Eso3 06 Lesson12 Elementary Functions

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LessonLesson 1212

Elementary FunctionsElementary Functions

3rd ESO Carlos Alberto Cortijo Bon

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Table of Contents

1.Constant Functions..................................................................................................................................1

2.Linear functions.......................................................................................................................................1

3.Functions for direct and inverse proportionality.....................................................................................2

3.1.Direct proportionality (linear functions)................................................................................................2

3.2.Inverse proportionality (hyperbolic functions).......................................................................................3

4.Hyperbolic functions................................................................................................................................4

5.Quadratic functions.................................................................................................................................5

5.1.Quadratic functions of the form ax2......................................................................................................5

5.2.Quadratic functions of the form ax2 q................................................................................................6

5.3.Quadratic functions of the form a(xp)2..............................................................................................7

5.4.Quadratic functions of the form a(xp)2q.........................................................................................7

5.5.Quadratic functions of the form ax2+bx+c...........................................................................................9

5.5.1.General equation for the vertex of a quadratic function..............................................................10

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Lesson 12: Elementary Functions

1. Constant Functions

A constant function is of the form:

y=c ,wherec

The graphical representation of a constant function is an horizontal

straight line.

For example, the functions y=3 and y=1 are constant functions.

2. Linear functions

A linear function is of the form:

y=m xn ,wherem , n

The graphical representation of a linear function is a straight line.

Type1: If n=0 , then the function has the form

y=m x ,wherem .

These functions pass through the point 0,0 .

m is the slope of the line. It is the increase of y when

x increases by 1 .

If the slope is positive, then the line goes upwardsas x increases.

If the slope is negative, then the line goes

downwards as x increases.

Carlos Alberto Cortijo Bon 1


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In this graph, four functions of this type are shown. Three of them have

a positive slope, and one of them has a negative slope.

Type2: Functions of the form y=m xn ,wherem , n

The graphical representation of a function of the form y=m xn ,wherem , n , is a straightline with y-intersect equal to n and slope equal to m .

These lines have slope equal to 2 and different x-intersects These lines have x-intersect equal to 1 and different slopes

3. Functions for direct and inverse proportionality

The relation between two directly or inversely proportional magnitudes can be described with a function.One of the magnitudes can be represented by the independent variable x , and the other by the dependent

variable y .

3.1. Direct proportionality (linear functions)

If two magnitudes have a relation of direct proportionality, then a multiplication of the first magnitude by a

given factor implies a multiplication of the second magnitude by the same factor. Thus, the quotient of the

values for both magnitudes is equal to a constant, that is called the direct proportionality constant.



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For instance, let x be the weight of a fruit in kg and y its cost in . x and y are two directly

proportional magnitudes. Double the weight of fruit, for instance, costs double the price. If a value of 4 for x

corresponds to a value of 8 for y , theny

xis constant and equal to

8

4=2 (the direct proportionality

constant of y with respect to x ). In this case, this is the price in per kg).y

x=2 y=2 x .

y

4 8

8 16

1 2

3 6

5 10

x

y

x=2 y=2 x

In general, the function that describes the relation between two directly proportional magnitudes is a

linear function of the form y=c x where c is the direct proportionality constant.

3.2. Inverse proportionality (hyperbolic functions)

If two magnitudes have a relation of inverse proportionality, then a multiplication of the first magnitude by

a given factor implies a division of the second magnitude by the same factor. Thus, the product of the values

for both magnitudes is equal to a constant, that is called the inverse proportionality constant.

For example: a list of people is made to collect 36. We want to know how much money must each

person contribute ( y ), depending on the number of people ( x ). These are inversely proportional

magnitudes. Double the people, for instance, means that each person has to contribute half as much money.

And the number of people multiplied by the amount that each person has to contribute is equal to 36.

In other words, x y=36 y=36x

.



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y

1 36

2 18

3 12

4 9

6 6

x

x y=36 y=36

x

Notice that if we swap the coordinates of a point of this

function, then we obtain another point of the function.

For instance: if (3,12) is a point of the function, then (12,3)

is another point of the function.

The function that describes the relation between two directly proportional magnitudes is a function of the

form y=c

xwhere c is the inverse proportionality constant. This is an hyperbolic function.

4. Hyperbolic functions

An hyperbolic function y=c

xhas the following characteristics that can can help when drawing it:

It passes through the points 1,c and c ,1 . If it passes through the point a , b , then it also passes through b , a , a ,b and

b ,a . This means that the function has odd symmetry.

It passes through the point c ,c .

It has a vertical asymptote which is the vertical line y=0 , and an horizontal asymptote x=0

y=16

x

x y

1 16

2 8

4 4

8 2

16 1

Passes through 1,16 and 16,1

Passes through 16,16=4,4Passes through a , b passes through b , a ,

a ,b and b ,a . Odd symmetry.

Vertical asymptote y=0 and horizontal asymptote x=0.

The following graph shows hyperbolic functions for different values of c



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From inside to outside: y=9

x, y=

16

x, y=

32

x, y=

64

x

The graph separates more from the axes for larger

values of the constant.

From inside to outside: y=9

x, y=

16

x, y=

32

x, y=

64

x

The constant can be negative number.

5. Quadratic functions

A quadratic function is a polynomial function of x where the second power of x , e.g. x2 , appears,

but no higher power of x , such as x3

, x4

, x5

, etc., appear.

A quadratic function has the general form y=ax2b xc ,where a , b , c and a0

5.1. Quadratic functions of the formax2

The simplest quadratic function is y=x2 .



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The graph of a quadratic function is always a

parabola.

In this function we can observe the following

characteristics, common to all quadratic functions:

It is symmetric with respect to a vertical line.

This vertical line is the axis. In this case, the axis

is the vertical line given by x=0 .

An extreme point (the vertex) exists in the axis.

In this case, the vertex is a minimum of the

function.

It has infinite branches. In this case, they are

upward branches. This means that the function

grows forever to the left and to the right of the

vertex.

The following graph represents several quadratic functions of the form y=ax2 for several values of a .

The following conclusions can be drawn from

comparing these graphics.

The symmetry axis is the y-axis, or the axisgiven by x=0.

When a>0, the function opens upward. When

a0, the vertex is a minimum. When a


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This graph represents the functions y=1

2x

2,

y=

1

2 x

2

2

and y=

1

2 x

2

3

.

Otherwise stated, it represents the functions

y=1

2x

2q for q=0 , q=2 and q=3 .

The second is the result of shifting the fist a distance

of two upwards. The third is the result of shifting the

first a distance of three downwards.

5.3. Quadratic functions of the forma(xp)2

By replacing x by xp in a function, the function is shifted to the left (if the constant is positive) or tothe right (if the constant is negative).

This graph represents the functions y=1

2x2

,

y=1

2x2

2and y=

1

2x3

2.

Otherwise stated, it represents the functions

y=1

2xp

2for p=0 , p=2 and p=3 .

The second one is the result of shifting the fist one a

distance of two to the right. The third one is the result

of shifting the first one a distance of three to the left.

5.4. Quadratic functions of the forma(xp)2q

By adding or subtracting a constant q , where q0 , to a function, we shift the function a distanceof q upwards or downwards.

By replacing x by xp or by xp , where p0 , we shift the function a distance of p to the

right or to the left.



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This table summarizes all the possible combinations:

If

p , q0 ,

the

function

a) y=axp2q is the

result ofshifting

the

function

y=ax2

a distance of p to the

right

and a distance of q

upwards

b) y=axp2q downwards

c) y=axp2q

leftupwards

d) y=axp2q downwards

And these graphs show the relation between the original function and the new one in each case.

a) b)

c) d)



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5.5. Quadratic functions of the formax2+bx+c

The usual form of a quadratic function is y=ax2bxc .Let's see with examples how to draw these functions.

Example1: y=x28x7 .

We know that this is a parabola. We expect it to have a symmetry axis in which the axis is located.

1. Find the axis.

First we find the image of x=0 : y=7 . The function passes through 0,7 .

Then we find another value of x with image 7: x28x7=7 x

28x=0 x x8=0

{

x=0

or

x=8

. The function passes through 0,7 and 8,7 . Thus, the axis must be in the middle

point between x=0 and x=8 . The axis is in x=08

2=4 .

2. Calculate the vertex. The vertex is in the axis x=4 . The image of x=4 is

y=42847=16327=9 . Thus, the vertex is 4,9 .

3. We start the graph with the three points that we have.

4. From the drawing we see that this function intersects with

the x-axis. We can obtain the x-intersects from the

equation: x28x7=0 x=

882417

2

x=86428

2 x=

836

2=

86

2 x={71 .

Thus, we have two more points of the function: 1,0 and 7,0 . We also draw them.

5. For a more precise graph, we can compute more points

around the axis x=4 , for example x=2 , x=3 ,

x=5 and x=7 . With all these points we can approximately draw the function.



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Example 2: y=x22x3 .

1. Find the axis.

First we find the image of x=0 : y=3 . The function passes through 0,3 .

2. Then we find another value of x with image -3: x22x3=3 x

22x=0

x x2=0 {x=0or

x2=0x=2

. The function passes through 0,3 and 2,3 . Thus,

the axis must be in the middle point between x=0 and x=2 . The axis is in x=02

2=1 .

3. Calculate the vertex. The vertex is in the axis x=1 . The image of x=1 is

y=12213=123=2 . Thus, the vertex is

1,2 .

4. We already have three points and the axis. We start the graph

with them.

5. This function opens downward, because the coefficient of x2 is

-1. And the vertex is below the x-axis. Thus, it has no x-

intersects.

6. For a more precise graph, we compute several other points

around the axis, for instance x=1 , x=2 , x=3 andx=4 . With all these points we can approximately draw thefunction.

5.5.1. General equation for the vertex of a quadratic function

Let y=ax2bxc be a quadratic function for which we want to obtain the vertex.

We can generalize the procedure that we have used in the previous example to obtain the vertex.

x=0 y=c . One point of the function is 0, c .

Now we obtain another point for which the image is c .



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y=ax2bxc=c 0=a x

2bx 0=x a xb 0=x a xb

{

x=0

or

a xb=0x=

b

a

.

Both x=0 and x=b

ahave the same image. Thus, the vertex must be halfway between them. This

is0ba

2=

b

a

2=

b

2a

Thus, the vertex of the quadratic function y=ax2bxc is in x=

b

2 a



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Fill in the blanks to complete the glossaryLnea recta

Funcin lineal

Slope (of a straight line)

Interseccin con el eje x/Interseccin con el eje y

Direct/inverse proportionality

Constante de proporcionalidad directa/inversa

Hyperbolic function

Asntota vertical/horizontal

Eje

Ejes

Quadratic function

Parbola

Vertex (of a parabola)

Una parbola tiene ramas hacia arriba/abajo


Eso3 06 Lesson12 Elementary Functions

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