10. functions

MODULE - 10MODULE - 10
FUNCTIONSFUNCTIONS
Demonstrate the ability to work with various types of functions.Demonstrate the ability to work with various types of functions.
(LO 2 AS 1a)(LO 2 AS 1a)
Recognize relationships, between variables in terms of numerical,Recognize relationships, between variables in terms of numerical,
graphical, verbal and symbolic representations.graphical, verbal and symbolic representations.
(LO 2 AS 1 b)(LO 2 AS 1 b)
Generate as many graphs as necessary.Generate as many graphs as necessary.
b 0b 0
(LO 2 AS 2)(LO 2 AS 2)
qaby
q
x
a
y
qaxy
qaxy
+=
+=
+=
+=
2
2
≠

Identify characteristics as listed below:Identify characteristics as listed below:
 domain and rangedomain and range
 intercepts with the axesintercepts with the axes
 turning points, maxima and minimaturning points, maxima and minima
 asymptotesasymptotes
 shape and symmetryshape and symmetry
 periodicity/amplitudeperiodicity/amplitude
 intervals on which the function increases orintervals on which the function increases or
decreasesdecreases
 the discrete or continuous nature of thethe discrete or continuous nature of the
graph (LO 2 AS 3)graph (LO 2 AS 3)

 Solve linear equations in two variablesSolve linear equations in two variables
simultaneouslysimultaneously
(LO 2 AS 5e)(LO 2 AS 5e)

RelationsRelations
 A relation is any rule by means of which eachA relation is any rule by means of which each
element of a first set is associated with atelement of a first set is associated with at
least one element of a second set.least one element of a second set.
 For example, suppose that in a givenFor example, suppose that in a given
relation, a first set isrelation, a first set is {- 2; - 1; 0 ; 1; 3}{- 2; - 1; 0 ; 1; 3} andand
that the second set is obtained by using thethat the second set is obtained by using the
rulerule y = 2x.y = 2x.
 By substituting the given x-values into theBy substituting the given x-values into the
equationequation y = 2x,y = 2x, it will be possible toit will be possible to
determine the second set, which contains thedetermine the second set, which contains the
corresponding y-values.corresponding y-values.
 The relation can be represented in differentThe relation can be represented in different
ways.ways.

1. A table of values1. A table of values
xx -2-2 -2-2 00 11 22 33
yy -4-4 -2-2 00 22 44 66
x in this relation is called the independent
variable, since the values of x were chosen
randomly.
However, it is clear that the values of y
depended entirely on the values of x as well as
the rule used, namely,
y = 2x.
In this relation, y is called the dependent
variable.

2. A set of ordered pairs2. A set of ordered pairs
 Relation =Relation = {(- 2; -4); (-1; -2); (0; 0); (1; 2);{(- 2; -4); (-1; -2); (0; 0); (1; 2);
(2; 4); (3; 6)}(2; 4); (3; 6)}

3.Set-Builder Notation3.Set-Builder Notation
 In set- builder notation the aboveIn set- builder notation the above
relation can be represented as follows:relation can be represented as follows:
 This is read as “the set of allThis is read as “the set of all xx andand yy
such thatsuch that y = 2y = 2 and x is an elementand x is an element
ofof {-2 ;-1;0;l ;2;3}{-2 ;-1;0;l ;2;3}”.”.
(x; y)(x; y) states that there is a relationshipstates that there is a relationship
betweenbetween xx andand y.y.
y = 2xy = 2x is the rule connecting x and y.is the rule connecting x and y.

4. The Cartesian Number Plane4. The Cartesian Number Plane

 If we now had to increase the elementsIf we now had to increase the elements
of the first set to include all realof the first set to include all real
numbers fornumbers for x,x, there would be so manythere would be so many
points that could be represented on thepoints that could be represented on the
Cartesian number plane. In fact, theCartesian number plane. In fact, the
points would be so close that we wouldpoints would be so close that we would
get what is called the graph of aget what is called the graph of a
straight linestraight line y = 2xy = 2x for allfor all x.x.

 See if you can draw the graph of thisSee if you can draw the graph of this
relation for allrelation for all x.x.
 Use the diagram below.Use the diagram below.

Domain and RangeDomain and Range
 The set of numbers to which we apply the rule isThe set of numbers to which we apply the rule is
referred to as the domain. The set of numbersreferred to as the domain. The set of numbers
obtained as a result of using the rule is referred to asobtained as a result of using the rule is referred to as
the range. For the previous relationthe range. For the previous relation y = 2xy = 2x , the, the
domain is the set of x-values used, whereas thedomain is the set of x-values used, whereas the
range is the set of y-values obtained. The rangerange is the set of y-values obtained. The range
depends on the domain used.depends on the domain used.
 In the relation, the domainIn the relation, the domain
used is clearly the setused is clearly the set x {-2; -1; 0; 1; 2; 31}.x {-2; -1; 0; 1; 2; 31}.
The range is therefore the set of y-valuesThe range is therefore the set of y-values
y {-4;-2;0;2;4;6}.y {-4;-2;0;2;4;6}.
}3;2;1;0;1;2{;2/);{ −−∈= xxyyx
∈
∈

Finding the domain and rangeFinding the domain and range
1.1. Given a set of ordered pairsGiven a set of ordered pairs
 Example:Example: {(-2; 16); (0; 4); (1; 4); (3; 7)}{(-2; 16); (0; 4); (1; 4); (3; 7)}
 Domain =Domain = {-2; 0; 1; 3}{-2; 0; 1; 3}
 Range =Range = {16; 4; 7}{16; 4; 7}

2. Given a graph2. Given a graph
 Use a clear plastic ruler.Use a clear plastic ruler.
 For domain:For domain: Keep the edge of the ruler verticalKeep the edge of the ruler vertical
and slide it across the graph from left to right.and slide it across the graph from left to right.
Where the edge starts cutting the graph, theWhere the edge starts cutting the graph, the
domain starts (as read off from the x-axis).domain starts (as read off from the x-axis).
 Where it stops cutting the graph the domainWhere it stops cutting the graph the domain
ends.ends.
 For range:For range: Keep the edge of the ruler horizontalKeep the edge of the ruler horizontal
and slide it across the graph from bottom to top.and slide it across the graph from bottom to top.
 Where the edge starts cutting the graph, theWhere the edge starts cutting the graph, the
range startsrange starts
(as read off from the y-axis).(as read off from the y-axis).
 Where it stops cutting the graph the range ends.Where it stops cutting the graph the range ends.

Example 1Example 1
 For each of the following graphs ofFor each of the following graphs of
given relations, determine the domaingiven relations, determine the domain
and rangeand range
(a)(a)

 {(-2;16); (0; 4); (1; 4); (3; 7)}{(-2;16); (0; 4); (1; 4); (3; 7)}
 Here, each element of the domain is associated withHere, each element of the domain is associated with
only one element of the range. The numbersonly one element of the range. The numbers 00 andand 11
are associated with the same element of the rangeare associated with the same element of the range
(namely 4). In this case, the relation is said to be a(namely 4). In this case, the relation is said to be a
function.function.

 {(-2; 16); (4; 1); (4; 6); (3; 7)}{(-2; 16); (4; 1); (4; 6); (3; 7)}
 Here, the numberHere, the number 44 in the domain isin the domain is
associated with more than one element of theassociated with more than one element of the
rangerange (1 and 6).(1 and 6). In this case, the relation isIn this case, the relation is
not a function.not a function.

2. Given a graph2. Given a graph
 We use a ruler to perform the “verticalWe use a ruler to perform the “vertical
line test” on a graph to see whether itline test” on a graph to see whether it
is a function or not. Hold a clear plasticis a function or not. Hold a clear plastic
ruler parallel to the y-axis, i.e. vertical.ruler parallel to the y-axis, i.e. vertical.
Move it from left to right over the axes.Move it from left to right over the axes.
If the ruler cuts the curve in one placeIf the ruler cuts the curve in one place
only, then the graph is a function.only, then the graph is a function.

Example 2Example 2
 Determine whether the followingDetermine whether the following
relations are functions or not.relations are functions or not.
(a)(a)

MappingMapping and functional notationand functional notation
 Since functions are special relations,Since functions are special relations,
we reserve certain notation strictly forwe reserve certain notation strictly for
use when dealing with functions.use when dealing with functions.
 Consider the functionConsider the function f = {(x; y)/ y = 3x),f = {(x; y)/ y = 3x),
 This function may be represented byThis function may be represented by
means of mapping notation ormeans of mapping notation or
functional notation.functional notation.

Mapping notationMapping notation
 This is read as “f mapsThis is read as “f maps xx ontoonto 3x3x 2”.2”.
 If x = 2If x = 2 is an element of the domain,is an element of the domain,
then the corresponding element in thethen the corresponding element in the
range isrange is 3(2) = 6.3(2) = 6.
 We say thatWe say that 66 is the image ofis the image of 22 in thein the
mapping ofmapping of f.f.
xxf 3: →

Functional notationFunctional notation
 This is read as “of x is equal to 3x”.This is read as “of x is equal to 3x”.
 The symbol f (x) is used to denote the element of theThe symbol f (x) is used to denote the element of the
range to which x maps.range to which x maps.
 In other words, the y-values corresponding to the xIn other words, the y-values corresponding to the x
-values are given by f (x), i.e. y = f (x).-values are given by f (x), i.e. y = f (x).
 For example, ifFor example, if x = 4,x = 4, then the corresponding y-valuethen the corresponding y-value
is obtained by substitutingis obtained by substituting x = 4x = 4 intointo 3x.3x.
 ForFor x = 4,x = 4, the y - value is fthe y - value is f (4) = 3(4) = 12.(4) = 3(4) = 12.
 The brackets in the symbol f (4) do not mean f timesThe brackets in the symbol f (4) do not mean f times
4, but rather the y-value at4, but rather the y-value at x = 4.x = 4.
( ) xxf 3=

Example 3Example 3
 Consider the functionConsider the function
 Suppose that the domain is given bySuppose that the domain is given by
 Determine the rangeDetermine the range
 Represent the function graphically.Represent the function graphically.
2: +→ xxf
}2;1;0;1;2{ −−∈x

 (a)(a)
(b)(b)
2)( += xxf
4222)2(
3212)1(
22)0(2)0(
12)1(2)1(
02)2(2)2(
=+=+=
=+=+=
=+=+=
=+−=+=−
=+−=+=−
xf
xf
xf
xf
xf

Example 4Example 4
 IfIf
 Determine the value of:Determine the value of:
(a)(a) ff (2)(2)
(b)(b) ff (-3)(-3)
(c)(c) ff
13)( 2
−= xxf






3
1

(a)(a)
(b)(b)
(c)(c)
(d)(d)
13)( 2
−= xxf
111121)2(3)2( 2
=−=−=f
13)( 2
−= xxf
261271)3(3)3( 2
=−=−−=−f
13)( 2
−= xxf
3
2
1
3
1
1
9
1
31
3
1
3)
3
1
(
2
−=−=−





=−





=f
13)( 2
−= xxf

EXERCISEEXERCISE
1. Determine the domain and range of the1. Determine the domain and range of the
following relations.following relations.
 State whether the relation is a functionState whether the relation is a function
or not.or not.

 (a) (b)(a) (b)
(c) (d)(c) (d)

2.2. Consider the functionConsider the function
 Determine and then representDetermine and then represent
graphically:graphically:
(a)(a) f (-1)f (-1) (b)(b) f (0)f (0)
(c)(c) (d)(d) f (2)f (2)





3
2
f

3. If3. If
Determine the value of:Determine the value of:
(a)(a) f (1)f (1) (b)(b) f(- 1)f(- 1)
(c)(c) f (2)f (2) (d)(d) f(- 2)f(- 2)
(e)(e) (f)(f)
(g)(g) f (a)f (a) (h)(h) f(2x)f(2x)
(i)(i) f (- x)f (- x) (j)(j) f (x - 1)f (x - 1)






2
1
f 





−
2
1
f
12)( 2
+−= xxxf

THE LINEAR FUNCTIONTHE LINEAR FUNCTION
 The graph of a linear function is aThe graph of a linear function is a
straight line.straight line.
 The equation of a linear function takesThe equation of a linear function takes
the formthe form y = m x + c.y = m x + c.

The Table MethodThe Table Method
 Example 1Example 1
Sketch the graph ofSketch the graph of y = x - 2y = x - 2 by using the tableby using the table
method.method.
xx -1-1 00 11 22
yy -3-3 -2-2 -1-1 00

 The x-values were randomly chosen.The x-values were randomly chosen.
 The y-values were found byThe y-values were found by
substituting thesubstituting the x -x - values into thevalues into the
equationequation y = x - 2.y = x - 2.

 Note:Note:
 The line cuts the x-axis at the pointThe line cuts the x-axis at the point (2; 0).(2; 0).
 This point represents the coordinates of theThis point represents the coordinates of the
x-intercept.x-intercept.
 For the x - intercept of any line, it is clear thatFor the x - intercept of any line, it is clear that
the y-value is alwaysthe y-value is always 0.0.
 The line cuts the y-axis at the pointThe line cuts the y-axis at the point (0; - 2).(0; - 2).
 This point represents the coordinates of theThis point represents the coordinates of the
y-intercept.y-intercept.
 For the y - intercept of any line, it is clear thatFor the y - intercept of any line, it is clear that
the x- value is alwaysthe x- value is always 0.0.

The Dual-Intercept MethodThe Dual-Intercept Method
Sketch the graph ofSketch the graph of 2x – 3y = 62x – 3y = 6 by using the dual-by using the dual-
intercept method.intercept method.
 This method involves determining the interceptsThis method involves determining the intercepts
with the axes using the above note.with the axes using the above note.
 x-intercept:x-intercept: y = 0y = 0
2x - 3(0) = 62x - 3(0) = 6
2x = 62x = 6
x = 3x = 3
(3; 0)(3; 0)

 y - intercept:y - intercept: letlet x = 0x = 0
- 3y = - 2x + 6- 3y = - 2x + 6
y =y =
y =y =
(0; -2)(0; -2)
2
3
2
−
x
22
3
)0(2
−=−

 The x-intercept is the pointThe x-intercept is the point (3; 0)(3; 0)
 The y-intercept is the pointThe y-intercept is the point (0; - 2)(0; - 2)
 Note:Note:
 The dual-intercept method will not work withThe dual-intercept method will not work with
lines that cut the axes at the origin, i.e. at thelines that cut the axes at the origin, i.e. at the
pointpoint
(0; 0).(0; 0).
 In these cases, use the table method toIn these cases, use the table method to
sketch the graph.sketch the graph.

Example 3Example 3
 Sketch the graph ofSketch the graph of y - 2x = 0.y - 2x = 0.
 Using the dual-intercept method:Using the dual-intercept method:
 x-intercept:x-intercept: y = 0y = 0
0 - 2x = 00 - 2x = 0
- 2x = 0- 2x = 0
x = 0x = 0
 y-intercept:y-intercept: x = 0x = 0
y – 2x = 0y – 2x = 0
y = 2xy = 2x
y = 2(0)y = 2(0)
y = 0y = 0

 The x-intercept is the pointThe x-intercept is the point (0; 0).(0; 0).
 The y-intercept is the pointThe y-intercept is the point (0; 0)(0; 0)
 This line cuts the axes at the origin. We willThis line cuts the axes at the origin. We will
now need to use the table method in this case.now need to use the table method in this case.
We will rewrite the line asWe will rewrite the line as y = 2x.y = 2x.
xx -1-1 00 11
yy -2-2 00 22

Discovery Exercise 1Discovery Exercise 1
 Sketch the graphs of the following linear functions on the sameSketch the graphs of the following linear functions on the same
set of axes using either the dual-intercept method or the tableset of axes using either the dual-intercept method or the table
method where the line cuts the axes at the origin.method where the line cuts the axes at the origin.
xxf −→:
1: +−→ xxg
4: +−→ xxh
2: −−→ xxj

 Now answer the following questions basedNow answer the following questions based
on your graphs:on your graphs:
(a)(a) What do you notice about the slope of eachWhat do you notice about the slope of each
line drawn?line drawn?
(b) What do you notice about the coefficient of(b) What do you notice about the coefficient of
x in each equation?x in each equation?
(c)(c) So what can you conclude about theSo what can you conclude about the
coefficient ofcoefficient of x in each equation?x in each equation?
(d) What is the y - intercept of each line and(d) What is the y - intercept of each line and
how do these y - intercepts relate to thehow do these y - intercepts relate to the
equations?equations?
(e)(e) In terms of translations, how can the graphsIn terms of translations, how can the graphs
of g, h and j be drawn using the graph off?of g, h and j be drawn using the graph off?

 Sketch the graphs of the followingSketch the graphs of the following
linear functions on the same set oflinear functions on the same set of
axes using either the dual - interceptaxes using either the dual - intercept
method or the table method where themethod or the table method where the
line cuts the axes at the origin.line cuts the axes at the origin.

xxf 2: → 22: +→ xxg
32: +→ xxh 22: −→ xxj

 Now answer the following questions basedNow answer the following questions based
on your graphs:on your graphs:
(a)(a) What do you notice about the slopeWhat do you notice about the slope
of each line drawn?of each line drawn?
(b)(b) What do you notice about theWhat do you notice about the
coefficient of x in each equation?coefficient of x in each equation?
(c)(c) So what can you conclude about theSo what can you conclude about the
coefficient of x in each equation?coefficient of x in each equation?
(d)(d) What is the y - intercept of each lineWhat is the y - intercept of each line
and how do these y - interceptsand how do these y - intercepts
relate to the equations?relate to the equations?
(e)(e) In terms of translations, how can theIn terms of translations, how can the
graphs of g, h and j be drawn usinggraphs of g, h and j be drawn using
the graph of f ?the graph of f ?

ConclusionConclusion
 For any linear function written in the formFor any linear function written in the form y = m x + c:y = m x + c:
 The constant c represents the y - intercept of the graph.The constant c represents the y - intercept of the graph.
 The coefficient of x, namely in, represents the slope or gradientThe coefficient of x, namely in, represents the slope or gradient
of the line.of the line.
IfIf m > 0,m > 0, the lines slope to the right.the lines slope to the right.
IfIf m < 0,m < 0, the lines slope to the left.the lines slope to the left.
 The graph of y = mx + c is the translation of the graph ofThe graph of y = mx + c is the translation of the graph of
y = m x.y = m x.
IfIf c > 0c > 0, the graph of y =, the graph of y = m x + cm x + c shifts byshifts by cc
units upwardsunits upwards
IfIf c < 0,c < 0, the graph of y =the graph of y = m x + cm x + c shifts byshifts by cc
units downwardsunits downwards

Example 4Example 4
 Rewrite the following equations in the form y = m x + c andRewrite the following equations in the form y = m x + c and
then write downthen write down
 the y - intercept and the gradient. State whether the lines slopethe y - intercept and the gradient. State whether the lines slope
to the left or right.to the left or right.
(a)(a) 3y - 4x – 3 = 03y - 4x – 3 = 0
3y = 4x - 33y = 4x - 3
y =y = (standard form : y = m x + c(standard form : y = m x + c ))
 y – intercept x = 0y – intercept x = 0
y =y =
The y - intercept is the pointThe y - intercept is the point (0; 1)(0; 1)
The gradient is m = which is positive.The gradient is m = which is positive.
The line slopes to the right.The line slopes to the right.
11)0(
3
4
−=−
3
4
1
3
4
−x

(b)(b) 4y + 3x – 8 = 04y + 3x – 8 = 0
4y = - 3x + 84y = - 3x + 8
y =y =
y – intercept x = 0y – intercept x = 0
y =y =
 The y-intercept is the pointThe y-intercept is the point (0; 2).(0; 2).
 The gradient isThe gradient is m =m = which iswhich is
negative.negative.
 The line slopes to the left.The line slopes to the left.
22)0(
4
3
2
4
3
=+
−
=+− x
4
3
−
2
4
3
+− x

Example 5Example 5
(a)(a) Sketch the graph ofSketch the graph of f (x) = - 2xf (x) = - 2x on the axes provided below.on the axes provided below.
(b)(b) Now draw the graph of the line formed if the graph off isNow draw the graph of the line formed if the graph off is
translated 2 units upwards. Indicate the coordinates of thetranslated 2 units upwards. Indicate the coordinates of the
intercepts with the axes as well as the equation of the newlyintercepts with the axes as well as the equation of the newly
formed line.formed line.
(c)(c) Now draw the graph of the line formed if the graph off isNow draw the graph of the line formed if the graph off is
translated 3 units downwards. Indicate the coordinates of thetranslated 3 units downwards. Indicate the coordinates of the
intercepts with the axes as well as the equation of the newlyintercepts with the axes as well as the equation of the newly

 EXERCISE 1EXERCISE 1
1.1. Draw neat sketch graphs of the followingDraw neat sketch graphs of the following
linear functions on separate axes.linear functions on separate axes.
 Use the dual-intercept method or whereUse the dual-intercept method or where
necessary, the table method.necessary, the table method.
(a)(a) f (x) = 3x- 6f (x) = 3x- 6 (f)(f) y - 3x = 6y - 3x = 6
(b)(b) g (x) = - 2x + 2g (x) = - 2x + 2 (g)(g) y = 3x + 2y = 3x + 2
(c)(c) h (x) = - 4xh (x) = - 4x (h)(h) 2x + 3y + 6 = 02x + 3y + 6 = 0
(d)(d) 5x + 2y = 105x + 2y = 10 (i)(i) 3x + 3y = 03x + 3y = 0
(e)(e) y – x = 0y – x = 0 (j)(j) 3x = 2y3x = 2y

2.2. Without actually sketching the graphs ofWithout actually sketching the graphs of
thethe following linear functions,following linear functions,
determine, by rewriting the equations indetermine, by rewriting the equations in thethe
formform y = m x + c,y = m x + c, the gradient and y -the gradient and y -
intercept of each line and state theintercept of each line and state the directiondirection
of the slope of each line.of the slope of each line.
(a)(a) 2y - 4x = 02y - 4x = 0 (c)(c) 6x - 3y = 16x - 3y = 1
(b)(b) 2y + 4x = 22y + 4x = 2 (d)(d) x - 2y = 4x - 2y = 4

3.3. (a)(a) Sketch the graph ofSketch the graph of f (x) = x - 1f (x) = x - 1 on aon a
set of axes.set of axes.
(b)(b) Now draw the graph of the lineNow draw the graph of the line
formed if the graph of f isformed if the graph of f is
translated 2 units upwards. Indicatetranslated 2 units upwards. Indicate
the coordinates of the intercepts with thethe coordinates of the intercepts with the
axes as well as the equation of the newlyaxes as well as the equation of the newly
(c)(c) Now draw the graph of the lineNow draw the graph of the line
translated 3 units downwards.translated 3 units downwards.
Indicate the coordinates of theIndicate the coordinates of the
intercepts with the axes as well asintercepts with the axes as well as
the equation of the newlythe equation of the newly

 4.4. (a)(a) Sketch the graph ofSketch the graph of
f (x) = - 2x + 1f (x) = - 2x + 1 on a set of axes.on a set of axes.
(b)(b) Now draw the graph of the lineNow draw the graph of the line
translated 2 units upwards.translated 2 units upwards.
(c) Indicate the coordinates of the(c) Indicate the coordinates of the
intercepts with the axes.intercepts with the axes.
(d)(d) Now draw the graph of the lineNow draw the graph of the line
translated 4 units downwards.translated 4 units downwards.
Indicate the coordinates of theIndicate the coordinates of the
intercepts with the axes.intercepts with the axes.

The gradient of a straight lineThe gradient of a straight line
 Consider the linear functionConsider the linear function y = 3x.y = 3x.
 Consider the pointsConsider the points (0; 0)(0; 0) andand (1; 3)(1; 3)
 change in y – valueschange in y – values
 change in x - valueschange in x - values
 change in x – valueschange in x – values
 change in x - valueschange in x - values
 Consider the pointsConsider the points (-1; -3)(-1; -3) andand (2; 6)(2; 6)
 change in x – valueschange in x – values
 What can you concludeWhat can you conclude
aboutabout change in y – valueschange in y – values
change in x – valueschange in x – values
(called the gradient) between(called the gradient) between
any two points on the lineany two points on the line y = 3x?y = 3x?

The Gradient-Intercept MethodThe Gradient-Intercept Method
 We can easily sketch lines using theWe can easily sketch lines using the
concept of gradient and the y-interceptconcept of gradient and the y-intercept
of the line.of the line.

Example 6Example 6
 Sketch the following lines using theSketch the following lines using the
gradient-intercept methodgradient-intercept method
andand
 Consider:Consider:
clearlyclearly ::
xxf
2
1
)( = xxg
2
1
)( −=
2
1
+=m
xy
2
1
=

 The positive sign indicates that the lineThe positive sign indicates that the line
slopes to the right.slopes to the right.
 The y-intercept is 0.The y-intercept is 0.
 The numerator tells us to rise up 1 unitThe numerator tells us to rise up 1 unit
from the y - intercept.from the y - intercept.
 The denominator tells us to run 2 unitsThe denominator tells us to run 2 units
to the right.to the right.
 Clearly m =.Clearly m =.
xy
2
1
−=
2
1
−

The negative sign indicates that the line slopes to the left.
The y-intercept is 0.
The numerator tells us to rise up 1 unit from the y -
intercept.
The denominator tells us to run 2 units to the left.

Example 7Example 7
 Sketch using the gradient – intercept methodSketch using the gradient – intercept method
(a)(a) 2x = 3y2x = 3y rewrite the equation in the formrewrite the equation in the form
y = m x + cy = m x + c
3y = 2x3y = 2x
y =y =
 Rise up 2 unitsRise up 2 units
 Run 3 units to the rightRun 3 units to the right
x
3
2

(b)(b) 3x + y = 03x + y = 0
y = - 3xy = - 3x
 Rise up 3 unitsRise up 3 units
 Run 1 unit to the leftRun 1 unit to the left

Horizontal and Vertical LinesHorizontal and Vertical Lines
 Discovery Exercise 3Discovery Exercise 3
 Sketch the graph of the following relation:Sketch the graph of the following relation:
{(x; y)/ y = 2; x, y]}{(x; y)/ y = 2; x, y]}
 In this relation, it is clear that the x-values canIn this relation, it is clear that the x-values can
vary but the y - values must always remain 2.vary but the y - values must always remain 2.
xx -1-1 00 11 22
yy 22 22 22 22

 It is clear from the above graph that theIt is clear from the above graph that the
line is horizontal.line is horizontal.
 Lines which cut the y - axis and areLines which cut the y - axis and are
parallel to the x - axis have equationsparallel to the x - axis have equations
of the form:of the form:
y = numbery = number

 Sketch the graph of the following relation:Sketch the graph of the following relation:
{(x; y)/ x = 2; x, ]}{(x; y)/ x = 2; x, ]}
 In this relation, it is clear that the y - values can varyIn this relation, it is clear that the y - values can vary
but the x - values must alwaysbut the x - values must always
remain 2.remain 2.
xx 22 22 22 22
yy -1-1 00 11 22
It is clear from the above graph that the
line is vertical.
Lines which cut the x - axis and are
parallel to the y - axis have equations of
the form:
x = numberx = number

Example 8Example 8
 Sketch the graphs of the following linesSketch the graphs of the following lines
on the same set of axes:on the same set of axes:
 x + l = 0x + l = 0 andand y – 3 = 0y – 3 = 0
 x + l = 0x + l = 0 y = 3y = 3
 x = -1x = -1

Remember:Remember:
 The gradient of a horizontal line isThe gradient of a horizontal line is
always zero.always zero.
 The gradient of a vertical line is alwaysThe gradient of a vertical line is always
undefined.undefined.

EXERCISE 2EXERCISE 2
1. Sketch the graphs of the following linear1. Sketch the graphs of the following linear
functions by using the gradient - interceptfunctions by using the gradient - intercept
method:method:
(a)(a) (d)(d) x – 5y = 0x – 5y = 0
(b)(b) (e)(e) y – x = 0y – x = 0
(c)(c) (f)(f)
2. Sketch the graphs of the following on the2. Sketch the graphs of the following on the
same set of axes:same set of axes:
(a)(a) x = - 3x = - 3 (d)(d) y + 2 = 0y + 2 = 0
(b)(b) x – 3 = 0x – 3 = 0 (e)(e) x + 1 = 0x + 1 = 0
(c)(c) y = 5y = 5 (f)(f) y – l = 3y – l = 3
xy
4
3
=
yx
3
1
−=
xy
4
3
−= 1
2
1
+−= xy

 3. Draw neat sketch graphs of the following3. Draw neat sketch graphs of the following
using any method of your choice:using any method of your choice:
(a)(a) x + y = 2x + y = 2 (f)(f) x + 4 = 0x + 4 = 0
(b)(b) x – y = 3x – y = 3 (g)(g) y + 2 = 0y + 2 = 0
(c)(c) x + 2y = 6x + 2y = 6 (h)(h) f (x) =f (x) =
(d)(d) x = – 2yx = – 2y (i)(i) 2x + 2y = - 22x + 2y = - 2
(e)(e) y =y = (j)(j) 3x - 2y + 6 = 03x - 2y + 6 = 0x
4
1
−
1
3
2
+x

Finding the equation of a lineFinding the equation of a line
 Determine the equation of theDetermine the equation of the
following line in the formfollowing line in the form
y = m x + c :y = m x + c :
 ThereforeTherefore c = 3.c = 3.
 y = m x + 3y = m x + 3
 Substitute the pointSubstitute the point (8 ; - 1)(8 ; - 1)
 - l = m(8) + 3- l = m(8) + 3
 - 1 = 8m + 3- 1 = 8m + 3
 - 8m = 4- 8m = 4
 m =m =
Therefore the equation is :Therefore the equation is :
y = x +3y = x +3
2
1
−
2
1
−

 Determine the equation of the following lineDetermine the equation of the following line
in the form y = mx + c:in the form y = mx + c:
 Method 1Method 1
 ThereforeTherefore c = 4.c = 4.
 y = mx + 4y = mx + 4
 Substitute the pointSubstitute the point (x; y) = (- 2 ; 0)(x; y) = (- 2 ; 0)
 into the equationinto the equation y = mx + 4y = mx + 4 to get m:to get m:
 0 = m(- 2) + 40 = m(- 2) + 4
 0 = - 2m + 40 = - 2m + 4
 2m = 42m = 4
 m = 2m = 2
 Therefore the equation isTherefore the equation is y = 2x + 4y = 2x + 4

 ThereforeTherefore c =4.c =4.
 y = mx + 4y = mx + 4
 The gradient of the line is:The gradient of the line is:
 RiseRise
 RunRun
 ==

 Therefore the equation isTherefore the equation is y = 2x + 4y = 2x + 4
2
2
4
=

 Use the formula for gradient fromUse the formula for gradient from
Analytical Geometry.Analytical Geometry.
 (- 2; 0)(- 2; 0) andand (0; 4)(0; 4)
 Gradient =Gradient =
 The y – intercept is 4.The y – intercept is 4.
 So,So, y = 2x + 4y = 2x + 4
2
2
4
)2(0
04
==
−−
−

 Determine the equations of theDetermine the equations of the
following lines:following lines:
 (a)(a) (b)(b)
 (c )(c ) (d)(d)

 2.2. (a)(a) Determine the equation of the lineDetermine the equation of the line
passing through the pointpassing through the point
(-1; - 2)(-1; - 2) and cutting the y-axis at 1.and cutting the y-axis at 1.
(b)(b) Determine the equation of the line with aDetermine the equation of the line with a
gradient of - 2 and passing through thegradient of - 2 and passing through the
pointpoint (2; 3).(2; 3).
(c)(c) Determine the equation of the line whichDetermine the equation of the line which
cuts the x-axis at 5 and the y - axis at - 5.cuts the x-axis at 5 and the y - axis at - 5.
(d)(d) Determine the equation of the line whichDetermine the equation of the line which
cuts the x-axis at – 3 and the y - axis atcuts the x-axis at – 3 and the y - axis at 99..

Intersecting linesIntersecting lines
 If you have the equations of two linearIf you have the equations of two linear
functions and you sketch the graphs of thefunctions and you sketch the graphs of the
two lines, it is possible to determine thetwo lines, it is possible to determine the
coordinates of the point of intersection ofcoordinates of the point of intersection of
these lines by either:these lines by either:
 reading off the solution graphicallyreading off the solution graphically
oror
 using simultaneous equations to determineusing simultaneous equations to determine
the solution algebraically.the solution algebraically.
 The following exercise will illustrate this forThe following exercise will illustrate this for
you.you.

 1.1. (a)(a) Draw neat sketch graphs of theDraw neat sketch graphs of the
following lines on the same set offollowing lines on the same set of
axes:axes: x + y = 3x + y = 3 andand x - y = -1.x - y = -1.
 Hence write down the coordinates ofHence write down the coordinates of
the point of intersection of thesethe point of intersection of these
lines.lines.
 (b)(b) SolveSolve x + y = 3x + y = 3 andand x - y = - 1x - y = - 1 usingusing
the method of simultaneousthe method of simultaneous
equations.equations.
What do you notice about theWhat do you notice about the
solutions?solutions?

 2.2. Determine the coordinates of theDetermine the coordinates of the
point of intersection of the followingpoint of intersection of the following
pairs of lines:pairs of lines:
x + 2y = 5x + 2y = 5 andand x- y = - 1x- y = - 1

 3.3. The graphs of two linear functionsThe graphs of two linear functions
are represented below.are represented below.
 Determine the coordinates of theDetermine the coordinates of the
point of intersection.point of intersection.

2
≠2
2
2
THE QUADRATIC FUNCTIONTHE QUADRATIC FUNCTION
 The graphs of theseThe graphs of these
functions are calledfunctions are called
parabolas and have theparabolas and have the
general equationgeneral equation y =y =
ax + qax + q where a 0.where a 0.
 Investigation 1Investigation 1
 Complete the followingComplete the following
table and then draw thetable and then draw the
graphs of each functiongraphs of each function
on the set of axeson the set of axes
provided below.provided below.
XX -2-2 -1-1 00 11 22
XX
2X2X
3X3X

2
2
Conclusion:
(a) How does the value of a in the equation y = ax
+ q affect the shape of the parabolas?
(b) What is the y-intercept of each graph drawn and what variable in the equation y = ax
+ q tell you what it is?
(c) What it the turning point of each parabola?
(d) Write down the equation of the line of symmetry of the parabolas.
(e) As read from left to right, for which values of x will the graphs of the parabolas
decrease and increase?
(f) Do the parabolas have a maximum or minimum y-value and what is this
maximum or minimum value?

Investigation 2Investigation 2
 Complete the followingComplete the following
table and then draw thetable and then draw the
graphs of eachgraphs of each
function on the set offunction on the set of
axes provided below.axes provided below.
xx -2-2 -1-1 00 11 22
xx
2x2x
3x3x
2
2

 (a) How does the value of a in the equation(a) How does the value of a in the equation y = ax + qy = ax + q
affect the shape of the parabolas?affect the shape of the parabolas?
 (b) What is the y-intercept of each graph drawn and(b) What is the y-intercept of each graph drawn and
what variable in the equationwhat variable in the equation y = ax + qy = ax + q tells youtells you
what it is?what it is?
 (c)(c) What it the turning point of each parabola?What it the turning point of each parabola?
 (d) Write down the equation of the line of symmetry(d) Write down the equation of the line of symmetry
of theof the parabolas.parabolas.
 (e)(e) As read from left to right, for which values of x willAs read from left to right, for which values of x will
the graphs of the parabolas decrease and increase?the graphs of the parabolas decrease and increase?
 (f) Do the parabolas have a maximum or minimum y -(f) Do the parabolas have a maximum or minimum y -
value and what is this maximum or minimumvalue and what is this maximum or minimum value?value?
2
2

2
2
2
2
Investigation 3Investigation 3
 Complete theComplete the
following table andfollowing table and
then draw thethen draw the
graphs of eachgraphs of each
function on the setfunction on the set
of axes providedof axes provided
below.below.
xx -1-1 00 11
xx
x + 2x + 2
x - 1x - 1
x - 4x - 4

(a) How does the value of a in the equation(a) How does the value of a in the equation y = ax+ qy = ax+ q
affect the shape of the parabolas?affect the shape of the parabolas?
(b) What is the y - intercept of each graph drawn and(b) What is the y - intercept of each graph drawn and
what variable in the equationwhat variable in the equation y = ax+ qy = ax+ q tells you whattells you what
it is?it is?
(c)(c) What it the turning point of each parabola?What it the turning point of each parabola?
(d) Write down the equation of the line of symmetry of(d) Write down the equation of the line of symmetry of
the parabolas.the parabolas.
(e) As read from left to right, for which values of x will(e) As read from left to right, for which values of x will
the graphs of the parabolas decrease and increase?the graphs of the parabolas decrease and increase?
(f)(f) Do the parabolas have a maximum or minimum y-Do the parabolas have a maximum or minimum y-
value and what is this maximum or minimum valuevalue and what is this maximum or minimum value
for each graph?for each graph?
(g) How does the graph of(g) How does the graph of y = x+ 2, y = x- 1y = x+ 2, y = x- 1 andand y = x- 4y = x- 4
relate to the “mother” graph y = x in terms ofrelate to the “mother” graph y = x in terms of
translations?translations?
(h) Show algebraically how to determine the x -(h) Show algebraically how to determine the x -
intercepts of the graphsintercepts of the graphs y = x- 1y = x- 1 andand y = x- 4.y = x- 4.

Example 1Example 1
 Consider the functionConsider the function
 (a)(a) Write down the coordinates of the y - intercept.Write down the coordinates of the y - intercept.
 (b)(b) Determine algebraically the coordinates of the x -Determine algebraically the coordinates of the x -
intercepts.intercepts.
 (c)(c) Sketch the graph of on a set of axes.Sketch the graph of on a set of axes.
 (d)(d) Determine:Determine:
 (1)(1) the turning pointthe turning point
 (2)(2) the minimum valuethe minimum value
 (3)(3) the domain and rangethe domain and range
 (4)(4) the axis of symmetrythe axis of symmetry
 (5)(5) the values of x for which f increasesthe values of x for which f increases
 (e)(e) Write down the equation of the graph formed if isWrite down the equation of the graph formed if is
shifted 10 units upwards.shifted 10 units upwards.
 (f)(f) Write down the equation of the graph formed if isWrite down the equation of the graph formed if is
reflected about the x - axis.reflected about the x - axis.
 Draw this newly formed graph on the same set of axes asDraw this newly formed graph on the same set of axes as
9)( 2
−= xxf
9)( 2
−= xxf
9)( 2
−= xxf
9)( 2
−= xxf

Solutions to exampleSolutions to example
 (a)(a) In the equation , the y-intercept is - 9.In the equation , the y-intercept is - 9.
 The coordinates of the y - intercept are thusThe coordinates of the y - intercept are thus (0; -9).(0; -9).
 (b)(b) For the x - intercepts,For the x - intercepts,
 letlet y = 0y = 0
 0 = (x + 3)(x - 3)0 = (x + 3)(x - 3)
 x + 3 = 0 or x - 3=0x + 3 = 0 or x - 3=0
 x= - 3 or x = 3x= - 3 or x = 3
 The coordinates of the x-intercepts areThe coordinates of the x-intercepts are
(- 3; 0)(- 3; 0) andand (3; 0)(3; 0)
 (c)(c) The “mother” graph in this example is andThe “mother” graph in this example is and
the graph ofthe graph of
 is the translation of by 9 units downwards.is the translation of by 9 units downwards.
 Therefore sketch the graph of firstTherefore sketch the graph of first
and then translate it downward by 9 units.and then translate it downward by 9 units.
 Indicate the intercepts with the axes.Indicate the intercepts with the axes.
9)( 2
−= xxf
92
−= xy
92
−= xy
2
xy =
2
xy =
2
xy =

 (d)(d) (1)(1) Turning point isTurning point is (0; - 9)(0; - 9)
 (2)(2) Minimum value is - 9Minimum value is - 9
 (3)(3) Domain Range isDomain Range is
(4)(4) The axis of symmetry is the y - axis which has an equation x = 0The axis of symmetry is the y - axis which has an equation x = 0
 (5)(5) The graph of f increases forThe graph of f increases for x > 0x > 0
 (6)(6) The graph of f decreases forThe graph of f decreases for x < 0x < 0
 (e)(e) If the graph of is shifted upward by 10 units, the newIf the graph of is shifted upward by 10 units, the new
graph formed will have an equation (you added 10 unitsgraph formed will have an equation (you added 10 units
to the y intercept)to the y intercept)
 (f)(f) If the graph of is reflected about the x- axis, then theIf the graph of is reflected about the x- axis, then the
 y coordinates of the points will change sign as was doney coordinates of the points will change sign as was done
in transformation geometry.in transformation geometry.
The equation of the reflected graph is obtained byThe equation of the reflected graph is obtained by
changing the sign of the y in the equation of :changing the sign of the y in the equation of :
or we can write this asor we can write this as ::
92
−=− xy
92
+−= xy

10. functions

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