Download Quadratic Function

SYMBOL
N, Z, Q, R

x A
+

:
1 3 5 6
,
,
,
2 4 7 41
=
≠
()
The sets of natural, integer, rational and real numbers
The empty set
“x belongs to A”
Adding Sign. Often referred to as the “plus” sign
Subtracting Sign. Often referred to as the “minus” sign
Multiplication Sign. Often referred to as the “times” sign
Division Sign
A half, three quarter, five seventh, six over forty-one
Equal Sign. “equal to”
“not equal to”
Parenthesis

Brackets
2
“x square” or “x (raised) to the second power”
to rise
x
√
Square root sign : x “the square root of x”
<
“less than”
≤
“less than or equal to”
Inequality Signs
>
“greater than”
≥
“greater than or equal to”
∆
Greek letter Delta
f x 
The function “f of x”
2
The polynomial “a x square plus b x plus c”
ax  bx  c
2
ax  bx  c  0 The equation “a x square plus b x plus c equal to 0”
The (quadratic) inequality “a x square plus b x plus c grater than or
ax 2  bx  c  0
equal to 0”
1/20
GLOSSARY
VARIABLE
EQUATION
INTERVAL
A quantity that is allowed to represent any element of a given set. Any
number in the set is called a value of the variable and the set itself is
called the domain of the variable
An assertion of equality, usually between two mathematical expressions f,
g involving numbers, parameters, and variables. We write f = g. When the
equation involves one or more variables, the equality asserted may be true
for some or all values of the variable(s).
A natural question then arises: for which values of the variable(s) is the
equality true?
The task of answering this question is referred to as solving the equation
Any continuous portion of the real number line. The numbers that belong
to an interval can be indicated by writing the left and the right boundary
points of the interval as an ordered pair.
BOUNDED
If a < b, than the set of all real numbers from a to b forms a
INTERVALS bounded interval with endpoints a and b
open
half-open
(open at the
right)
half-open
(open at the
left)
closed
a xb
a, b
a xb
a, b
a xb
a, b
a xb
a, b
UNBOUNDED Intervals that extend without bound on one or both sides
INTERVALS can be represented by the symbol for “infinity”  . The
symbol  does not represent a real number
 , a
xa
xa
xa
xa
INEQUALITY
QUADRATIC
INEQUALITY
 , a
a,
a,
An assertion of inequality, usually between two mathematical expressions
f, g involving numbers, parameters, and variables. It is in general neither
true nor false; rather, its truth depends on the value(s) of the variable(s).
For inequality in one variable, a value of the variable that makes the
statement true is a solution to the inequality. The set of all solution is
called the solution set of inequality
Any inequality of one of the forms:
ax 2  bx  c  0  ax 2  bx  c  0  ax 2  bx  c  0  ax 2  bx  c  0
2/20
CARTESIAN
A system in which ordered pairs of real numbers of the form x, y 
COORDINATE
represent the locations of points from two perpendicular axes that
intersect at origin. The first member of the ordered pair x, y  is called
SYSTEM
the x-coordinate or abscissa. The second member of the pair x, y  is
MONOMIAL
POLYNOMIAL
FACTORING
POLYNOMIALS
called the y-coordinate or ordinate.
A monomial is a number, a variable, or the indicated product of a number
and one or more variables with positive exponents. The numerical factor
of a monomial is called its COEFFICIENT.
A polynomial is a monomial or the sum of two or more monomials. Each
monomial in the sum is called TERM of the polynomial. A polynomial having
two unlike terms is called BINOMIAL, three unlike terms TRINOMIAL.
A polynomial in one variables is in STANDARD FORM when the powers of the
variables decrease in value as the terms are read from left to right
Factoring a polynomial means writing the polynomial as the product of two
or more lower degree polynomials each of which is called a FACTOR of the
original polynomial. We can factor a polynomial in different ways:
25x 3  15x 2  5x 2 5x  3
removing the greatest common
monomial factor
factoring by grouping

special factoring patterns:
o difference or two squares
o sum or difference of two cubes
o square of a binomial
o cube of a binomial
factoring x 2  sx  p ( a  b  s , a  b  p )
FUNCTION
GRAPH OF A


x 5  x 3  2x 2  2x  x 3 x 2  1 
2 x 2 1  x 2 1 x3  2
 


A2  B 2   A  B  A  B 

A3  B 3   A  B  A2  AB  B 2

 A  B 2  A 2  2 AB  B 2
 A  B 3  A3  3 A 2 B  3 AB 2  B 3
x 2  sx  p  x  a x  b
Functions are powerful mathematical tools for describing how two (or
more) variables are related to each other. A function may take many
forms. Regardless of the form it takes, a function must satisfy the
following two conditions:
o there is a “rule” that tells how each of the possible values of one
variable is matched with a value of the other variables
o for each value of x corresponds one and only one value for y.
The DOMAIN of the function is the set of all real numbers for which the
expression is defined. The set consisting of all images of elements of the
domain is called the RANGE of the function.
The graph of a function f is the set of all points x, y  such that x is in the
FUNCTION
domain of f, and y  f x 
QUADRATIC
A polynomial function of degree two: y  ax 2  bx  c
FUNCTION
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YOUR GLOSSARY AND NOTES
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EXERCISES USING THE WORDS IN THE GLOSSARY
Exercise 1.
In pair, use the glossary to ask your neighbour the definition of one of the words in it.
Then change your shoes: he/she asks and you give the definition. You may ask and answer
five words each other.
You can use questions like…
o What is a …
o Give me the definition of …
Exercise 2.
Work in pair. Do you think you need the definition of some other words about solving
quadratic inequalities? Which ones? List them and try to give the English definition. If you
can’t, ask to your teacher.
Do the same work with symbols.
Exercise 3.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
Monomials are constants or the _____________ of a constant and one or more
variables raised to whole-number powers. The constant is called the ___________
of the monomial.
To solve an _________________ means to find the value(s) of the
______________ that makes it an identity.
The solution of an _______________ is one or more intervals of values.
The set of real number x  a is a ___________________ open interval.
The set of real number a  x  b is a ________________ bounded interval.
ax 2  bx  c  0 is a quadratic ______________.
Factoring a polynomial means writing the polynomial as the _____________ of two
or more lower degree polynomials each of which is called a _____________ of the
original polynomial.
In ordered pairs of real numbers of the form x, y  , we call the x-coordinate
___________and the y-coordinate _________________.
The set of acceptable values for the independent variable of a function is called
_________________.
The set consisting of all images of elements of the domain is called the _________
of the function.
The ____________ of a function is the set of all ________ whose coordinates are
corresponded by the function.
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Exercise 3bis
Complete in pairs the following questions with the mathematical words you have learned during the lesson:
1) A______________________of__________numbers of the form (x,y) represents the
__________________of points in the plan.
X is called__________________, y is called___________________.
2) The axes of a Cartesian _____________________________ are ________________________ each
other and intersect in the ____________.
3) A monomial is the __________________of a number and one or more ________________with
positive____________________.
4) A _____________ is the sum of one or more monomial.
5)The _______________form of a polynomial in one ________________is when the powers of the
variables _______________in value when the terms are read from
____________to_________________.
6)___________________a polynomial means to write it with the product of one or more
_____________degree polynomials.
7)A _________________ is a ____________which describes how one or more variables are related to
each other.
8) The _______________is the set of ____________numbers that are possible to assign to the x value.
9) The values of y ______________________to the x values is called _______________of the function.
10) The ____________of a function is drowned in a ________________________________system,
linking with a line all points (x,y)
11) y=2x-5 is a ________________function of ___________degree.
12) y=4x2-7x+2 is a __________________function of second degree.
13) 4x2-y2 is the difference of two __________________
14) 4x2-32x+64 is the ______________________of the following binomial ____________________
15) 9x3-27 is factored like the _______________of_________________________
16)factoring ____________________is (x-5)(x+3)
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Homework (dopo glossary)
How do you call the set in which variables change? (domain)
Is it necessary to assign the domain of a variable?
How many variables make an equation of first degree true ?
What does it mean to solve an equation?
What is the characteristic symbol to write an equation?
Is this an equation or an identity?
x  12  x 2  2 x  1
Why? (because the equality is true for every values of x )
What are the solutions of this equations? x  5 x  6  0
2
Can you find real values which make this equality true ?
x  12  5
What can you say in this case? (The equality is impossible)
Can you write on the blackboard an inequality of first degree? ( 3x  5  0 )
What are the solutions? x 
5
;
3
How many are there?;
What are the symbols used to write an inequality?
Define a Cartesian coordinate system
What is a monomial?
What is a polynomial?
What is the standard form of the polynomial ?
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Is the solutions like an interval?
QUADRATIC INEQUALITIES
In these lessons we want to explain how to solve a quadratic inequality using the graph of a
parabola (quadratic function). Our work is going to be done in two steps:
1. we are going to learn how to sketch the graph of a parabola
2. we are going to study the sign of a quadratic polynomial using parabolas
Quadratic Function
The basic quadratic function is the function y  x 2 . If we form a table of values we find
x 0 1 -1 2 -2 3 -3 4 -4
y 0 1 1 4 4 9 9 16 16
Now we can plot the points and connect them and we find the graph of a parabola.
-5
-4
-3
-2
16
14
12
10
8
6
4
2
0
-1 0
1
2
3
4
5
For each value of independent variable x, the image y  x 2 is a positive number (or zero if
x  0 ). We can see graphically the solutions of every inequality involving only x 2 and we can
follow this table:
inequality solutions
x2  0
x  R
2
x 0
x0
2

x 0
2
x 0
x0
8/20
Functions y  ax 2 are parabolas too. Their graph, for positive a, is the same of the basic
function y  x 2 stretched by a factor of a (if a  1 ) or compressed by a factor of 1 / a (if
0  a  1 ). All of them are parabolas opening up.
-5
-4
-3
-2
16
14
12
10
8
6
4
2
0
-1 0
inequality solutions
ax 2  0
x  R
2
ax  0
x0
a0
2

ax  0
2
ax  0
x0
a=2
a=1/2
1
2
3
4
5
If a is negative, parabolas will be opening down.
2
-5
-4
-3
-2
0
-1 -2 0
1
2
3
4
inequality
ax 2  0
ax 2  0
a0
ax 2  0
ax 2  0
5
-4
a=-1
-6
a=-2
-8
a=-1/2
-10
-12
solutions
x0

x0
x  R
-14
-16
All of them have a symmetry axis: y-axis is it; and they have the point O as their vertex.
We can observe that if k is a constant and g x  f x  k , then the graph of g x is the
graph of f x  shifted vertically k units. The graph of f x  is shifted up if k  0 , and is
shifted down if k  0 .
Functions y  ax 2  k are parabolas
16
shifted vertically . In these
cases y-axis is symmetry
axis yet, but the vertex is the
point P 0, k 
14
12
10
8
y=x^2
6
4
y=(x^2)-2
y(x^2)+2
2
-5
-4
-3
9/20
-2
0
-1-2 0
-4
1
2
3
4
5
We can also see that if h is a constant and g x  f x  h , then the graph of g x is the
graph of f x  shifted horizontally h units. The graph of f x  is shifted to the right if
h  0 , and is shifted to the left if h  0 .
Functions y  ax  h  are parabolas
2
16
shifted horizontally. In these cases
the symmetry axis is the line x  h ,
and the vertex is the point P h,0
14
12
y=3(x^2)
10
8
y="3(x-2)^2"
6
y="3(x+2)^2"
4
2
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
what happens to the symmetry axis? does it shift to the rigtht or to the left?
Every time we have a quadratic function y  ax 2  bx  c we can always write it in the form
y  ax  h   k :
2
b

y  ax 2  bx  c  a x 2  x 
a

 2 b
c
b2


a
x

x



a
a
4a 2

2
 b2
c
b 
4ac  b 2

  2    a x 


a
2a 
4a

 4a
and so we can easily say that every quadratic function is a parabola,
o opening up if a  0 or opening down if a  0 ,
o with line x  h as symmetry axis
o and V h, k  as its vertex,
where h  
b

b 2  4ac

and k  
2a
4a
4a
 y  ax 2  bx  c
To find where a parabola intercepts x-axis we need to solve the system 
;
y  0
so the solutions of the quadratic equation are the values we are looking for.
In the same way we can find where a parabola intercepts y-axis solving the system
 y  ax 2  bx  c
.

x  0
10/20
Let’s do some examples:
1) y  x 2  5x  6 is a parabola opening up, with x 
5
as symmetry axis and the point
2
5 1
V  ,  as its vertex. The parabola intercepts x-axis at the points A2, 0 , B3, 0
2 4
and y-axis at the point C0, 6 .
8
7
6
5
4
3
2
1
-5
-4
-3
0
-1 -1 0
-2
1
2
3
4
5
-2
2) y   x 2  2 x  8 is a parabola opening down, with x  1 as symmetry axis and the point
V  1, 9 as its vertex. The parabola intercepts x-axis at the points A 4, 0 , B2, 0
and y-axis at the point C 0, 8.
10
8
6
4
2
-5
-4
-3
-2
0
-1 -2 0
-4
-6
-8
-10
11/20
1
2
3
4
5
Exercise 1.
10
0
-5
-4
-3
-2
-1 -2 0
8
1
2
3
4
5
6
4
-4
2
-6
-8
-5
-4
-3
-2
0
-1 -2 0
-10
-4
-12
-6
1
2
3
4
5
-8
-14
-10
-16
8
10
9
8
7
6
5
4
3
2
1
0
-2
-1
-5
7
6
5
4
3
2
1
-5
-3
-3
-2
0
-1 -1 0
1
2
3
4
-2
0
-4
-4
1
-2
15
13
11
9
7
5
3
1
-1
-1 -3 0
-5
2
3
1
2
4
3
5
4
5
Match each parabola’s graph with one of these equations:
1. y  x  3
4. y  x  4x  2
2
2. y   x  2 
2
5. y  x 2  4
3. y  x 2  5 x  6
Compare your answers which those of the student near you.
When you are not agree, explain why you think you are right (or why your friend are not).
12/20
5
Exercise 2.
Find the symmetry axis, the vertex and the points at which each parabola intercepts xaxis and y-axis. Then draw their graph:
2
1. y  2 x  3
4. y  x  4x  2
2. y  4 x  2 
5. y   x 2  9
3. y   x 2  7 x  8
6. y  x 2  9
2
Compare your results with ones of your classmate
13/20
Exercise 3.
For each parabola complete the table writing the intervals in which the function has
positive or negative values.
8
y0
7
6
y0
y0
5
4
3
2
1
-5
-4
-3
-2
0
-1 -1 0
1
2
3
4
5
-2
-5
-4
-3
-2
y0
15
13
11
9
7
5
3
1
-1
-1 -3 0
-5
1
2
3
4
5
-1
0
1
2
3
4
y0
8
6
4
2
-4
-3
-2
0
-1 -2 0
1
2
3
4
5
-4
-6
-8
-10
14/20
y0
y0
y0
y0
y0
C
5
10
-5
y0
B
y0
10
9
8
7
6
5
4
3
2
1
0
-2
A
D
-5
-4
-3
-2
0
-1 -2 0
y0
1
2
3
4
y0
y0
5
-4
-6
-8
-10
-12
-14
E
-16
Exercise 4.
Try to find the steps solve the inequality 4 x 2  12 x  7  0 using the graph of the parabola
y  4 x 2  12 x  7 .
Can you generalize?
How can you use parabola’s graph to solve quadratic inequalities?
15/20
Exercise 5.
Complete the table putting the interval(s) in which you find solutions of equations or
inequalities given. Naturally we have always y  ax 2  bx  c
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
y0
16/20
Exercise 6.
Make a spidergram of words you need to solve quadratic inequalities using parabolas and
make you sure you can define them.
Exercise 7.
Solve the following inequalities using parabola’s method
1. 6 x 2  x  1  0
2.  2 x 2  3x  1  0
1
1
3. x 2  x   0
2
2
2

2
x

3
x
0
4.
2
5.  x  1  0
6. 9 x 2  6 x  1  0
7. x 2  9  0
8.  x 2  9  0
9. 25 x 2  16  40 x  0
10.  x 2  4  0
17/20
Test
Fill the gaps withthe mathematical words you have learned:
1) The axes of a Cartesian _____________________________ are ________________________to
each other and intersect in the ____________.
2) A monomial is the __________________of a number and one or more ________________with
positive____________________.
3) A _____________ is the sum of one or more monomial.
4)___________________a polynomial means to write it with the product of one or more
_____________degree polynomials.
5)A _________________ is a ____________which describes how one or more variables are related to
each other.
6) The _______________is the set of ____________numbers that are possible to assign to the x value.
7) The values of y ______________________to the x values is called _______________of the function.
8) The ____________of a function is sketched in a ________________________________system,
linking with a line all points (x,y)
9) y=2x-5 is a ________________function of ___________degree.
10) y=4x2-7x+2 is a __________________function of second degree.
Solve the following inequalities using parabola’s method step by step
Inequalities
Calculate  and if
  0, solve the
Sketch the parabola
equation
6x2  x  1  0
18/20
Write the solutions of
enequalities using both
the symbols of
enequalities both the
intervals
 2 x 2  3x  1 >0
2 x 2  3x  0
 x2  1  0
9x2  6x  1  0
x2  9  0
 x2  9  0
19/20
Bibliography
http://colbycc.edu/www/math/oddsends/symbolism.gif
http://math.about.com/library/weekly/aa052502a.htm
Lawrence S. Leff – College Algebra – BARRON’S
Steven G. Krantz – Dictionary of algebra, arithmetic, and trigonometry – CRC PRESS
Fred Safier – Theory and problems of precalculus – McGRAW-HILL
Persano, Riboldi, Zanoli – Matematica per il biennio delle superiori - JUVENILIA
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