Module 6: Functions · 2020. 8. 12. · GRADE 10 FUNCTIONS 1 Module 6: Functions TEXTBOOK PAGE 121...
Transcript of 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:
GRADE 10 FUNCTIONS 2
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.
GRADE 10 FUNCTIONS 3
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
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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.
Example: Sketch graph
-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.
Example: Sketch graph
-intercept (set )
Gradient but first get the equation in the standard form of a straight line
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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
GRADE 10 FUNCTIONS 6
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.
𝑦 𝑎𝑥𝑖𝑠
𝑥 𝑎𝑥𝑖𝑠
𝑦 𝑎𝑥𝑖𝑠
𝑥 𝑎𝑥𝑖𝑠
GRADE 10 FUNCTIONS 7
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.
After going through these notes, you can now do:
Exercise 6 page 139
Exam aid book.
GRADE 10 FUNCTIONS 8
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
After going through these notes, you can now do:
Exercise 7 page 140.
Exam aid book.
Summary page 141
GRADE 10 FUNCTIONS 9
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
GRADE 10 FUNCTIONS 10
EXAMPLE 2 page 143
Use the table method to sketch the function and then sketch the graph.
Characteristics of the parabola
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.
Characteristics of the parabola
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
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Effects of the ‘ ’ value
positive then happy face
As ‘ ’ increases, the arms of the parabola
move closer to the -axis
Effects of the ‘ ’ value
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 ( )
After going through these notes, you can now do:
Exercise 8 page 144 nr 1
Exam aid book.
GRADE 10 FUNCTIONS 12
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’?
You will note that the graph moved
down with one unit.
𝑦 𝑥 → 𝑦 𝑥 𝟏
Note the turning point as well
𝐴 → 𝐵 𝟏
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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
Effects of the ‘ ’ value
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 )
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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
Consider the function
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
It indicates vertical shifting.
If graph shifts down
If graph shifts up
GRADE 10 FUNCTIONS 15
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
After going through these notes, you can now do:
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
It indicates vertical shifting.
If graph shifts down
If graph shifts up
GRADE 10 FUNCTIONS 16
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:
To use this method you must have:
Two -intercepts
One coordinate that lies on the graph
GRADE 10 FUNCTIONS 17
EXAMPLE 2 page 150
Determine the equation of the graph in the form
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 ‘ ’
Now you have determined the equation
(Notice the brackets are Difference of Two Squares)
) (Multiply the brackets out FOIL)
(Simplify further and get the standard form
)
METHOD 3:
To use this method you must have:
Two coordinates and that lie on the graph
OWN EXAMPLE
Determine the equation of the graph in the form
From this sketch we can see that we have two coordinates and
Now first substitute into to find equation one.
GRADE 10 FUNCTIONS 18
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.
After going through these notes, you can now do:
Exercise 11 page 150
Exam aid book.
GRADE 10 FUNCTIONS 19
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
GRADE 10 FUNCTIONS 20
EXAMPLE 2 page 152
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 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.
Characteristics of the hyperbola
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
solutions in the following table which will make the studying a bit easier.
Undefined
Two arms in opposite
quadrants (two and four)
𝒙-axis and 𝒚-axis are the
asymptotes
Axis of symmetry:
Line 𝒚 𝒙
GRADE 10 FUNCTIONS 21
Effects of the ‘ ’ value
positive then quadrant 1 and 3
As ‘ ’ increases, the graph moves further
away from the axes
Effects of the ‘ ’ value
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:
After going through these notes, you can now do:
Exercise 12 page 155 nr 1
Exam aid book.
GRADE 10 FUNCTIONS 22
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
Consider the functions:
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’
Now let us see what happened from the
‘mother graph’ to the other graph.
GRADE 10 FUNCTIONS 23
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 hyperbola.
or
Effects of the ‘ ’ value
This ‘ ’ value is the horizontal asymptote ‘q’ value
It indicates vertical shifting.
If graph shifts down
If graph shifts up
EXAMPLE 6 page 157
Consider the function
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
It indicates vertical shifting.
If graph shifts down
If graph shifts up
GRADE 10 FUNCTIONS 24
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:
Horizontal asymptote ‘q’ value
It indicates vertical shifting.
If graph shifts down
If graph shifts up
GRADE 10 FUNCTIONS 25
After going through these notes, you can now do:
Exercise 13 page 159
Exam aid book.
GRADE 10 FUNCTIONS 26
TEXTBOOK PAGE 160
Determine the equation of a hyperbola
There is one method to determining the equation of a hyperbola. You must have:
Horizontal asymptote ‘q’ value
One coordinate that lies on the graph
EXAMPLE 1 page 160
Determine the equation of the graph in the form
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 ‘ ’
Now you have determined the equation
GRADE 10 FUNCTIONS 27
EXAMPLE 2 page 160
Determine the equation of the graph in the form
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 ‘ ’
Now you have determined the equation
After going through these notes, you can now do:
Exercise 14 page 161
Exam aid book.
GRADE 10 FUNCTIONS 28
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 ∞
GRADE 10 FUNCTIONS 29
EXAMPLE 2 page 163
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 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 ∞
GRADE 10 FUNCTIONS 30
Characteristics of the exponential function
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
solutions in the following table which will make the studying a bit easier.
Effects of the ‘ ’ value
If ‘ ’ is positive then graph is above the
asymptote
As ‘ ’ increases, the closer the arm gets to
the -axis
Effects of the ‘ ’ value
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 ( )
After going through these notes, you can now do:
Exercise 16 page 168
Exam aid book.
GRADE 10 FUNCTIONS 31
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
Consider the functions:
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
Note the horizontal asymptote also
moved up two units.
𝑦 → 𝑦 𝟐
𝑓 𝑥 is the ‘mother graph’ where the
horizontal asymptote is 𝑦 (𝑥-axis)
Now let us see what happened from the
‘mother graph’ to the other graphs.
What happened from the 𝑓 𝑥 to 𝑥 ?
You will note that the graph moved
down with one unit.
𝑓 𝑥 𝑥 → 𝑔 𝑥 𝑥 𝟏
Note the horizontal asymptote also
moved down one unit.
𝑦 → 𝑦 𝟏
GRADE 10 FUNCTIONS 32
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 exponential function.
or
Effects of the ‘ ’ value
This ‘ ’ value is the horizontal asymptote ‘q’ value
It indicates vertical shifting.
If graph shifts down
If graph shifts up
It also affects the range
∞ when is positive or
∞ when is negative
After going through these notes, you can now do:
Exercise 17 page 171 nr 1.
Exam aid book.
GRADE 10 FUNCTIONS 33
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:
Horizontal asymptote ‘q’ value
One coordinate that lies on the graph
EXAMPLE 1 page 172
Determine the equation of the graph in the form
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 ‘ ’
Now you have determined the equation
EXAMPLE 2 page 172
Determine the equation of the graph in the form
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
GRADE 10 FUNCTIONS 34
Now substitute the coordinate to determine the value of ‘ ’
Now you have determined the equation
(
)
After going through these notes, you can now do:
Exercise 18 page 173.
Exam aid book.
GRADE 10 FUNCTIONS 35
TEXTBOOK PAGE 173
Reflections of graphs about the axes
After going through these notes, you can now do:
Exercise 19 page 174.
Exam aid book.
GRADE 10 FUNCTIONS 36
TEXTBOOK PAGE 176
Graph interpretation
GRADE 10 FUNCTIONS 37
GRADE 10 FUNCTIONS 38
GRADE 10 FUNCTIONS 39
After going through these notes, you can now do:
Mixed Exercise page 175.
Exam aid book.