PreCalculus Class Notes P5 Sketching Graphs of Polynomial Functions
Worked Example
Sketch the graph of y = 1 ( x + 2 ) ( x − 1) ( x − 3) without using graphing technology.
4
2
Zeros
set factors = 0
solve for x
y-intercept
set x = 0, find y
y = 1 ( 0 + 2 ) ( 0 − 1) ( 0 − 3) = 3
4
2
3
3
x = −2, x = 1 and x = 3
Behavior at zeros
Eventangent
Oddcrossing
x = −2; tangent
x = 1; crossing like x3
x = 3; crossing
End Behavior (summary chart in P3 notes)
Leading term
2
3
y = 1 ( x ) ( x ) ( x ) = 1 x6
4
4
Leading coefficient
1
> 0 , positive
4
Degree
6, even
Describe end behavior: as x → +∞, y → +∞; as x → −∞, y → +∞
y
y
x
x
For each polynomial below,
1) Find all real roots, including multiplicities and type (tangent or crossing)
2) Find the y-intercept
3) Describe the end behavior
4) Sketch and label y-intercept and roots with values
A.
2
y = ( x − 1) ( x + 2 )( x − 3)
y-intercept
Zeros
Behavior at zeros
Leading term
Leading coefficient
Degree
Describe end behavior:
Sketch
B. f ( x ) = − x ( x + 2 )( x − 3)
2
y-intercept
Zeros
Behavior at zeros
Leading term
Leading coefficient
Degree
Describe end behavior:
Sketch
2
C. y = ( x 2 − 4 ) ( 2 x − 3)( x − 1) ( x + 2 )
y-intercept
Zeros
Behavior at zeros
Leading term
Leading coefficient
Degree
Describe end behavior:
Sketch
Write a possible polynomial function for the graph below.
y
3
x
−2
−9