relationships between variables
Relationships between variables Before we attempt to discuss the relationships between variables, it is important for you to know how to construct a function, or an Algebraic Formula. Study the following example, and then do the exercises afterwards. Example One bar of chocolates costs 50c Then two bars will cost R1 (2 X 50c) Then x bars will cost x X 50c = 50x Complete the following If Dwayne baths 2 Then in 3 days he will bath And in d days he will bath
times per day _______ times (2 X 3) _______ times ( 2 X ______ )
In one week you receive 3 hours’ homework In 3 weeks you receive _____ hours’ homework (3 X 3) In q weeks you receive _____ hours’ homework If John swims 2 He will swim And he will swim
times in one day ____ times in 3 days ____ times in y days
(2 X 3)
Sibongile is 10 years old One year from now she will be _____ years old (10 + 1) Four years from now she will be ____ years old (10 + 4) k years from now she will be ___ years old (10 + ….)
Functions
The sum of Pete and Mpho’s ages is 20 years If Pete is 5 years old, Mpho will be ___ years old (20 – 5) If Pete is 9 years old, Mpho will be ___ years old (20 – 9) If Pete is x years old, Mpho will be ___ years old If 10 sweets cost 50c then one sweet costs ____ (50 ÷ 10) 5 sweets cost 50c then one sweet costs ______ (50 ÷ 5) y sweets cost 50 c then one sweet will cost ________ An algebraic term is something like : 2x² or 4y or 3xyz² or 4a3 Observe carefully that terms may contain algebraic symbols, but are ALWAYS connected by multiplication or division. Something such as 3x + 4y² consist of two terms, as there is a “+” between them, and then this combination is called an algebraic expression. REMEMBER: Plus and Minus divide terms, and multiplication and division connect to form one term
Study the following definitions well: 1) Another name for an algebraic expression is a polynomial 2)
A monomial contains only one term, a binomial contains 2 terms, and a trinomial contains 3 terms. Functions
3)
The constant (number) in front of a variable is called the coefficient In 5x , 5 is the coefficient of x In 13xy, 13 is the coefficient of xy, but 13x is the coefficient of y In 2a + 3b , 2 is the coefficient of a
It could be required from you write an algebraic expression that is a formula and adheres to certain conditions: Write down an algebraic formula for a certain number increased by 6. You must put this “certain number” equal to x, and notice that this number is increased by 6, thus 6 is added. The algebraic formula is then : x + 6. Complete the following exercise 1) Write algebraic formulas for the following a) A certain number decreased by 6 b) A certain number increased 5 times c) A certain number divided by 10 d) The sum of twice a certain number and 10 e) 8 subtracted from 3 times a certain number f) The square of a certain number 2)
A man is a years old, and his wife is b years old. Write algebraic expressions for the following: a) The man’s age 6 years from now b) The man’s age if it is doubled Functions
c) The sum of their ages d) The woman’s age 10 years ago e) The sum of their ages 15 years from now 3)
Write algebraic formulas for a) Any even number b) Any odd number
4)
I have sold a bicycle that cost a for b. What is my profit?
5)
What is the length of a rectangle if it is 100mm more than the width, which is b?
6)
Which Natural number follows on a + 2?
If we take the following example: To obtain a mark for the final exam, the marks for the paper must be divided by 2, 25 added, and then this answer must be 20 multiplied by 13
x 20 The formula for these specifications is: 25 2 13 If the teacher now has 38 students in the class, she can use a computer program to do these calculations for her. A computer will, however, not know what to do with this formula.
Functions
To solve this problem, we define another variable, usually “y”, and x 20 write the formula as follows: y 25 2 13 x is each learner in the class’s exam mark, and each of them will have a different final mark, which is y. It could then be plotted as a graph, by taking the learners one-by-one by finding the x-value on the axis, and draw a thin line vertically upwards. Take the corresponding y-value and draw a horizontal line, until it meets the vertical line. Make a dot where these two meet, and call it by its number, as in the sketch. Carry on in this fashion, until you have done all the learners’ marks. Then connect all the dots, and colour in the area under the graph, i.e.
This shaded area will then give you an indication of how the class did compared to each other. By just evaluating the graph, try to find the class-average. Functions
Next assume that you are doing an experiment to determine the weight somebody can lift according to the number of glasses Wondershake he has consumed (a new type of milkshake which apparently gives one enormous strength). You will then obviously measure the weight lifted for one glass, then for two, etc., and the results can also be given in a table format: Glasses 0 1 2 3 Weight 50kg 100kg 150kg 200kg You will observe that the values in the table for the weight are dependent on the number of glasses Wondershake the person has consumed You will also observe that there appears to be a fixed relationship between the weight lifted and the number of glasses consumed. It seems as if every glass that is added, enables the person to lift an extra 50kg. This way we can calculate what weight could be lifted after 10 glasses, or 6 glasses Wondershake. The relationship is then given by: Weight = 50 X Glasses We do observe that the first weight that can be lifted without the person having drunk a glass of Wondershake first, is 50 kg. This means that our formula must be given as Weight = 50 X Glasses + 50 (if the number of glasses = 0, the weight lifted will be 50, as per the table) Functions
As the weight is dependent on the value we feed in for the glasses, we say that the glasses are the independent variable , and weight is the dependent variable. As we have done with the example of the learners’ marks, we will follow convention and take note that, in Mathematics, we often use symbols instead of words, and the above example will become: y = 50x + 50 where x normally indicates the independent variable, and y the dependent variable. We have a special way of writing these types of equations, namely f : x 50x + 50 and this is known as mapping This equation with which we have just worked is known as a function. Another way of thinking about a function is to represent it as a machine. Think about it as follows: A couple of goods are entering the machine. Inside it is shuffled, processed and reworked. When it exits on the other side it looks completely different from the original. Functions
It is obvious that, if nothing enters the machine, nothing can come out at the other end. The output then is dependent on what goes in. The same thing will happen when we work with functions For example: y = 3x + 4 If we feed a value in for “x” we see that it will first be multiplied by 3, before 4 is added, and the result of this computation is given as the answer. As we had with the first example, let us take a look at another practical example: A ball is dropped from a certain height, and the height to which it jumps back is measured, and tabulated: Height fallen (h) Height jumped back (H)
800
900
1100
400
450
550
1300
1500 700
h 2 Where h is the independent variable, H the dependent variable, and h H is the function. 2 We can now build a “machine” that will look like this:
Functions
H
Determine a “machine” for each of the following: 1) Water is heated and the temperature is measured at certain time intervals: Time (t) 0 10 20 35 50 Temp (°C) 15 25 35 55 (T) T = __________________ 2)
A trolley is pushed and then left to move forward on a smooth horizontal surface on its own. The distance that the trolley moves since it has been let go is measured Time (t) 0 1 2 4 Distance 0 1,1 2,2 7,7 (m) ______________________
3)
A ball is rolling downward along an incline. The distance covered by the ball is measured at certain time intervals Time (s) 1 2 3 5 Distance 1 4 9 49 (mm) _____________________
Functions
By now you should have a fair idea of a function as a number machine. Later we will learn more about functions, as well as the conditions for being a function, as well as some more terminology and definitions. It is important that you get well acquainted with all the theory, without learning it parrot-style.
You may have learned, when you started with Algebra, that a number machine could be portrayed in the following way: for the equation y = 2x.
This, however, is not always practical, especially if you have more than one value to feed into the x, such as 1, 2, 3, 4, 5, as you will then have to redraw the machine every time, or you will have to write all the input values underneath each other, and the output values in the right sequence underneath each other on the right.
Functions
In this specific instance it is not too bad, but what if the machine looked like this
Now you will get the same answer whether you feed in – 2 or 2 or – 4 and 4, and if you do not want to repeat the answer every time, how will you indicate which output belongs to which input? In Mathematics we then prefer to use Venn-diagrams. A Venndiagram looks almost like two potatoes, with lines connecting the inputs to the respective outputs. Thus, for y = 2x, and the inputs – 1 , – 2 , – 3 , 1 , 2 the Venn diagram will look as follows:
For y = x², en and the inputs – 2 , – 1 , 1 , 2 the Venn diagram will be: We can also do a presentation of these collections by using a set of ordered number pairs, where we write each input with its output in a bracket { (– 2 ; 4) ; (– 1 ; 1) ; ( 1 ; 1) ; (2 ; 4) } Functions
times per day _______ times (2 X 3) _______ times ( 2 X ______ )
In one week you receive 3 hours’ homework In 3 weeks you receive _____ hours’ homework (3 X 3) In q weeks you receive _____ hours’ homework If John swims 2 He will swim And he will swim
times in one day ____ times in 3 days ____ times in y days
(2 X 3)
Sibongile is 10 years old One year from now she will be _____ years old (10 + 1) Four years from now she will be ____ years old (10 + 4) k years from now she will be ___ years old (10 + ….)
Functions
The sum of Pete and Mpho’s ages is 20 years If Pete is 5 years old, Mpho will be ___ years old (20 – 5) If Pete is 9 years old, Mpho will be ___ years old (20 – 9) If Pete is x years old, Mpho will be ___ years old If 10 sweets cost 50c then one sweet costs ____ (50 ÷ 10) 5 sweets cost 50c then one sweet costs ______ (50 ÷ 5) y sweets cost 50 c then one sweet will cost ________ An algebraic term is something like : 2x² or 4y or 3xyz² or 4a3 Observe carefully that terms may contain algebraic symbols, but are ALWAYS connected by multiplication or division. Something such as 3x + 4y² consist of two terms, as there is a “+” between them, and then this combination is called an algebraic expression. REMEMBER: Plus and Minus divide terms, and multiplication and division connect to form one term
Study the following definitions well: 1) Another name for an algebraic expression is a polynomial 2)
A monomial contains only one term, a binomial contains 2 terms, and a trinomial contains 3 terms. Functions
3)
The constant (number) in front of a variable is called the coefficient In 5x , 5 is the coefficient of x In 13xy, 13 is the coefficient of xy, but 13x is the coefficient of y In 2a + 3b , 2 is the coefficient of a
It could be required from you write an algebraic expression that is a formula and adheres to certain conditions: Write down an algebraic formula for a certain number increased by 6. You must put this “certain number” equal to x, and notice that this number is increased by 6, thus 6 is added. The algebraic formula is then : x + 6. Complete the following exercise 1) Write algebraic formulas for the following a) A certain number decreased by 6 b) A certain number increased 5 times c) A certain number divided by 10 d) The sum of twice a certain number and 10 e) 8 subtracted from 3 times a certain number f) The square of a certain number 2)
A man is a years old, and his wife is b years old. Write algebraic expressions for the following: a) The man’s age 6 years from now b) The man’s age if it is doubled Functions
c) The sum of their ages d) The woman’s age 10 years ago e) The sum of their ages 15 years from now 3)
Write algebraic formulas for a) Any even number b) Any odd number
4)
I have sold a bicycle that cost a for b. What is my profit?
5)
What is the length of a rectangle if it is 100mm more than the width, which is b?
6)
Which Natural number follows on a + 2?
If we take the following example: To obtain a mark for the final exam, the marks for the paper must be divided by 2, 25 added, and then this answer must be 20 multiplied by 13
x 20 The formula for these specifications is: 25 2 13 If the teacher now has 38 students in the class, she can use a computer program to do these calculations for her. A computer will, however, not know what to do with this formula.
Functions
To solve this problem, we define another variable, usually “y”, and x 20 write the formula as follows: y 25 2 13 x is each learner in the class’s exam mark, and each of them will have a different final mark, which is y. It could then be plotted as a graph, by taking the learners one-by-one by finding the x-value on the axis, and draw a thin line vertically upwards. Take the corresponding y-value and draw a horizontal line, until it meets the vertical line. Make a dot where these two meet, and call it by its number, as in the sketch. Carry on in this fashion, until you have done all the learners’ marks. Then connect all the dots, and colour in the area under the graph, i.e.
This shaded area will then give you an indication of how the class did compared to each other. By just evaluating the graph, try to find the class-average. Functions
Next assume that you are doing an experiment to determine the weight somebody can lift according to the number of glasses Wondershake he has consumed (a new type of milkshake which apparently gives one enormous strength). You will then obviously measure the weight lifted for one glass, then for two, etc., and the results can also be given in a table format: Glasses 0 1 2 3 Weight 50kg 100kg 150kg 200kg You will observe that the values in the table for the weight are dependent on the number of glasses Wondershake the person has consumed You will also observe that there appears to be a fixed relationship between the weight lifted and the number of glasses consumed. It seems as if every glass that is added, enables the person to lift an extra 50kg. This way we can calculate what weight could be lifted after 10 glasses, or 6 glasses Wondershake. The relationship is then given by: Weight = 50 X Glasses We do observe that the first weight that can be lifted without the person having drunk a glass of Wondershake first, is 50 kg. This means that our formula must be given as Weight = 50 X Glasses + 50 (if the number of glasses = 0, the weight lifted will be 50, as per the table) Functions
As the weight is dependent on the value we feed in for the glasses, we say that the glasses are the independent variable , and weight is the dependent variable. As we have done with the example of the learners’ marks, we will follow convention and take note that, in Mathematics, we often use symbols instead of words, and the above example will become: y = 50x + 50 where x normally indicates the independent variable, and y the dependent variable. We have a special way of writing these types of equations, namely f : x 50x + 50 and this is known as mapping This equation with which we have just worked is known as a function. Another way of thinking about a function is to represent it as a machine. Think about it as follows: A couple of goods are entering the machine. Inside it is shuffled, processed and reworked. When it exits on the other side it looks completely different from the original. Functions
It is obvious that, if nothing enters the machine, nothing can come out at the other end. The output then is dependent on what goes in. The same thing will happen when we work with functions For example: y = 3x + 4 If we feed a value in for “x” we see that it will first be multiplied by 3, before 4 is added, and the result of this computation is given as the answer. As we had with the first example, let us take a look at another practical example: A ball is dropped from a certain height, and the height to which it jumps back is measured, and tabulated: Height fallen (h) Height jumped back (H)
800
900
1100
400
450
550
1300
1500 700
h 2 Where h is the independent variable, H the dependent variable, and h H is the function. 2 We can now build a “machine” that will look like this:
Functions
H
Determine a “machine” for each of the following: 1) Water is heated and the temperature is measured at certain time intervals: Time (t) 0 10 20 35 50 Temp (°C) 15 25 35 55 (T) T = __________________ 2)
A trolley is pushed and then left to move forward on a smooth horizontal surface on its own. The distance that the trolley moves since it has been let go is measured Time (t) 0 1 2 4 Distance 0 1,1 2,2 7,7 (m) ______________________
3)
A ball is rolling downward along an incline. The distance covered by the ball is measured at certain time intervals Time (s) 1 2 3 5 Distance 1 4 9 49 (mm) _____________________
Functions
By now you should have a fair idea of a function as a number machine. Later we will learn more about functions, as well as the conditions for being a function, as well as some more terminology and definitions. It is important that you get well acquainted with all the theory, without learning it parrot-style.
You may have learned, when you started with Algebra, that a number machine could be portrayed in the following way: for the equation y = 2x.
This, however, is not always practical, especially if you have more than one value to feed into the x, such as 1, 2, 3, 4, 5, as you will then have to redraw the machine every time, or you will have to write all the input values underneath each other, and the output values in the right sequence underneath each other on the right.
Functions
In this specific instance it is not too bad, but what if the machine looked like this
Now you will get the same answer whether you feed in – 2 or 2 or – 4 and 4, and if you do not want to repeat the answer every time, how will you indicate which output belongs to which input? In Mathematics we then prefer to use Venn-diagrams. A Venndiagram looks almost like two potatoes, with lines connecting the inputs to the respective outputs. Thus, for y = 2x, and the inputs – 1 , – 2 , – 3 , 1 , 2 the Venn diagram will look as follows:
For y = x², en and the inputs – 2 , – 1 , 1 , 2 the Venn diagram will be: We can also do a presentation of these collections by using a set of ordered number pairs, where we write each input with its output in a bracket { (– 2 ; 4) ; (– 1 ; 1) ; ( 1 ; 1) ; (2 ; 4) } Functions
