Archive for the ‘Lesson 12: Functions’ Category

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Lesson 12: Functions. The Last Joke (III)

June 24, 2012

 

First some notes to appreciate the joke:

Log means  tronco but it also represents the abbreviation of the logarithmic function to the base 10 you will study in your fourth year.

Timber  means viga madero para construcción.

To hesitate means vacilar, dudar

To aid and abet somebody  means instigar y secundar a alguien (en la comisión de un delito)

This is the joke:

A math student is pestered by a classmate who wants to copy his homework assignment. The student hesitates, not only because he thinks it’s wrong, but also because he doesn’t want to be sanctioned for aiding and abetting.
His classmate calms him down: “Nobody will be able to trace my homework to you: I’ll be changing the names of all the constants and variables: a to b, x to y, and so on.”
Not quite convinced, but eager to be left alone, the student hands his completed assignment to the classmate for copying.
After the deadline, the student asks: “Did you really change the names of all the variables?”
“Sure!” the classmate replies. “When you called a function f, I called it g; when you called a variable x, I renamed it to y; and when you were writing about the log of x+1, I called it the timber of x+1…”

Enjoy your Summer!!

 

 

 

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Lesson 12: Functions (II)

June 22, 2012

Now we are going to define one of the more important concepts in Maths

2. Definition of function

A function is a relation between two numeric variables that assigns to each input number  EXACTLY ONE output number .

We name  independent variable (we choose it is fixed by us; and we name  dependent variable because it depends from the value of  we choose.

If we represent the pairs of values related by the function  on the Cartesian plane we obtain the graph of the function.

 Vertical line test

If no vertical line can be drawn so that it intersects a graph more than once, then it is a graph of a function.

Think about it, if a vertical line intersects a graph in more than one place, then the x value (input) would associate with more than one y value (output), and you know what that means.  The relation is not a function.

The next two examples illustrate this concept.

We are going to use the vertical line test to determine whether each graph is a graph of a function.

 
For  we can find two points in the graph (3,4)  and (3,6). So to 3 corresponds to two values of , 4 and 6. It is not a function
 
To each value of  x corresponds only one value of y.This graph would pass the vertical line test, because there would not be any place on it that we could draw a vertical line and it would intersect it in more than one place.Therefore, this is a graph of a function.

Function Notation

  • We use for functions lower case letters like f, g, h.
  • We write f(x) = y .
  • f(x)  read “f of x”.

This a very good video about how to use the vertical test:

3. Different ways of displaying a function

A function can be displaying through a table of values, an algebraic expression, or words.

3.1. Representing a function with a table of values

To plot a function expressed by a table of values we show the pairs of corresponding values of the variables independent and dependent as if they were the coordinates of points in the Cartesian plane. Take into account that:

  • The input values, the values of the independent variable, are represented on the x–axis.
  • The output values, the values of the dependent variable, are represented on the y–axis.

Example

1. A shop sells mobile phones with a price of 45€ pee unit. Write the function that relates the number of mobiles bought and the price in €.

 If I buy 1 mobile I will pay 45€, if I buy 2 mobiles I will pay 90€, if I buy 3 mobiles I will pay 135€..

Num of mobiles 1 2 3 4 5 6
Price in € 45 90 135 180 225 265

 We plot the coordinates of these points in the Cartesian plane and we obtain the graph of the function (“the picture”).

The number of mobiles is represented on the x–axis (abscissas).

The price is represented on the y–axis (ordinates).

 

We can’t join the points because we can’t buy 1.5 mobiles or 2.3 mobiles. This type of variables is named discrete variables.

3.2. Representing a function through an equation

The algebraic expression of a function is written y = f(x) and it is named function equation.

  • Recall the function notation f(x)  read “f of x”.
  • f is the function name.
  • Output values are also called functional values. Note that you can use any letter to represent a function name, fis a very common one used.
  • xis your input variable.
  • y is your output variable.

Think of functional notation as a fancy assignment statement.   When you need to evaluate the function for a given value of x, you simply replace x with that given value and simplify.  Just like you do when you are finding the value of an expression when you are given a number for your variable as shown in Lesson 5: Algebraic Expressions. For example, if we are looking for f(0), we would plug in 0 for the value of x in our function f.

Example

2. Find the functional values f(-2), f(-1),  f(0), f(1) and f(2) for the function f(x) = 2x – 1. Draw the graph of the function.

Again, think of functional notation as a fancy assignment statement.  For example, when we are looking for f(0), we are going to plug in 0 for the value of x in our function f and so forth.

Values of the independent variable Values of y
 f(x) = 2x – 1  Plug in -2 for xf(x) = 2 · (-2)- 1 Evaluate: f(-2) = -5
 f(x) = 2x – 1  Plug in -1 for x f(x) = 2 · (-1)- 1 Evaluate: f(-1) = -3
 f(x) = 2x – 1  Plug in 0 for x f(x) = 2 · 0 – 1 Evaluate: f(0) = -1 
 f(x) = 2x – 1  Plug in  1 for x f(x) = 2 · 1 – 1 Evaluate: f(1) = 1
 f(x) = 2x – 1  Plug in  2 for xf(x) = 2 · 2 – 1 Evaluate: f(2) = 3 

 We can build a table of values

 -2  -1  0  1  2
y  -5  -3  -1  1  3

 Representing the points we obtain the graph of the function.

In this case we can join the points because you can always work out triple x minus one unit, even if the number is not an integer.

In the links below you can practice how to evaluate a function

Evaluate a linear function (this is a function whose graph is a line)

Complete a function table  (of a linear function, this is a function whose graph is a line)

Evaluate a nonlinear function

In the following link you can practice how to get an equation (only a linear expression as, for instance,  y = 45 · x or y = 45 · x +1)  for a table.

Write a rule for a function table  

3.3. Representing a function using words

This situation is usually found in word problem where you are asked for the graph and the equation of the function. In case of having to find out the equation of a function given in words the process is the same as when solving word problems with equations. This is you have to translate words into algebra.

Let’s return to example 1.

Example

3.A shop sells mobile phones with a price of 45€ per unit. Write the function that relates the number of mobiles bought and the price in €. Find out the equation of this function.

If I buy 1 mobile I will pay 45€, if I buy 2 mobiles I will pay 90€, if I buy 3 mobiles I will pay 135€.. if I buy x mobiles I will pay 45 · x €. Therefore the equation of the function will be

 y = 45 · x

We use to write it  y = 45  x.

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Lesson 12: Functions (I)

June 22, 2012

 Vocabulary on Functions

Lesson 12: Functions (Notes)

Exercises for Functions

Linear Functions Word Problems

I Have to Know by the End of this Lesson

 

A little of History

The French philosopher and mathematician, René Descartes (1596-1650) developed his philosophy on the idea of needing a starting point on which all other knowledge can be based: I think, therefore I am. In the world of mathematics he is the founder of analytic geometry, which uses a pair of perpendicular lines that cross at a point called the origin as its basis, and is also referred to as the Cartesian coordinates system.

First recall some concepts about Cartesian coordinates on the plane.

1. The Cartesian plane

We will begin with vocabulary.

First, a coordinate. A coordinate is a number  that labels a point on a line.

The coordinate 0 is called the origin of coordinates.

Points to the right of 0 are labelled with positive numbers: 1, 2, 3, etc.  Points to the left of the origin are labelled with negative numbers: −1, −2, −3, etc. Those coordinates are the “addresses” of those points.

A coordinate axis is a line with coordinates.

Now, to label the points in a plane, we will need more than one coordinate axis, and we place a second at right angles to the first.  

Points above the origin have positive coordinates; points below have negative coordinates.

Those lines are called rectangular coordinate axes, because they are at right angles to one another; the coordinates on them are called rectangular coordinates.

Hence we have the name coordinate geometry or, as it is often called, analytic geometry.

The rectangular coordinates of a point are an ordered pair, (x, y).

The coordinates of the origin O are (0, 0). We don’t move right or left and we don’t move up or down. We will see that 0 is an extremely important coordinate.  It means that the point is on one of the axes.

The horizontal axis is called the X-axis or axis of abscissas, the vertical axis is called the Y-axis or axis of ordinates  and the fixed point O is called the origin.

This system of reference can be used to determine the position of any point on a set of axes by an ordered pair of numbers, which are usually written in brackets and separated by a comma.

The numbers in each ordered pair of numbers are called the coordinates of the corresponding point. The first number is called the x-coordinate (or abscissa) and the second the y-coordinate (or ordinate).

We always write (x, y); x is named abscissa and y ordinate

Quadrants

To make it easy to talk about where on the coordinate plane a point is, we divide the coordinate plane into four sections called quadrants, labelled counter-clockwise: The first, the second, the third, and the fourth

   

       In Roman numbers 

   Arabic numbers

                                                

Points in Quadrant 1 have positive x and positive y coordinates.

Points in Quadrant 2 have negative x but positive y coordinates

Points in Quadrant 3 have negative x and negative y coordinates.

Points in Quadrant 4 have positive x but negative y coordinates.

In the following links you can practice several skills:

1. Points on coordinate graphs

2. Quadrants and axes

3. Coordinate graphs as maps

4. Distance between two points