, –3
x – intercepts: _________
Example #5: Sketch the graph of
3
2
f ( x)  x  3x  4 x 12
2
a. Max # of turns: _________
↑
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
–2, 2, –3
d. x – intercepts: __________
–12
e. y – intercept(s): __________
f ( x)  x  3x  4 x 12
3
2
2
a. Max # of turns: _________
↑
b. Right End Behavior: _____
↓
c. Left End Behavior: ______
–2, 2, –3
d. x – intercepts: __________
–12
e. y – intercept(s): __________