Exercises Using the Words in the Glossary

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…

• What is a …
• 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) 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.

2) To solve an ______means to find the value(s) of the ______that makes it an identity.

3) The solution of an ______is one or more intervals of values.

4) The set of real number is a ______open interval.

5) The set of real number is a ______bounded interval.

6) is a quadratic ______.

7) 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.

8) In ordered pairs of real numbers of the form , we call the x-coordinate ______and the y-coordinate ______.

9) The set of acceptable values for the independent variable of a function is called ______.

10) The set consisting of all images of elements of the domain is called the ______of the function.

11) The ______of a function is the set of all ______whose coordinates are corresponded by the function.

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)

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?

Why? (because the equality is true for every values of x )

What are the solutions of this equations?

Can you find real values which make this equality true ?

What can you say in this case? (The equality is impossible)

Can you write on the blackboard an inequality of first degree? ()

What are the solutions? ; How many are there?; Is the solutions like an interval?

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 ?

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 . 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.

For each value of independent variable x, the image is a positive number (or zero if ). We can see graphically the solutions of every inequality involving only and we can follow this table:

inequality / solutions
/ 

Functions are parabolas too. Their graph, for positive a, is the same of the basic function stretched by a factor of a (if ) or compressed by a factor of (if ). All of them are parabolas opening up.

inequality / solutions
/ 

If a is negative, parabolas will be opening down.

inequality / solutions
/ 

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 , then the graph of is the graph of shifted vertically units. The graph of is shifted up if , and is shifted down if .

Functions are parabolas

shifted vertically . In these

cases y-axis is symmetry

axis yet, but the vertex is the

point

We can also see that if h is a constant and , then the graph of is the graph of shifted horizontally units. The graph of is shifted to the right if , and is shifted to the left if .

Functions are parabolas

shifted horizontally. In these cases

the symmetry axis is the line ,

and the vertex is the point

what happens to the symmetry axis? does it shift to the rigtht or to the left?

Every time we have a quadratic function we can always write it in the form :

and so we can easily say that every quadratic function is a parabola,

• opening up if or opening down if ,
• with line as symmetry axis
• and as its vertex,

where and

To find where a parabola intercepts x-axis we need to solve the system ;

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 .

Let’s do some examples:

1) is a parabola opening up, with as symmetry axis and the point as its vertex. The parabola intercepts x-axis at the points ,

and y-axis at the point .

2) is a parabola opening down, with as symmetry axis and the point as its vertex. The parabola intercepts x-axis at the points ,

and y-axis at the point .

Exercise 1.

Match each parabola’s graph with one of these equations:

1. 4.

2. 5.

3.

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).

Exercise 2.

Find the symmetry axis, the vertex and the points at which each parabola intercepts x-axis and y-axis. Then draw their graph:

1. 4.

2. 5.

3. 6.

Compare your results with ones of your classmate

Exercise 3.

For each parabola complete the table writing the intervals in which the function has positive or negative values.

1/16

1/16

Exercise 4.

Try to find the steps solve the inequality using the graph of the parabola .

Can you generalize?

How can you use parabola’s graph to solve quadratic inequalities?

Exercise 5.

Complete the table putting the interval(s) in which you find solutions of equations or inequalities given. Naturally we have always

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.

2. 7.

3. 8.

4. 9.

5. 10.

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 solve the equation / Sketch the parabola / Write the solutions of enequalities using both the symbols of enequalities both the intervals
>0

Bibliography

http://colbycc.edu/www/math/oddsends/symbolism.gif

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

1/16