This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
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In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
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# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission β Simplifying Students Life
Our Belief β βThe great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.β
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How to Split Bills in the Odoo 17 POS ModuleCeline George
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Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Model Attribute Check Company Auto PropertyCeline George
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In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
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It is possible to hide or invisible some fields in odoo. Commonly using βinvisibleβ attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Palestine last event orientationfvgnh .pptxRaedMohamed3
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An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Operation βBlue Starβ is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
How to Create Map Views in the Odoo 17 ERPCeline George
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The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as βdistorted thinkingβ.
1. Polynomial functions
mc-TY-polynomial-2009-1
Many common functions are polynomial functions. In this unit we describe polynomial functions
and look at some of their properties.
In order to master the techniques explained here it is vital that you undertake plenty of practice
exercises so that they become second nature.
After reading this text, and/or viewing the video tutorial on this topic, you should be able to:
β’ recognise when a rule describes a polynomial function, and write down the degree of the
polynomial,
β’ recognize the typical shapes of the graphs of polynomials, of degree up to 4,
β’ understand what is meant by the multiplicity of a root of a polynomial,
β’ sketch the graph of a polynomial, given its expression as a product of linear factors.
Contents
1. Introduction 2
2. What is a polynomial? 2
3. Graphs of polynomial functions 3
4. Turning points of polynomial functions 6
5. Roots of polynomial functions 7
www.mathcentre.ac.uk 1 c mathcentre 2009
2. 1. Introduction
A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, involving
only non-negative integer powers of x. We can give a general defintion of a polynomial, and
define its degree.
2. What is a polynomial?
A polynomial of degree n is a function of the form
f(x) = anxn
+ anβ1xnβ1
+ . . . + a2x2
+ a1x + a0
where the aβs are real numbers (sometimes called the coefficients of the polynomial). Although
this general formula might look quite complicated, particular examples are much simpler. For
example,
f(x) = 4x3
β 3x2
+ 2
is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic
polynomial, or just a cubic. And
f(x) = x7
β 4x5
+ 1
is a polynomial of degree 7, as 7 is the highest power of x. Notice here that we donβt need every
power of x up to 7: we need to know only the highest power of x to find out the degree. An
example of a kind you may be familiar with is
f(x) = 4x2
β 2x β 4
which is a polynomial of degree 2, as 2 is the highest power of x. This is called a quadratic.
Functions containing other operations, such as square roots, are not polynomials. For example,
f(x) = 4x3
+
β
x β 1
is not a polynomial as it contains a square root. And
f(x) = 5x4
β 2x2
+ 3/x
is not a polynomial as it contains a βdivide by xβ.
Key Point
A polynomial is a function of the form
f(x) = anxn
+ anβ1xnβ1
+ . . . + a2x2
+ a1x + a0 .
The degree of a polynomial is the highest power of x in its expression. Constant (non-zero)
polynomials, linear polynomials, quadratics, cubics and quartics are polynomials of degree 0, 1,
2 , 3 and 4 respectively. The function f(x) = 0 is also a polynomial, but we say that its degree
is βundefinedβ.
www.mathcentre.ac.uk 2 c mathcentre 2009
3. 3. Graphs of polynomial functions
We have met some of the basic polynomials already. For example, f(x) = 2 is a constant function
and f(x) = 2x + 1 is a linear function.
f(x)
x
1
2 f(x) = 2
f(x) = 2x + 1
It is important to notice that the graphs of constant functions and linear functions are always
straight lines.
We have already said that a quadratic function is a polynomial of degree 2. Here are some
examples of quadratic functions:
f(x) = x2
, f(x) = 2x2
, f(x) = 5x2
.
What is the impact of changing the coefficient of x2
as we have done in these examples? One
way to find out is to sketch the graphs of the functions.
f(x)
x
f(x) = x2
f(x) = 2x2
f(x) = 5x2
You can see from the graph that, as the coefficient of x2
is increased, the graph is stretched
vertically (that is, in the y direction).
What will happen if the coefficient is negative? This will mean that all of the positive f(x) values
will now become negative. So what will the graphs of the functions look like? The functions are
now
f(x) = βx2
, f(x) = β2x2
, f(x) = β5x2
.
www.mathcentre.ac.uk 3 c mathcentre 2009
4. f(x)
x
f(x) = βx2
f(x) = β2x2
f(x) = β5x2
Notice here that all of these graphs have actually been reflected in the x-axis. This will always
happen for functions of any degree if they are multiplied by β1.
Now let us look at some other quadratic functions to see what happens when we vary the
coefficient of x, rather than the coefficient of x2
. We shall use a table of values in order to plot
the graphs, but we shall fill in only those values near the turning points of the functions.
x β5 β4 β3 β2 β1 0 1 2
x2
+ x 6 2 0 0 2 6
x2
+ 4x 0 β3 β4 β3 0
x2
+ 6x β5 β8 β9 β8 β5
You can see the symmetry in each row of the table, demonstrating that we have concentrated
on the region around the turning point of each function. We can now use these values to plot
the graphs.
f(x)
x
f(x) = x2
+ 4x
f(x) = x2
+ 6x
f(x) = x2
+ x
As you can see, increasing the positive coefficient of x in this polynomial moves the graph down
and to the left.
www.mathcentre.ac.uk 4 c mathcentre 2009
5. What happens if the coefficient of x is negative?
x β2 β1 0 1 2 3 4 5
x2
β x 6 2 0 0 2 6
x2
β 4x 0 β3 β4 β3 0
x2
β 6x β5 β8 β9 β8 β5
Again we can use these tables of values to plot the graphs of the functions.
f(x)
x
f(x) = x2
β 4x
f(x) = x2
β 6x
f(x) = x2
β x
As you can see, increasing the negative coefficient of x (in absolute terms) moves the graph
down and to the right.
So now we know what happens when we vary the x2
coefficient, and what happens when we
vary the x coefficient. But what happens when we vary the constant term at the end of our
polynomial? We already know what the graph of the function f(x) = x2
+ x looks like, so how
does this differ from the graph of the functions f(x) = x2
+ x + 1, or f(x) = x2
+ x + 5, or
f(x) = x2
+ x β 4? As usual, a table of values is a good place to start.
x β2 β1 0 1 2
x2
+ x 2 0 0 2 6
x2
+ x + 1 3 1 1 3 7
x2
+ x + 5 7 5 5 7 11
x2
+ x β 4 β2 β4 β4 β2 2
Our table of values is particularly easy to complete since we can use our answers from the x2
+ x
column to find everything else. We can use these tables of values to plot the graphs of the
functions.
www.mathcentre.ac.uk 5 c mathcentre 2009
6. f(x)
x
f(x) = x2
+ x + 5
f(x) = x2
+ x β 4
f(x) = x2
+ x + 1
f(x) = x2
+ x
As we can see straight away, varying the constant term translates the x2
+ x curve vertically.
Furthermore, the value of the constant is the point at which the graph crosses the f(x) axis.
4. Turning points of polynomial functions
A turning point of a function is a point where the graph of the function changes from sloping
downwards to sloping upwards, or vice versa. So the gradient changes from negative to positive,
or from positive to negative. Generally speaking, curves of degree n can have up to (n β 1)
turning points.
For instance, a quadratic has only one turning
point.
A cubic could have up to two turning points, and
so would look something like this.
However, some cubics have fewer turning points: for
example f(x) = x3
. But no cubic has more than two
turning points.
www.mathcentre.ac.uk 6 c mathcentre 2009
7. In the same way, a quartic could have up to three
turning turning points, and so would look some-
thing like this.
Again, some quartics have fewer turning points, but none has more.
Key Point
A polynomial of degree n can have up to (n β 1) turning points.
5. Roots of polynomial functions
You may recall that when (x β a)(x β b) = 0, we know that a and b are roots of the function
f(x) = (x β a)(x β b). Now we can use the converse of this, and say that if a and b are roots,
then the polynomial function with these roots must be f(x) = (x β a)(x β b), or a multiple of
this.
For example, if a quadratic has roots x = 3 and x = β2, then the function must be f(x) =
(xβ3)(x+2), or a constant multiple of this. This can be extended to polynomials of any degree.
For example, if the roots of a polynomial are x = 1, x = 2, x = 3, x = 4, then the function must
be
f(x) = (x β 1)(x β 2)(x β 3)(x β 4),
or a constant multiple of this.
Let us also think about the function f(x) = (x β 2)2
. We can see straight away that x β 2 = 0,
so that x = 2. For this function we have only one root. This is what we call a repeated root,
and a root can be repeated any number of times. For example, f(x) = (x β 2)3
(x + 4)4
has
a repeated root x = 2, and another repeated root x = β4. We say that the root x = 2 has
multiplicity 3, and that the root x = β4 has multiplicity 4.
The useful thing about knowing the multiplicity of a root is that it helps us with sketching the
graph of the function. If the multiplicity of a root is odd then the graph cuts through the x-axis
at the point (x, 0). But if the multiplicity is even then the graph just touches the x-axis at the
point (x, 0).
For example, take the function
f(x) = (x β 3)2
(x + 1)5
(x β 2)3
(x + 2)4
.
β’ The root x = 3 has multiplicity 2, so the graph touches the x-axis at (3, 0).
www.mathcentre.ac.uk 7 c mathcentre 2009
8. β’ The root x = β1 has multiplicity 5, so the graph crosses the x-axis at (β1, 0).
β’ The root x = 2 has multiplicity 3, so the graph crosses the x-axis at (2, 0).
β’ The root x = β2 has multiplicity 4, so the graph touches the x-axis at (β2, 0).
To take another example, suppose we have the function f(x) = (x β 2)2
(x + 1). We can see
that the largest power of x is 3, and so the function is a cubic. We know the possible general
shapes of a cubic, and as the coefficient of x3
is positive the curve must generally increase to
the right and decrease to the left. We can also see that the roots of the function are x = 2 and
x = β1. The root x = 2 has even multiplicity and so the curve just touches the x-axis here,
whilst x = β1 has odd multiplicity and so here the curve crosses the x-axis. This means we can
sketch the graph as follows.
2
β1
x
f (x)
Key Point
The number a is a root of the polynomial function f(x) if f(a) = 0, and this occurs when (xβa)
is a factor of f(x).
If a is a root of f(x), and if (x β a)m
is a factor of f(x) but (x β a)m+1
is not a factor, then we
say that the root has multiplicity m.
At a root of odd multiplicity the graph of the function crosses the x-axis, whereas at a root of
even multiplicity the graph touches the x-axis.
Exercises
1. What is a polynomial function?
2. Which of the following functions are polynomial functions?
(a) f(x) = 4x2
+ 2 (b) f(x) = 3x3
β 2x +
β
x (c) f(x) = 12 β 4x5
+ 3x2
(d) f(x) = sin x + 1 (e) f(x) = 3x12
β 2/x (f) f(x) = 3x11
β 2x12
www.mathcentre.ac.uk 8 c mathcentre 2009
9. 3. Write down one example of each of the following types of polynomial function:
(a) cubic (b) linear (c) quartic (d) quadratic
4. Sketch the graphs of the following functions on the same axes:
(a) f(x) = x2
(b) f(x) = 4x2
(c) f(x) = βx2
(d) f(x) = β4x2
5. Consider a function of the form f(x) = x2
+ax, where a represents a real number. The graph
of this function is represented by a parabola.
(a) When a > 0, what happens to the parabola as a increases?
(b) When a < 0, what happens to the parabola as a decreases?
6. Write down the maximum number of turning points on the graph of a polynomial function of
degree:
(a) 2 (b) 3 (c) 12 (d) n
7. Write down a polynomial function with roots:
(a) 1, 2, 3, 4 (b) 2, β4 (c) 12, β1, β6
8. Write down the roots and identify their multiplicity for each of the following functions:
(a) f(x) = (x β 2)3
(x + 4)4
(b) f(x) = (x β 1)(x + 2)2
(x β 4)3
9. Sketch the following functions:
(a) f(x) = (x β 2)2
(x + 1) (b) f(x) = (x β 1)2
(x + 3)
Answers
1. A polynomial function is a function that can be written in the form
f(x) = anxn
+ anβ1xnβ1
+ anβ2xnβ2
+ . . . + a2x2
+ a1x + a0 ,
where each a0, a1, etc. represents a real number, and where n is a natural number (including 0).
2.
(a) f(x) = 4x2
+ 2 is a polynomial
(b) f(x) = 3x3
β 2x +
β
x is not a polynomial, because of
β
x
(c) f(x) = 12 β 4x5
+ 3x2
is a polynomial
(d) f(x) = sin x + 1 is not a polynomial, because of sin x
(e) f(x) = 3x12
β 2/x is not a polynomial, because of 2/x
(f) f(x) = 3x11
β 2x12
is a polynomial
3.
(a) The highest power of x must be 3, so examples might be f(x) = x3
β 2x + 1 or f(x) = x3
β 2.
(b) The highest power of x must be 1, so examples might be f(x) = x or f(x) = 6x β 5.
(c) The highest power of x must be 4, so examples might be f(x) = x4
β 3x3
+ 2x or f(x) =
x4
β x β 5.
(d) The highest power of x must be 2, so examples might be f(x) = x2
or f(x) = x2
β 5.
4.
www.mathcentre.ac.uk 9 c mathcentre 2009
10. f(x)
x
f(x) = 4x2
f(x) = β4x2
f(x) = x2
f(x) = βx2
5.
(a) When a > 0, the parabola moves down and to the left as a increases.
(b) When a < 0, the parabola moves down and to the right as a decreases.
6.
(a) 1 turning point (b) 2 turning points (c) 11 turning points
(d) (n β 1) turning points.
7.
(a) f(x) = (x β 1)(x β 2)(x β 3)(x β 4) or a multiple
(b) f(x) = (x β 2)(x + 4) or a multiple
(c) f(x) = (x β 12)(x + 1)(x + 6) or a multiple.
8.
(a) x = 2 odd multiplicity
x = β4 even multiplicity
(b) x = 1 odd multiplicity
x = β2 even multiplicity
x = 4 odd multiplicity
www.mathcentre.ac.uk 10 c mathcentre 2009