MHF4U0
Polynomial Function Investigation
Name:______________
PART 1A: END BEHAVIOUR- ODD DEGREE Use a graphing app (i.e. https://www.desmos.com/calculator)
Polynomial
Sketch the function
Degree # of
Leading From / To
function equation
Notice the graph near the zero(es)
Zeroes coeffici Quadrants
ent
y=x
y = −x
From
quadrant II
Graph crosses the x-axis
at its zero.
To
quadrant IV
y = x( x − 1)( x + 2)
y = − x( x − 1)( x + 2)
Page 1 of 10
First
degree
(Odd)
One
Negativ
e
From Q II
to
Q IV
MHF4U0
Polynomial Function Investigation
Polynomial
function equation
ODD DEGREE
Sketch of the function
Notice the graph near the zero(es)
Name:______________
Degree
# of
Zeroes
Leading
coeffici
ent
(+ or -)
Start / End
Quadrants
(left →
right)
y = ( x + 1) 2 ( x − 2)
y = −( x + 1) 2 ( x − 2)
Come up with your own equation of a polynomial function of odd degree: ____________________________
Predict its graph, then check with a graphing software.
CONCLUSIONS: Complete the following:
1. As x increases from left to right, a polynomial function of odd degree which has:
a positive leading coefficient, starts in quadrant _____ and ends in quadrant ______.
2. As x increases from left to right, a polynomial function of odd degree which has:
negative leading coefficient, starts in quadrant _____ and ends in quadrant ______.
3. A polynomial function of odd degree n, has a minimum of ___zeroes and a maximum of_____ zeroes.
Page 2 of 10
MHF4U0
Polynomial Function Investigation
Name:______________
PART 1B: END BEHAVIOUR- EVEN DEGREE
Polynomial
Sketch of the function
function equation
Notice the graph near the zero(es)
EVEN DEGREE
y = x2
𝑦 = −𝑥 2
From
quadrant III
Graph
bounces
off the xaxis at its
zero
To
quadrant IV
y = ( x − 1) 2 ( x + 2) 2
y = −( x − 1) 2 ( x + 2) 2
Page 3 of 10
Degree
# of
Zeroes
Leading
coefficie
nt
From / To
Quadrants
Second
(Even)
degree
One
Negative
From
Q III
To
Q IV
MHF4U0
Polynomial Function Investigation
Polynomial
function equation
EVEN DEGREE
Sketch of the function
Name:______________
Degree
# of
Zeroes
Leading
coefficie
nt
(+ or -)
Start / End
Quadrants
(left →
right)
y = x 2 ( x + 1) 2
𝑦 = −𝑥 2 (𝑥 + 1)2
Come up with your own equation of a polynomial function of even degree: ____________________________
Predict its graph, then check with a graphing software.
CONCLUSIONS: Complete the following:
1. As x increases from left to right, a polynomial function of even degree which has:
a positive leading coefficient, starts in quadrant _____ and ends in quadrant ______.
2. As x increases from left to right, a polynomial function of even degree which has:
negative leading coefficient, starts in quadrant _____ and ends in quadrant ______.
3. A polynomial function of even degree n, has a minimum of ___zeroes and a maximum of_____ zeroes..
Page 4 of 10
MHF4U0
Polynomial Function Investigation
Name:______________
PART 2: Multiplicity of the roots and the shape of the graph near a zero of the function
Polynomial
function equation
Sketch the function
Notice the graph near the zero(es)
State what happens to the graph near each
zero of the function
y = ( x − 1)( x + 2)3
𝑦 = (𝑥 − 1)2 (𝑥 + 2)3
At the triple root of x = -2,
the graph crosses the x-axis, and
flattens out.
Note: A triple root is a root of ODD
MULTIPLICITY.
At the double root of x=1, the graph
bounces off the x-axis.
Note: A double root is a root of EVEN
MULTIPLICITY.
y = − x 2 ( x + 1) 2
𝑦 = −(𝑥 − 3)3 (𝑥 + 2)4
Page 5 of 10
MHF4U0
Polynomial Function Investigation
Name:______________
Predict the graph of each of these functions using “End Behaviour” and “Graph near zeroes”, then check.
y = x 2 ( x + 2) 2 ( x − 3)
y = −( x + 2)5 ( x − 3) 2
Graph behaviour at a zero (x-intercept) of the function. Complete the following sentences:
1. When a polynomial function has an even multiplicity root, its graph ______________________ at that root.
(i.e. at a double root,…)
2. When a polynomial function an odd multiplicity root, its graph __________________________ at that root.
(i.e. at a single root, at a triple root…)
Page 6 of 10
MHF4U0
Polynomial Function Investigation
Name:______________
APPLY YOUR KNOWLEDGE: NO CALCULATOR!
What is the minimum and maximum number of zeroes possible for each of the polynomial functions with
the given degrees: a) 2
(b) 3
(c) 5
(d) 6
(e) 7
(f) n
1. Identify the sign of the leading coefficient and describe the end behaviour. Using this information,
decide if each function is cubic or quartic
Page 7 of 10
MHF4U0
Polynomial Function Investigation
Name:______________
2. Without using any graphing technology, graph the following polynomial functions:
Function:
y = − x( x − 2) 4 ( x + 3)
y = −x(x − 2)4 (x + 3)
Function:
y = x 3 ( x − 4)( x + 5) 2
y = x3(x − 4)(x + 5)2
Degree:
Degree:
Sign:
Sign:
Roots:
Roots:
y-intercept:
y-intercept:
domain:
domain:
range:
range:
end behaviours: as x → −, y → _____
as x → , y → _____
end behaviours: as x → −, y → _____
as x → , y → _____
Page 8 of 10
MHF4U0
Function:
Polynomial Function Investigation
y = −( x + 1)5 ( x − 3) 4
y = −(x + 1)5 (x − 3)4
Function:
Name:______________
y = (x + 3)2(x −1)3
y = ( x + 3) 2 ( x − 1)3
Degree:
Degree:
Sign:
Sign:
Roots:
Roots:
y-intercept:
y-intercept:
domain:
domain:
range:
range:
end behaviours: as x → −, y → _____
as x → , y → _____
end behaviours: as x → −, y → _____
as x → , y → _____
Page 9 of 10
MHF4U0
Function:
Polynomial Function Investigation
y = − x 4 ( x + 3)( x − 4)3
y = −x4(x + 3)(x − 4)3
Function:
Name:______________
y = (x−3)(x+5)(x−2)2
y = ( x − 3)( x + 5)( x − 2) 2
Degree:
Degree:
Sign:
Sign:
Roots:
Roots:
y-intercept:
y-intercept:
domain:
domain:
range:
range:
end behaviours: as x → −, y → _____
as x → , y → _____
end behaviours: as x → −, y → _____
as x → , y → _____
Page 10 of 10