Sketching Polynomials
Sketching
Polynomials
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Sketching
Polynomials
SKETCHING
POLYNOMIALS
Up until now you may have sketched quadratic polynomials (parabolas) or linear polynomials (straight
lines). This unit deals with how to find intercepts and sketch graphs of polynomials with higher degrees.
Answer these questions, before working through the chapter.
I used to think:
What is a multiple root of a polynomial?
What do the zeros of a polynomial represent in its graph?
What is a point of inflection and when do they occur?
Answer these questions, after working through the chapter.
But now I think:
What is a multiple root of a polynomial?
What do the zeros of a polynomial represent in its graph?
What is a point of inflection and when do they occur?
What do I know now that I didn’t know before?
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Basics
Zeros and Multiplicity
The solutions to a polynomial equation P^ xh = 0 are called x-intercepts of P^ xh . They are also called 'zeros' or
'roots'. Remember, to solve the equation P^ xh = 0 it is necessary to factorise. Here is an example:
Solve the equation x3 - 2x 2 - 7x - 4 = 0
Let P^ xh = x3 - 2x2 - 7x - 4
P^-1h = 0 ` x + 1 is a factor
x 2 - 3x - 4
x + 1 g x -2x2 - 7x - 4
x3 + x2
3
- 3x 2 - 7x
- 3x 2 - 3x
- 4x -4
- 4x -4
0
` P^ xh = ^ x + 1h^ x2 - 3x - 4h = 0
= ^ x + 1h^ x + 1h^ x - 4h = 0
` x = -1 or x = -1 or x = 4
These are the 'zeros' of P^ xh
If the same zero appears twice (like x = -1 in the above example), then it is called a double root. The number
of times a zero appears is called the multiplicity of the zero. For example, x = -1 in the above example has a
multiplicity of 2. The multiplicity of x = 4 is 1
Here is another example:
Find the multiplicity of the zeros of Px = x 4 + 5x3 + 6x 2 - 4x - 8
P^ xh can be factorised into
So the zeros of P^ xh are
2
P^ xh = ^ x - 1h^ x + 2h^ x + 2h^ x + 2h
x = 1 or x =-2 or x =-2 or x =-2
•
x = 1 which is a single root, since it appears only once (multiplicity 1)
•
x = -2 which is a triple root, since it appears three times (multiplicity 3)
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Types of Polynomials
Here are some important types of polynomials based on the degree.
•
A linear polynomial has a degree of 1. It has the form P^ xh = ax + b . These have at most 1 zero.
•
A quadratic polynomial has a degree of 2. It has the form P^ xh = ax2 + bx + c .
These have at most 2 unique zeros.
•
A cubic polynomial has a degree of 3. It has the form P^ xh = ax3 + bx2 + cx + d .
These have at most 3 unique zeros.
•
A quartic polynomial has a degree of 4. It has the form P^ xh = ax4 + bx3 + cx2 + dx + e .
These have at most 4 unique zeros.
If the leading coefficient is 1 then the polynomial is called monic.
Solve these equations:
a
x2 + 2x - 8 = 0
x2 - 6x + 9 = 0
b
` ^ x - 2h^ x + 4h = 0
` ^ x - 3h^ x - 3h = 0
This has 2 distinct zeros
This equation has 2 solutions, but they are not unique
` x =2 or x = -4
` x = 3 or x = 3
y
-4
y
2
x
3
x
Two unique solutions
Double zero
Sometimes, there may be no real roots. Look at this example
A quadratic polynomial with no zeros
y
2x2 + 3x + 4 = 0
9 = b2 - 4ac
= ^3 h2 - 4^2h^4h
= -23 1 0
` No real solutions ^no x-interceptsh
x
No x-intercepts
These examples show that it is possible for a quadratic polynomial to have 2, 1 or 0 zeros. This is why we say that
it can have at most 2 unique zeros. In general, a polynomial of degree n will have at most n zeros.
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Questions
1. Let P^ xh = x3 - 4x 2 - 3x + 18
a
Factorise P^ xh using long division
b
Identify the zeros of P^ xh
c
Identify the multiplicity of each of the zeros of P^ xh
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Basics
Sketching Polynomials
Questions
Basics
2. Let P^ xh = x 4 - 2x3 - 7x 2 + 20x - 12
a
Factorise P^ xh given that P^2h = 0
b
Does P^ xh have single roots, double roots or triple roots?
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Questions
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3. Identify the type of each of these polynomials and state how many possible zeros each polynomial may have.
a
x 2 + 3x + 2
b
x3 - 3x2 + 4x + 1
c
3x - 3
d
3x4 - 2x3 + x2 - 5x + 6
e
2x2 + 3x3 - 1 + x
f
1 + 2x2 - 5x4 + 3x3 + 9x
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Sketching Polynomials
The first two things to look for when sketching polynomials are:
•
The zeros. These will be the points where the curve cuts the x-axis. (The x-intercepts)
•
The constant term. This will be the point where the curve cuts the y-axis. (The y-intercept)
Once the x-intercepts and y-intercepts are known, the degree of the polynomial and the leading coefficient need
to be checked at the same time. This information will determine the shape of the curve.
Let's say the degree of the polynomial is n and the leading coefficient is a.
Here are the rules (always from left to right)
a is positive ^a 2 0h
Finishes by increasing
a is negative ^a 1 0h
a 2 0, n is even
a 1 0, n is even
y
n is
even
Finishes by decreasing
y
Finish by moving up
Start by moving down
x
x
Start by moving up
Finish by moving down
a 1 0, n is odd
a 2 0, n is odd
y
y
Finish by moving up
n is
odd
Start by moving down
x
x
Start by moving up
Finish by moving down
Remember, a polynomial is a continuous function and so the curve will have no breaks for any x-value.
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Here are some examples. Notice the difference between the polynomials and how it affects their graphs.
Sketch the cubic polynomial: A^ xh = x3 - 3x 2 - x + 3
Step 1: Factorise to identify the zeros.
A^ xh = x3 - 3x2 - x + 3 = ^ x - 1h^ x + 1h^ x - 3h
` A^ xh has x-intercepts x = 1, x = -1 and x = 3
Step 2: On a set of axes, mark off the y-intercept (the constant term) and the x-intercepts (the zeros).
y
3
y-intercept
Zeros
-1
1
x
3
Step 3: Use the degree and leading coefficient to find the shape of the graph
Remember, the leading coefficient is the number in front of the highest power of x. The constant
is the term without x. The leading coefficient is 1 (positive) and the degree of A^ xh is 3 (odd).
` a 2 0 and n is odd.
y
Finish by moving up
3
-1
1
3
x
Start by moving up
The curve must be drawn through all the intercepts and have the correct shape based on a and n.
The arrows on the ends of the curve mean that it continues forever toward negative infinity ^-3h on the left
and positive infinity ^3h on the right.
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Here are two quartic polynomials with different shapes:
Sketch the following Quartic Polynomials:
a
P^ xh = x4 + 2x3 - 9x2 - 2x + 8
b
Step 1: Factorise to identify the zeros.
Q^ xh = -x4 - 4x3 - x2 + 6x
Step 1: Factorise to identify the zeros.
P^ xh = ^ x - 1h^ x - 2h^ x + 1h^ x + 4h
Q^ xh = -x^ x - 1h^ x + 2h^ x + 3h
P^ xh has x-intercepts at
x = 1, x = 2, x = -1 and x = -4
Q^ xh has x-intercepts at
x = 0, x = 1, x = -2 and x = -3
Step 2: On a set of axes, mark off the
y-intercept (the constant term) and
the x-intercepts (the zeros).
Step 2: On a set of axes, mark off the
y-intercept (the constant term) and
the x-intercepts (the zeros).
y
y
8
Zeros
-4
Zeros
y-intercept
-1
1
x
2
-3
-2
0
x
1
y-intercept
Step 3: Use the degree and leading coefficient
to find the shape of the graph
Step 3: Use the degree and leading coefficient
to find the shape of the graph
The leading coefficient is 1 (positive)
The degree of P^ xh is 4 (even)
` a 2 0, n is even (end by increasing)
y
Start by moving down
The leading coefficient is -1 (negative)
The degree of Q^ xh is 4 (even)
` a 1 0, n is even (end by decreasing)
y
Finish by moving up
8
-4
-1
1
2
x
-3
Start by moving up
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-2
0
x
1
Finish by moving down
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Questions
Knowing More
1. Use the set of axes at the bottom of the page to answer these questions.
a
The polynomial P^ xh = -x3 + 2x2 + 5x - 6 can be factorised into the brackets -^ x + 2h^ x - 3h^ x - 1h .
What are the zeros of this polynomial? Mark these zeros off on the x-axis.
b
What is the constant term? Mark this off as the y-intercept on the axes.
c
What is the degree of the polynomial? Is this even or odd?
d
What is the leading coefficient?
e
Will the curve start by moving up or down?
f
Will the curve finish by moving up or down?
g
Draw the curve for P^ xh on the axes below.
y (or P^ xh )
7
6
5
4
3
2
1
-6
-5
-4
-3
-2
-1 1
-1
-2
-3
-4
-5
-6
-7
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3
4
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Sketching Polynomials
Questions
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2. Use the set of axes at the bottom of the page to answer these questions.
a
The polynomial Q^ xh = x4 + 5x3 + 5x2 - 5x - 6 can be factorised into the brackets
^ x + 3h^ x - 1h^ x + 1h^ x + 2h . What are the zeros of this polynomial? Mark these zeros off on the x-axis.
b
What is the constant term? Mark this off as the y-intercept on the axes.
c
What is the degree of the polynomial? Is this even or odd?
d
What is the leading coefficient?
e
Will the curve start by moving up or down?
f
Will the curve finish by moving up or down?
g
Draw the curve for Q^ xh on the axes below.
y (or Q^ xh )
7
6
5
4
3
2
1
-6
-5
-4
-3
-2
-1 1
-1
2
3
4
5
6
x
-2
-3
-4
-5
-6
-7
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Questions
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3. Use this curve to answer questions about the polynomial function it came from.
Not to scale
y
2
-2 -1
x
-4
a
How many zeros does this polynomial have? How do you know?
b
Is the degree of the polynomial even or odd?
c
What is the constant term of the polynomial? How do you know this?
d
Is the leading coefficient positive or negative? How do you know this?
e
Find the polynomial P^ xh given that it is monic.
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4. Use this curve to answer questions about the polynomial function it came from.
Not to scale
y
-4
-1
2
4
x
-32
a
How many zeros does this polynomial have? How do you know?
b
Is the degree of the polynomial even or odd?
c
What is the constant term of the polynomial? How do you know this?
d
Is the leading coefficient positive or negative? How do you know this?
e
Find the polynomial P^ xh given that the leading coefficient is -1.
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5. Use this curve to answer questions about the polynomial it came from.
Not to scale
y
-4
-1
3
5
x
a
Is the degree even or odd? Is the leading coefficient positive or negative?
b
Write the polynomial P^ xh in factored form given that it is monic.
c
What is the y-intercept if the polynomial is monic?
d
Write the polynomial P^ xh in factored form given the leading coefficient is 2.
e
Find the y-intercept if the leading coefficient is 2.
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6. Factorise and draw a rough sketch of P^ xh = x3 - x 2 - 24x - 36 given that P^-2h = 0 .
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Multiple Roots
Sometimes a root may be repeated in a polynomial:
•
For example, using long division, P^ xh = 3x3 - 16x2 + 28x - 16 can be factorised in the following way:
P^ xh = 3x3 - 16x2 + 28x - 16
= ^ x - 2h^ x - 2h^3x - 4h
The bracket ^ x - 2h appears twice and so x = 2 is called a double root of P^ xh .
•
In another example Q^ xh = x4 + 9x3 + 12x2 - 80x - 192 can be factorised to form
Q^ xh = x4 + 9x3 + 12x2 - 80x - 192
= ^ x + 4h^ x + 4h^ x + 4h^ x - 3h
The bracket ^ x + 4h appears three times and so x = -4 is called a triple root of Q^ xh .
•
If a bracket appears only once (is not repeated) then the root is called a single root.
Here are some examples:
Find polynomials with these roots
a
A polynomial with a single root at 5 and a double root at -3, and a leading coefficient of -2
A^ xh = -2^ x - 5h^ x + 3h^ x + 3h
Leading coefficient
Single root
Double root
= -2x3 - 2x2 + 42x + 90
b
A monic polynomial with single roots at 3 and 2 and a double root at -1.
P^ xh = ^ x - 3h^ x - 2h^ x + 1h^ x + 1h
Single root
Single root
Double root
= x 4 - 3x 3 - 3x 2 + 7x + 6
c
A polynomial with a triple root at 2 and a single root at -4, with a leading coefficient of 2.
Q^ xh = 2^ x - 2h^ x - 2h^ x - 2h^ x + 4h
Leading coefficient
Triple root
Single root
= 2x4 - 4x3 - 24x2 + 80x - 64
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Sketching Polynomials with Double Roots
Up until now only polynomials without multiple roots (single roots only) have been drawn. Single roots cause the
curve to simply "cut through" the x-axis. Multiple roots behave slightly differently on the x-axis.
A double root causes the curve to touch (or "bounce off") the x-axis at the zero.
y
y
Double root
or
x
x
Double root
Here is an example of sketching with a double root.
Sketch the polynomial of P^ xh = x3 - 3x + 2
•
•
x3 - 3x + 2 can be factorised so that P^ xh = ^ x + 2h^ x - 1h^ x - 1h
P^ xh has a single root at x = -2 and a double root at x = 1
•
P^ xh has degree 3 (odd) and the leading coefficient is 1 (positive). So the curve will start moving up, and
finish moving up.
•
The y-intercept is +2 (the constant term of P^ xh )
y
Finish by moving up
2
-2
1
x
Double root
Start by moving up
Notice how the curve "cuts through" the single root at x = -2 and only touches the x-axis at x = 1 .
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Sketching Polynomials with Triple Roots
A single root intercepts the x-axis. A double root causes the curve to only touch the x-axis. A triple root causes the
curve to do something different. At the point of the triple root over the x-axis, the curve intercepts at the root and
"flattens" as it passes through. The point where the curve does this is called a point of horizontal inflection (some
people spell it inflexion).
Here is an example
Sketch the curve of y = P^ xh where P^ xh = 2x 4 - 12x 2 + 16x - 6
•
•
•
•
P^ xh can be factorised (using long division) into the form P^ xh = 2^ x + 3h^ x - 1h^ x - 1h^ x - 1h .
P^ xh has a single root at x = -3 and a triple root at x = 1 .
The y-intercept is -6 (the constant term of P^ xh )
P^ xh has a degree of 4 (even) and a leading coefficient of 2 (positive).
So start by moving down, finish by moving up.
Not to scale
yy
Start by moving down
Finish by moving up
-3
x
1
Single root
Triple root
-6
Notice how the curve "cuts through" the single root at x = -3 and only has a point of horizontal inflection
at the triple root x = 1 .
If a polynomial has double or triple roots, it won't affect the shape of the polynomial. Multiple roots will
only affect the behaviour of a curve at the zeros on the x-axis (whether the curve bounces or has a point of
horizontal inflection). The shape is still based on the degree of the polynomial and the leading coefficient.
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Remember the rules for the shape of polynomial curves. If P^ xh has leading coefficient a and degree n then:
•
If n is even and a 2 0 then start moving down, finish moving up.
•
If n is even and a 1 0 then start moving up, finish moving down.
•
If n is odd and a 2 0 then start moving up, finish moving up.
•
If n is odd and a 1 0 then start moving down, finish moving down.
The rules for what the curve will do at the roots are:
•
Single root: The curve will pass through the x-axis at the root
•
Double root: The curve will touch ("bounce off") the x-axis at the root
•
Triple root: The curve will pass through the x-axis at the root as a point of horizontal inflection.
Here is an example of a polynomial with a double root and a triple root:
Sketch the curve of y = P^ xh where P^ xh = -3x5 - 12x 4 - 3x3 + 30x 2 + 12x - 24
P^ xh can be factorised P^ xh = -3^ x - 1h^ x - 1h^ x + 2h^ x + 2h^ x + 2h .
Leading coefficient
•
•
•
Double root
Triple root
P^ xh has a double root at x = 1 and a triple root at x = -2 .
The y-intercept is -24 (the constant term of P^ xh )
P^ xh has a degree of 5 (odd) and a leading coefficient of -3 (negative). Start by moving down,
finish by moving down.
Not to scale
y
Start by moving down
Triple root
Double root
-2
1
-24
Finish by moving down
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1. For each of the following find:
(i)
The roots and identify if each root is a multiple root.
(ii) The degree of the polynomial.
(iii) The constant term of the polynomial.
a
P^ xh = x^ x + 1h^ x - 2h^ x + 3h
b
T^ xh = ^ x + 4h^ x - 3h^ x + 4h
c
B^ xh = 4^ x - 1h2 ^ x + 2h3
d
M^ xh = -2^ x - 4h2 ^ x + 2h^ x - 4h
2. Write a polynomial P^ xh with these properties in factorised form.
a
Single roots at 2 and -3
Double root at 5
Leading coefficient of 2
b
Single root at -5
Double root at -2 and 4
Leading coefficient of 4
c
Double root at -4
Triple root at 2
Leading coefficient of -1
d
Triple root at 1
Double root at -7
Leading coefficient of 5
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3. The polynomial P^ xh = 2x3 - 16x 2 + 40x - 32 can be factorised into P^ xh = 2^ x - 4h^ x - 2h2 .
a
Are the roots single roots or multiple roots?
b
How would the graph of P^ xh behave at each root?
c
What is the degree and is the leading coefficient negative or positive?
d
On the diagram below, mark off each of the zeros on the x-axis and identify the type of root.
y (or P^ xh )
-4
e
-3
-2
-1 1
2
3
x
4
Draw a rough sketch of the graph of P^ xh .
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4. The polynomial A^ xh = -x 4 - x3 + 12x 2 + 28x + 16 can be factorised into A^ xh = -^ x - 4h^ x + 1h^ x + 2h^ x + 2h .
a
Are the roots single roots or multiple roots?
b
On the diagram below, mark off each of the zeros and identify the type of root.
y (or A^ xh )
-4
c
-3
-2
-1 1
Sketch the graph of A^ xh on the axes above.
5. What is a triple root, and how do graphs behave at triple roots?
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6. The polynomial T^ xh = 3x 4 + 9x3 - 18x 2 - 84x - 72 can be factorised into T^ xh = 3^ x - 3h^ x + 2h3 .
a
Are any of the roots triple roots?
b
On the diagram below, mark off each of the zeros and identify the type of root.
y (or T^ xh )
-4
-3
-2
-1 1
2
c
Where is the inflection point?
d
What is the degree of T^ xh and is the leading coefficient positive or negative?
e
Draw a rough sketch of T^ xh on the above set of axes.
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7. The polynomial M^ xh = - 2x 4 + 4x3 + 24x 2 - 80x + 64 can be factorised into M^ xh = -2^ x - 2h3 ^ x + 4h .
a
Are any of the roots triple roots?
b
On the diagram below, mark off each of the zeros and identify the type of root.
y (or M^ xh )
-4
-3
-2
-1 1
2
3
4
c
Where is the inflection point?
d
What is the degree of the polynomial M^ xh and is the leading coefficient positive or negative?
e
Draw a rough sketch of M^ xh on the above set of axes.
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Transformations of Polynomials
A transformation of a polynomial P^ xh is in one of these forms where a is a constant:
•
•
•
•
•
•
aP^ xh stretches the graph of P^ xh by a factor of a
-P^ xh is a vertical reflection of P^ xh and y = P^-xh is a horizontal reflection of P^ xh
P^ x + ah moves the graph a units to the left
P^ x - ah moves the graph a units to the right
P^ xh + a moves the graph a units up
P^ xh - a moves the graph a units down
Graphing y = aP(x)
The transformation aP^ xh is called a 'dilation' and stretches the graph of P^ xh by a factor of a.
Here are some examples:
The dashed line represents the graph for y = P^ xh . Draw y = 2P^ xh
y
#2
x
y = P(x)
y = 2P(x)
It's easy to see that P^ xh = ^ x + 2h^ x - 3h^ x + 4h and 2P^ xh = 2^ x + 2h^ x - 3h^ x + 4h have the same x-intercepts.
Dilations of polynomials have the same zeros. Only the height of the graph will change, not the zeros. If a 1 1 then
the graph will 'shrink' and not 'stretch'.
The dashed line represents the graph for y = P^ xh . Draw y = 1 P^ xh
2
y
#1
2
x
y = 1 P (x)
2
y = P(x)
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Vertical Reflections
'Vertical Reflection' is a fancy way of saying 'flipping from top-to-bottom'. These are similar to dilations y = aP (x) ,
the only difference is that a is negative.
Let's say a = -1, then the 'dilation' is y = -P(x). This means that when P(x) is positive, y = -P(x) is negative,
and when P(x) is negative, y = -P(x) will be positive. This means the graph will reflect around the x-axis.
The dashed line represents the graph for y = P(x). Draw y = -P(x)
y
y = -P(x)
reflect
x
y = P(x)
Sometimes the graph is reflected and dilated. If a is negative but not -1 then the graph is reflected about the
x-axis and dilated accordingly. Here is an example.
The dashed line represents the graph for y = P(x). Draw y = -2P(x)
y
This curve has been
reflected and dilated
x
y = P(x)
y = -2P(x)
Even if polynomials are dilated and reflected vertically, they will still have the same zeros as the original polynomial.
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1. The graph below represents y = P^ xh where P^ xh = ^ x + 3h^ x - 3h^ x + 6h .
y
Not to scale
108
81
54
27
-7
-6
-5
-4
-3
-2
-1 1
2
3
4
5
6
7
x
-27
-54
-81
-108
a
Use the graph of P(x) to sketch the graph of y = 2P(x) on the same axes. What is the y-intercept of y = 2P(x)?
How does this relate to the y-intercept of P(x)?
b
Use the graph of P(x) to sketch the graph of y = 1 P (x) on the same set of axes.
2
What is the y-intercept of y = 1 P (x) ? How does this relate to the y-intercept of P(x)?
2
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2. The graph below represents y = Q^ xh = -x 4 + 8x3 + 10x 2 - 104x - 105 = -^ x + 1h^ x - 5h^ x + 3h^ x - 7h .
y
Not to scale
400
300
200
100
-7
-6
-5
-4
-3
-2
-1 1
2
3
4
-100
-200
-300
-400
a
Use the graph of Q(x) to sketch the graph of y = 3Q(x) on the same axes.
b
Use the graph of 3Q(x) to sketch the graph of y = -3Q(x).
c
What are the zeros of -3Q(x)? Why?
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Horizontal Shifts
'Horizontal Shift' is just a fancy way of saying 'moving left or right'. These transformations look like this:
•
•
P (x + a) moves the graph a units to the left
P (x - a) moves the graph a units to the right
Here are some examples:
The dashed line represents the graph for
y = P(x). Draw y = P(x - 1)
The dashed line represents the graph for
y = P(x). Draw y = P(x + 2)
y
y
1 unit
2 units
x
y = P(x +2)
x
y = P(x -1)
y = P(x)
y = P(x)
Vertical Shifts
'Vertical Shift' is just a fancy way of saying 'moving up or down'. These transformations look like this:
•
•
P (x) + a moves the graph a units upwards
P (x) - a moves the graph a units downwards
Here are some examples:
The dashed line represents the graph for
y = P(x). Draw y = P(x) + 2
The dashed line represents the graph for
y = P(x). Draw y = P(x) - 3
y
y
y = P(x)
2 units
y = P(x+2)
x
x
3 units
y = P(x -3)
y = P(x)
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3. The graph of P^ xh = x^ x - 1h^ x + 3h is shown below:
y
12
Not to scale
10
8
6
4
2
-6
-5
-4
-3
-2
-1 1
2
3
4
5
-2
-4
-6
a
Use the graph of P(x) to sketch P(x + 3) on the above axes.
b
Use the graph of P(x) to sketch P(x) + 3 on the above axes.
c
Use the graph of P(x) to sketch P(x - 4) on the above axes.
d
Use the graph of P(x) to sketch P(x) - 2 on the above axes.
e
From the graphs above, do you think horizontal shifts of polynomials have the same zeros? Why?
f
From the graphs above, do you think vertical shifts of polynomials have the same zeros? Why?
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Horizontal Reflection
'Horizontal Reflection' is a fancy way of saying 'flipping from left-to-right'. This happens when the x inside the
bracket is negative. So P(-x) is a reflection of P(x) about the y-axis (horizontal reflection).
The dashed line represents the graph for y = P(x). Draw y = P(-x)
y
x
y = P(x)
y = P(-x)
reflect
Combining Transformations
Sometimes a polynomial may be transformed in more that one way.
The dashed curve below represents y = P^ xh = x3 + 2x 2 - x - 2 = ^ x - 1h^ x + 2h^ x + 1h
The solid line represents y = P^ x - 3h + 2
y
3 units
2
2 units
-5
-4
-3
-2
1
-1 1
-1
2
3
4
5
x
-2
-3
-4
The solid curve has been moved 2 units upwards and 3 units to the right.
If the transformations contain only shifts, then their order doesn't matter. It is different if there is a dilation
or reflection…then that is done first!
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Here is an example of a combination of transformations with a dilation and a shift.
The dashed line represents y = P(x). Draw y = 1 P (x) - 2
2
The dilation must be done first!
y
6
5
4
3
2
2 units
1
-5
-4
-3
Dialate by 1
2
-2
-1 1
-1
2
3
4
5
x
-2
-3
-4
-5
After the dilation the graph was moved 2 units downwards
Here is an example with a horizontal shift. Remember, the dilation is done first.
The dashed graph below represents y = P^ xh = -2x 2 + 2 = -2^ x + 1h^ x - 1h . Sketch y = 2P^ x - 1h
The dilation must be done first!
y
4
3
2
1
-4
-3
-2
-1 1
-1
2
-2
-3
1 unit
-4
After the dilation the graph was moved 1 unit to the right
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Sketching Polynomials
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4. Explain in words what happens to the graph of P(x) under these transformations:
a
y = 3P ^ x h
b
y = P^ x + 6h
c
y = P^ x - 6h
d
y = P^-xh + 2
e
y = 2P^ x - 4h
f
y = -P ^ x h - 5
g
y = P^ x + 3h + 3
h
y = -P ^ x + 2 h - 1
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5. The graph below is of y = P^ xh .
Not to scale
y
5
4
3
2
1
-5
-4
-3
-2
-1 1
-1
-2
-3
-4
-5
a
On the same set of axes, sketch the graph of y = P^ x + 2h + 1 .
b
On the same set of axes, sketch the graph of y = 2P^ xh - 1 .
c
On the same set of axes, sketch the graph of y = 2P^ x - 1h .
d
On the same set of axes, sketch the graph of y = P^-xh .
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Answers
Basics:
Knowing More:
1. a P^ xh = ^ x + 2h^ x - 3h^ x - 3h
b
x =-1 or
x = 3 or
c
x =-1 has a multiplicity of 1
x = 3 has a multiplicity of 2
1. d The leading coefficient is -1
x=3
e
The curve starts by moving down
f
The curve finishes by moving down
g
2. a
b
3. a
P^ xh = ^ x - 2h^ x - 1h^ x - 2h^ x + 3h
•
x = 2 is double root
•
x = 1 is single root
•
x =-3 is single root
Zero
Quadratic polynomial
Zero
Zero
y-intercept
This has at most two unique zeros
b
2. a Zeros at:
x =-3, x = 1, x =-1 and x =-2
Cubic polynomial
This has at most three unique zeros
c
b
The constant term is -6
c
The degree of the polynomial is 4.
This is even
d
The leading coefficient is 1.
The polynomial is monic.
e
The curve starts by moving down
f
The curve finishes by moving up
Linear polynomial
This has at most one zero
d
Quartic polynomial
This has at most four unique zeros
e
Cubic polynomial
This has at most three unique zeros
g
f
Quartic polynomial
This has at most four unique zeros
Zero Zero Zero
Zero
Knowing More:
1. a
Zeros at x =-2, x = 3 and x = 1
y-intercept
b
The constant term is -6
c
The degree of the polynomial is 3, ` odd
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Knowing More:
Knowing More:
3. a This polynomial has three unique zeros as
it crosses the x axis three times.
b
The degree of the polynomial is odd.
c
The constant term of the polynomial is -4.
The curve crosses the y axis at -4.
d
The leading coefficient is positive. The
degree of the polynomial is odd and the
curve starts by going up.
e
6. P^ xh = ^ x + 2h^ x + 3h^ x - 6h = 0
P^ xh = x3 - 4x - 4
4. a This polynomial has four unique zeros as it
crosses the x axis four times.
b
The degree of the polynomial is even.
c
The constant term of the polynomial is
-32. The curve crosses the y axis at -32.
Using Our Knowledge:
1. a (i)
d
e
The leading coefficient is negative. The
degree of the polynomial is even and
the curve starts by going up.
(ii) The degree of the polynomial is 4
P^ xh =-x4 + x3 + 22x2 - 32
(iii) The constant term of the polynomial is 0.
b
5.
a
x = 0, x =-1, x = 2 and x =-3
None of the roots are multiple roots.
(i)
The degree of the polynomial is even. The
coefficient of the polynomial is positive.
b
P^ xh = ^ x + 4h^ x + 1h^ x - 3h^ x - 5h
c
The y-intercept of the polynomial is 60.
d
P^ xh = 2^ x + 4h^ x + 1h^ x - 3h^ x - 5h
e
The y-intercept of the polynomial is 120.
x =-4, x = 3 and x =-4
T^ xh has a single root at x = 3 and a
double root at x =-4
(ii) The degree of the polynomial is 3
(iii) The constant term of the polynomial
is -48.
c
(i)
x = 1, x = 1, x =-2, x =-2
and x =-2
B^ xh has a double root at x = 1
and a triple root at x =-2
(ii) The degree of the polynomial is 5
(iii) The constant term of the polynomial
is 32
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Using Our Knowledge:
d
(i)
Using Our Knowledge:
x = 4, x = 4, x =-2 and x = 4
M^ xh has a triple root at x = 4
and a single root at x =-2
4. b and c
(ii) The degree of the polynomial is 4
(iii) The constant term of the polynomial
is 256.
2. a P^ xh = 2^ x -2h^ x +3h^ x -5h^ x -5h
b
c
d
P^ xh = 4^ x +5h^ x +2h^ x +2h^ x -4h^ x -4h
P^ xh =-1^ x -2h^ x -2h^ x -2h^ x +4h^ x +4h
P^ xh = 5^ x -1h^ x -1h^ x -1h^ x + 7h^ x + 7h
3. a P^ xh has a double root at x = 2 and a
single root at x = 4 .
b
At x = 2 the graph of P^ xh will touch
the x-axis.
c
At x = 4 the graph of P^ xh will cross
the x-axis.
5. A triple root is a solution to a polynomial
equation at appears three times. A grpah of
a polynomial with a triple root has a point
of horizontal inflection on the x-axis at the
triple root.
6. a T^ xh has a triple root at x =-2 and
a single root at x = 3
b
and e
The degree of P^ xh is 3
(the highest order is x3 ).
The leading coefficient is poisitive (2).
d
and
c
The inflection point is at x =-2
d
The degree of T^ xh is 4
(the highest order is x4 ).
e
The leading coefficient is positive (3).
7. a
4. a
M^ xh has a triple root at x = 2 and
a single root at x =-4
A^ xh has a double root at x =-2 , a single
root at x = 4 and a single root at x =-1
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Thinking More:
Using Our Knowledge:
7.
b
and e
2. a
y = Q^ x h
c
The inflection point is at x = 2
d
The degree of M^ xh is 4
(the highest order is x4 ).
y = 3Q ^ x h
b
The leading coefficient is negative (-2).
Thinking More:
y =-3Q^ xh
y = 2P^ xh
1. a
y = Q^ x h
y = P^ x h
c
This is because the whole polynomial has
been multiplied and multiplying zero by
any factor will always equal zero.
The y-intercept of y = 2P^ xh =-108 is 2
times the y-intercept of y = P^ xh =-54
3. a
b
y = P^ x h
y = 1 P^ x h
2
The y-intercept of y = 1 P^ xh =-27 is 1
2
2
times the y-intercept of y = P^ xh =-54
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The zeros of both -3Q^ xh and Q^ xh are
x =-1, x = 5, x =-3 and x = 7
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Answers
Thinking More:
Thinking More:
4. d The graph of P^ xh is reflected (flipped)
horizontally and shifted vertically 2 units
upwards.
3. b
e
f
c
g
h
The graph of P^ xh is dilated (stretched
by a factor of 2) and is then shifted
horizontally 4 units to the right.
The graph of P^ xh is reflected (flipped)
vertically and shifted vertically 5 units
downwards.
The graph of P^ xh is shifted horizontally
3 units to the left and shifted vertically 3
units upwards.
The graph of P^ xh is reflected (flipped)
vertically, shifted horizontal 2 units to the
left and shifed vertically 1 unit downward.
d
5. a
e
f
4. a
Horizontal shifts of polynomials do not
have the same zeros. This is because
every point of the polynomial is shifted
horizontally including the zeros.
b
Vertical shifts of polynomials do not
have the same zeros. This is because
every point of the polynomial is shifted
vertically including the zeros.
The graph of P^ xh is dilated (stretched by
a factor of 3. The zeros remain the same.)
b
The graph of P^ xh is shifted horizontally
6 units to the left.
c
The graph of P^ xh is shifted horizontally
6 units to the right.
y = 2P^ xh - 1
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Thinking More:
5.
c
d
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