Module 6: Functions · 2020. 8. 12. · GRADE 10 FUNCTIONS 1 Module 6: Functions TEXTBOOK PAGE 121...

GRADE 10 FUNCTIONS 1

Module 6: Functions

TEXTBOOK PAGE 121

Introduction to functions

A function is a rule that takes an input ( value) and returns an output . A function assigns exactly one

output to each input. Functions can be given names like or etc. For example or .

When a name is not needed the form or is used.

Mathematically, a function is a set of ordered pairs ( ) in which no two ordered pairs has the

same -coordinate.

The process can be represented as a machine diagram where the calculator can serve the purpose of

the machine.

Look on page 121 and 122 for a different type of representation of a function.

Given a graph page 122

We use a ruler to perform the “vertical line test” on a graph to see whether it is a function or not.

Instructions:

1. Hold a clear plastic ruler parallel to the -axis (vertical).

2. Move it from left to right over the Cartesian plane.

3. If the ruler cuts the graph in only one place, then the graph is a function.

Now use these instructions and work through Example 2 on page 122.

Mapping and functional notation page 123

There are two ways that a function may be represented by means of mapping or functional notation. We will

only be dealing with functional notation. Let us look at the following example:


This reads as “ of is equal to 3 ”

is used to denote the range, in other words the -values corresponding to the -values are

given by , i.e. .

For example, if , then the corresponding -value is obtained by substituting into .

or

Domain and range page 120

Domain – set of numbers to which we apply the rule (input or -values)

Range – set of numbers obtained as a result of using the rule (output or -values)

Let us use the example and let be .

From this we now know that is the function and we can use these -values (domain) to

substitute into the place of ‘ ’ to determine the -values (range)

Domain { }

Range, first determine these outputs by using your function and substituting in the -values.

{ }

Given a graph page 120

We use a clear plastic ruler to determine the domain and range of a graph.

Instructions:

For domain – keep the edge of the ruler vertical and slide it across the graph from left to right.

Where the edge starts cutting the graph, the domain starts (read from the -axis). Where it stops

cutting the graph the domain ends.

For range – keep the edge of the ruler horizontal and slide it across the graph from bottom to top.

Where the edge starts cutting the graph, the range starts (read from the -axis). Where it stops cutting

the graph the range ends.

Now use these instructions and work through Example 1 on page 120.

After going through today’s notes, you can now do:

Exercise 1 page 125.

Exam aid book.


TEXTBOOK PAGE 127

Linear function introduction

Linear functions are in other words straight lines and have the general standard form:

where gradient/slope and -intercept

The gradient ( ):

To determine the gradient of a linear function you use the gradient formula that you were taught in grade 9

If the gradient is positive, then the graph is increasing

if the gradient is negative, then the graph is decreasing

The -intercept ( ):

The -intercept is the point where the linear function cuts the -axis, this is then a coordinate in the

following form . NOTE that the -coordinate will always be zero and the -cordinate will be the ‘ ’

value.

If then the line goes through the origin, -axis.

If then the line cuts the -axis above the origin.

If then the line cuts the -axis below the origin.

𝒄 𝟎

𝒄 𝟎

𝑦 𝑎𝑥𝑖𝑠

𝑥 𝑎𝑥𝑖𝑠 𝑐


There are three ways to draw a straight line:

1. Table method

Substitute the following -values { } into the function and determine the output

values.

Example: Sketch graph

Substitute your { } to get the -values

These are now ordered pairs and plot the points (

) etc.

2. Dual intercept method

Find the -intercept by setting the equal to zero and -intercept by setting equal to zero.


-intercept (set )

-intercept (set )

3. Gradient-intercept method

Set equal to zero to find the -intecpt, then use the gradient to detrime another point that the graph

pass through.


-intercept (set )

Gradient but first get the equation in the standard form of a straight line


Make the subject

Now divide the coefficient of with every term to get standard form

Now you can see that the gradient is

and the -intercept is

When doing today’s homework, you don’t have to use all three methods to draw a straight line. Only choose

one method and stick to that method but the best method to use is definitely the DUAL INTERCEPT

METHOD. Your homework is the following:

Exercise 4 page 136 nr 1(only left column) and nr 2

Exam aid book.

𝑚 𝑦

𝑥

Translate 2 units down

Translate 3 units right


TEXTBOOK PAGE 137

Linear function – horizontal and vertical lines

Horizontal lines have the equation where you now know -intercept. The gradient of a

horizontal line is .

Vertical lines have the equation where -intercept. The gradient of a vertical line is

After going through these notes, you can now do:

Exercise 5 page 138 nr 1

Exam aid book.


𝑥 𝑎𝑥𝑖𝑠


𝑥 𝑎𝑥𝑖𝑠


Determining the equation of a linear function page 138

There are two ways to find the equation of a linear function, depending on what information is given:

1. The intercept and one coordinate are given.

EXAMPLE 1 page 138

The intercept is 3, therefore .

Then use the coordinate for substitution is and determine the gradient (

Now you can use this and determine the equation.

2. Two coordinates are given.

OWN EXAMPLE

Find the equation of the line through the points and

Find the gradient ( )

and use either point E or F as substitution in the equation. Let us use

Now you can use this and determine the equation.


Exercise 6 page 139

Exam aid book.


Intersecting lines page 140

Where two or more graphs intersect at a point, these points are called points of intersection (POI).

There are two ways to determine the coordinates of the points of intersection:

1. Reading off the coordinate graphically (on a sketch).

2. Using simultaneous equations to determine the coordinate algebraically.

OWN EXAMPLE

Find the point of intersection of the following two lines: and .

Out of equation 1 (make either or the subject)

Substitute eq. 3 into eq. 2

(solve for )

Substitute into eq. 3

(solve for )

Now this and value are the coordinates for the point of intersection



Exam aid book.

Summary page 141


TEXTBOOK PAGE 142

Introduction to quadratic function (PARABOLA)

Quadratic functions are in other words parabolas and have the general standard form:

or where .

EXAMPLE 1 page 142

Use the table method to sketch the function and then sketch the graph.

Characteristics of the parabola

1. Graph is above the -axis because the square ( is positive.

2. Graph is symmetrical about the -axis, this is the axis of symmetry.

3. Has a minimum turning point and is concave upwards, note the happy face.

Axis of symmetry is the

𝑦-axis

Minimum turning point

is point 𝐴

Concave upwards, note

happy face


EXAMPLE 2 page 143

Use the table method to sketch the function and then sketch the graph.


1. Graph is below the -axis because the square ( is negative.

2. Graph is symmetrical about the -axis, this is the axis of symmetry.

3. Has a maximum turning point and is concave downwards, note the sad face.


You must please study the following characteristics of the parabola (VERY IMPORTANT)

In example 1 and 2 on page 142-143 there are questions with the solutions, I have made a summary of these

solutions in the following table which will make the studying a bit easier.

Axis of symmetry is the

𝑦-axis

Concave downwards,

note sad face

Maximum turning point

is point 𝐴


Effects of the ‘ ’ value

positive then happy face

As ‘ ’ increases, the arms of the parabola

move closer to the -axis


negative then sad face

As ‘ ’ decreases, the arms of the parabola

move closer to the -axis

The graph:

Decreases when

Increases when

The graph:

Decreases when

Increases when

Domain:

Range:

Domain:

Range:

Minimum turning point (0;0) Maximum turning point (0;0)

Line of symmetry:

-axis ( )

Line of symmetry:

-axis ( )



Exam aid book.


TEXTBOOK PAGE 145

Vertical shifting for the parabola

For example 4 on page 145 it explains when we shift the graph up or down in other words vertical shifting.

Remember the standard form of a parabola where ‘ ’ is the -intercept. Note it’s exactly

the same as the linear function (the ‘ ’ value will always be the -intercept). In the table above I have

explained the ‘ ’ value now I will be explaining the effects of the ‘ ’ value.

EXAMPLE 4 page 145

Consider the functions:

𝑦 𝑥 (blue graph) is the ‘mother graph’

Now let us see what happened from the

‘mother graph’ to the other two graphs.

What happened from the ‘mother graph’ to

the ‘red graph’?

You will note that the graph moved

up with two units.

𝑦 𝑥 → 𝑦 𝑥 𝟐

Note the turning points as well

𝐴 → 𝐵 𝟐

What happened from the ‘mother graph’ to

the ‘purple graph’?


down with one unit.

𝑦 𝑥 → 𝑦 𝑥 𝟏

Note the turning point as well

𝐴 → 𝐵 𝟏


I have made a summary in the following table of the effects of the ‘ ’ value. This table is continuous of the

previous table with the characteristics of the parabola.

or


This ‘ ’ value is the -intercept

This ‘ ’ value is also the turning point of the parabola

It indicates vertical shifting.

If graph shifts down

If graph shifts up

EXAMPLE 5 page 147

Consider the function

a) Coordinates of the -intercept:

Now remember the standard form of a parabola is . This ‘ ’ value is the -intercept

. Therefore for the coordinates for the -intercept is

b) Determine algebraically the coordinates of the -intercepts:

Now from the linear function you can remember that finding the -intercept we set

in the function.

(Set )

(Solve for by means of factorisation,

for a parabola it will be

Difference of Two Squares)

(Set each bracket equal to zero and solve )

The coordinates for the -intercepts are and

For the coordinates of the -intercepts you will now

note that the values you solved for are the -

coordinates and the -coordinates are zero because we

set in the first step.

c) Sketch the graph )


d) Determine the following (for these questions use the table with the summary of the characteristics of

the parabola):

For this question first you need to know the shape of this parabola, is it the one that has a Happy or

Sad face.

1) Turning point

2) Minimum value

The minimum value is

3) Domain and range

Domain:

Range:

4) Axis of symmetry

-axis

5) Values of for which increases

6) Values of for which decreases

e) Write down the equation of the graph formed if is shifted 10 units upwards.

Remember the ‘ ’ value indicates vertical shifting. So in the function the ‘ ’ value is and

the graph is shifted 10 units upwards. The new equation will therefore be:

EXAMPLE 6 page 148


a) Sketch the graph )

positive then happy face

The graph:

Decreases when

Increases when

Domain:

Range:

Minimum turning point (0;0)

Line of symmetry:

-axis ( )

-intercept

turning point of the parabola



If graph shifts up


b) Determine the following (for these questions use the table with the summary of the characteristics of

the parabola):

For this question first you need to know the shape of this parabola, is it the one that has a Happy or

Sad face.

1) Turning point

2) Maximum value

The maximum value is

3) Domain and range

Domain:

Range:

4) Axis of symmetry

-axis

5) Values of for which increases

6) Values of for which decreases


Exercise 9 page 146

Exercise 10 page 149

Exam aid book.

negative then sad face

The graph:

Decreases when

Increases when

Domain:

Range:

Maximum turning point (0;0)

Line of symmetry:

-axis ( )

-intercept

turning point of the parabola



If graph shifts up


TEXTBOOK PAGE 149

Determine the equation of a parabola

There are three methods to determining the equation of a parabola. Each method will be explained with an

example.

METHOD 1:

To use this method you must have:

-intercept

One coordinate that lies on the graph

EXAMPLE 1 page 149

Determine the equation of the graph in the form

From this sketch we can see that we have one coordinate and

the -intercept

So if the -intercept is 3 then it’s also the ‘ ’ value

Now substitute the coordinate to determine the

value of ‘ ’

Now you have determined the equation

METHOD 2:


Two -intercepts



EXAMPLE 2 page 150


From this sketch we can see that we have one coordinate and the

two -intercepts and

The factorised form of the equation of the parabola can be used:

where represents the -intercepts.

Now you are going to substitute in your two values for the -intercepts.

[ ][ ]

(Multiply the brackets out inside the block brackets)

Now substitute the coordinate to determine the value of ‘ ’


(Notice the brackets are Difference of Two Squares)

) (Multiply the brackets out FOIL)

(Simplify further and get the standard form

)

METHOD 3:


Two coordinates and that lie on the graph

OWN EXAMPLE


From this sketch we can see that we have two coordinates and

Now first substitute into to find equation one.


Now substitute into to find equation two.

Now use simultaneous equations and solve the values for ‘ ’ and ‘ ’

Out of equation 1 (make either or the subject)

Substitute eq. 3 into eq. 2

(solve for )

Substitute into eq. 3

(solve for )

Now this and values you use for the standard form and determine the equation.



Exam aid book.


TEXTBOOK PAGE 151

Introduction to hyperbolic function (HYPERBOLA)

Hyperbolic functions are in other words hyperbola and have the general standard form:

where

EXAMPLE 1 page 151

Use the table method to sketch the function

and then sketch the graph.

Characteristics of the hyperbola

1. The hyperbola has two arms in opposite quadrants (quadrants one and three)

2. Graph is symmetrical about the line , this is the axis of symmetry.

3. The graph approaches the axes but does not touch the axis, this is called asymptotic behaviour. In

the case of

the and axes are the asymptotes.

Undefined

Two arms in opposite

quadrants (one and three)

𝒙-axis and 𝒚-axis are the

asymptotes

Axis of symmetry:

Line 𝒚 𝒙


EXAMPLE 2 page 152

Use the table method to sketch the function

and then sketch the graph.


1. The hyperbola has two arms in opposite quadrants (quadrants two and four)

2. Graph is symmetrical about the line , this is the axis of symmetry.

3. The graph approaches the axes but does not touch the axis, this is called asymptotic behaviour. In

the case of

the and axes are the asymptotes.


You must please study the following characteristics of the parabola (VERY IMPORTANT)

In example 1 and 2 on page 151-153 there are questions with the solutions, I have made a summary of these


Undefined

Two arms in opposite

quadrants (two and four)

𝒙-axis and 𝒚-axis are the

asymptotes

Axis of symmetry:

Line 𝒚 𝒙



positive then quadrant 1 and 3

As ‘ ’ increases, the graph moves further

away from the axes


negative then quadrant 2 and 4

As ‘ ’ increases, the graph moves further

away from the axes

The graph:

Decreases when and

Increases for no values

The graph:

Decreases for no values

Increases when and

Domain:

Range:

Domain:

Range:

Asymptotes

-axis ( )

-axis ( )

Asymptotes

-axis ( )

-axis ( )

Line of symmetry:

Line of symmetry:



Exam aid book.


TEXTBOOK PAGE 156

Vertical shifting for hyperbola

For example 5 and 6 on page 156-159 it explains when we shift the graph up or down in other words

vertical shifting. Remember the standard form of a hyperbola

where ‘ ’ is the horizontal

asymptote ( ). In the table above I have explained the ‘ ’ value now I will be explaining the effects of

the ‘ ’ value.

EXAMPLE 5 page 156


Undefined 2

Undefined

What happened from the ‘mother graph’ to the

‘red graph’?

You will note that the graph moved up

with one unit.

𝑦

𝑥 → 𝑦

𝑥 𝟏

Note the horizontal asymptote also

moved up one unit.

𝑦 → 𝑦

𝑦

𝑥 (green graph) is the ‘mother graph’


‘mother graph’ to the other graph.



previous table with the characteristics of the hyperbola.

or


This ‘ ’ value is the horizontal asymptote ‘q’ value



If graph shifts up

EXAMPLE 6 page 157


a) Determine the following (for these questions use the table with the summary of the characteristics of

the hyperbola):

1) Equation of the asymptote

2) Coordinates of the -intercept

Now from the linear or parabola function

you can remember that finding the -

intercept we set in the function.

(Set

(Take the ‘ ’ value over to the LHS)

(Multiply both sides by ‘ ’)

(Solve for )

b) Sketch the graph

Horizontal asymptote ‘q’ value



If graph shifts up


c) Write down (for these questions use the table with the summary of the characteristics of the

hyperbola):

For this question first you need to know the shape of this hyperbola, is it the one that is positive or

negative.

1) Domain and range

Domain:

Range:

2) Values of for which the graph

decreases

For no values

3) Values of for which the graph

increases

When and

negative then quadrant 2 and 4

The graph:

Decreases for no values

Increases when and

Domain:

Range:

Asymptotes

-axis ( )

-axis ( )

Line of symmetry:




If graph shifts up




Exam aid book.


TEXTBOOK PAGE 160

Determine the equation of a hyperbola

There is one method to determining the equation of a hyperbola. You must have:



EXAMPLE 1 page 160


From this sketch we can see that we have

one coordinate and the horizontal

asymptote

So if the horizontal asymptote is 2 then

it’s also the ‘ ’ value

Now substitute the coordinate

to determine the value of ‘ ’



EXAMPLE 2 page 160


From this sketch we can see that we

have one coordinate and the

horizontal asymptote

So if the horizontal asymptote is then it’s

also the ‘ ’ value

Now substitute the coordinate

to determine the value of ‘ ’




Exam aid book.


TEXTBOOK PAGE 162

Introduction to exponential function

The graphs of these functions are called exponential graphs and have the general standard form:

where

EXAMPLE 1 page 162

Use the table method to sketch the function and (

)

then sketch the graph.

Characteristics of the exponential function

1. The -intercept of the exponential functions is

2. The graph approaches the -axis but does not touch the axis, this is called asymptotic behaviour. In

the case -axis is the horizontal asymptote.

(

)

𝑓 𝑥 𝑥

Graph increases for all values of 𝑥

Domain: 𝑥

Range: 𝑦 ∞

𝒙-axis is the asymptote

𝑔 𝑥 (

)𝑥

Graph decreases for all values of 𝑥

Domain: 𝑥

Range: 𝑦 ∞


EXAMPLE 2 page 163

Use the table method to sketch the function and (

)

then sketch the graph.


1. The -intercept of the exponential functions is

2. The graph approaches the -axis but does not touch the axis, this is called asymptotic behaviour. In

the case -axis is the horizontal asymptote.

(

)

𝑓 𝑥 𝑥

Graph decreases for all values of 𝑥

Domain: 𝑥

Range: 𝑦 ∞

𝒙-axis is the asymptote

𝑔 𝑥 (

)𝑥

Graph increases for all values of 𝑥

Domain: 𝑥

Range: 𝑦 ∞



You must please study the following characteristics of the exponential function (VERY IMPORTANT)

In example 1 – 4 on page 162-167 there are questions with the solutions, I have made a summary of these



If ‘ ’ is positive then graph is above the

asymptote

As ‘ ’ increases, the closer the arm gets to

the -axis


If ‘ ’ is negative then graph is below the

asymptote

As ‘ ’ decreases, the closer the arm gets to

the -axis

If

Increases for all values of

If

Decreases for all values of

(

)

If

Decreases for all values of

If

Increases for all values of

(

)

Domain:

Range: ∞

Domain:

Range: ∞

Asymptote

-axis ( )

Asymptote

-axis ( )



Exam aid book.


TEXTBOOK PAGE 168

Vertical shifting for exponential function

For example 5 and 6 on page 168-170 it explains when we shift the graph up or down in other words

vertical shifting. Remember the standard form of an exponential function

where ‘ ’ is the horizontal asymptote. In the table above I have explained the ‘ ’

value now I will be explaining the effects of the ‘ ’ value.

EXAMPLE 5 page 156


What happened from the 𝑓 𝑥 to 𝑥 ?

You will note that the graph moved up

with two units.

𝑓 𝑥 𝑥 → 𝑔 𝑥 𝑥 𝟐


moved up two units.

𝑦 → 𝑦 𝟐

𝑓 𝑥 is the ‘mother graph’ where the

horizontal asymptote is 𝑦 (𝑥-axis)


‘mother graph’ to the other graphs.

What happened from the 𝑓 𝑥 to 𝑥 ?


down with one unit.

𝑓 𝑥 𝑥 → 𝑔 𝑥 𝑥 𝟏


moved down one unit.

𝑦 → 𝑦 𝟏



previous table with the characteristics of the exponential function.

or


This ‘ ’ value is the horizontal asymptote ‘q’ value



If graph shifts up

It also affects the range

∞ when is positive or

∞ when is negative


Exercise 17 page 171 nr 1.

Exam aid book.


TEXTBOOK PAGE 172

Determine the equation of an exponential function

There is one method to determining the equation of an exponential equation. You must have:



EXAMPLE 1 page 172


From this sketch we can see that we have one coordinate

and the horizontal asymptote

So if the horizontal asymptote is 3 then it’s also the ‘ ’

value

Now substitute the coordinate to

determine the value of ‘ ’


EXAMPLE 2 page 172


From this sketch we can see that we have one

coordinate and the horizontal asymptote

So if the horizontal asymptote is -3 then it’s also the

‘ ’ value


Now substitute the coordinate to determine the value of ‘ ’


(

)



Exam aid book.


TEXTBOOK PAGE 173

Reflections of graphs about the axes



Exam aid book.


TEXTBOOK PAGE 176

Graph interpretation





Mixed Exercise page 175.

Exam aid book.

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Module 6: Functions · 2020. 8. 12. · GRADE 10 FUNCTIONS 1 Module 6: Functions TEXTBOOK PAGE 121...

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