MODULE - 10 MODULE - 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

qx

ay

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 or intervals on which the function increases or

decreasesdecreases the discrete or continuous nature of the the discrete or continuous nature of the

graph (LO 2 AS 3)graph (LO 2 AS 3)

Solve linear equations in two variables Solve linear equations in two variables simultaneouslysimultaneously

(LO 2 AS 5e)(LO 2 AS 5e)

RelationsRelations A relation is any rule by means of which each A relation is any rule by means of which each

element of a first set is associated with at least element of a first set is associated with at least one element of a second set. one element of a second set.

For example, suppose that in a given relation, a For example, suppose that in a given relation, a first set is first set is {- 2; - 1; 0 ; 1; 3}{- 2; - 1; 0 ; 1; 3} and that the second and that the second set is obtained by using the rule set is obtained by using the rule y = 2x.y = 2x.

By substituting the given x-values into the By substituting the given x-values into the equation equation y = 2x,y = 2x, it will be possible to determine it will be possible to determine the second set, which contains the the second set, which contains the corresponding y-values. corresponding y-values.

The relation can be represented in different The 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 above In set- builder notation the above relation can be represented as follows:relation can be represented as follows:

This is read as “the set of all This is read as “the set of all xx and and yy such that such that y = 2y = 2 and x is an element and x is an element

of of {-2 ;-1;0;l ;2;3}{-2 ;-1;0;l ;2;3}”.”.

(x; y)(x; y) states that there is a relationship states that there is a relationship between between xx and and 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 elements If we now had to increase the elements of the first set to include all real of the first set to include all real numbers for numbers for x,x, there would be so many there would be so many points that could be represented on the points that could be represented on the Cartesian number plane. In fact, the Cartesian number plane. In fact, the points would be so close that we would points would be so close that we would get what is called the graph of a get what is called the graph of a straight line straight line y = 2xy = 2x for all for all x.x.

See if you can draw the graph of this See if you can draw the graph of this relation for all relation 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 is The set of numbers to which we apply the rule is

referred to as the domain. The set of numbers referred to as the domain. The set of numbers obtained as a result of using the rule is referred to as obtained as a result of using the rule is referred to as the range. For the previous relation the range. For the previous relation y = 2xy = 2x , the , the domain is the set of x-values used, whereas the domain is the set of x-values used, whereas the range is the set of y-values obtained. The range range is the set of y-values obtained. The range depends on the domain used.depends on the domain used.

In the relation, the domain In the relation, the domain used is clearly the set used 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-values The 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 vertical and Keep the edge of the ruler vertical and

slide it across the graph from left to right. Where the slide it across the graph from left to right. Where the edge starts cutting the graph, the domain starts (as edge starts cutting the graph, the domain starts (as read off from the x-axis).read off from the x-axis).

Where it stops cutting the graph the domain ends.Where it stops cutting the graph the domain ends. For range:For range: Keep the edge of the ruler horizontal and Keep the edge of the ruler horizontal and

slide it across the graph from bottom to top.slide it across the graph from bottom to top. Where the edge starts cutting the graph, the range Where the edge starts cutting the graph, the range

starts 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 of For each of the following graphs of given relations, determine the domain given relations, determine the domain and range and range

(a)(a)

(b)(b)

{(-2;16); (0; 4); (1; 4); (3; 7)}{(-2;16); (0; 4); (1; 4); (3; 7)} Here, each element of the domain is associated with Here, each element of the domain is associated with

only one element of the range. The numbers only one element of the range. The numbers 0 0 and and 11 are associated with the same element of the range are 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 number Here, the number 4 4 in the domain is in the domain is

associated with more than one element of the associated with more than one element of the range range (1 and 6).(1 and 6). In this case, the relation is In 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 “vertical We use a ruler to perform the “vertical line test” on a graph to see whether it is line test” on a graph to see whether it is a function or not. Hold a clear plastic 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 place If 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 following Determine whether the following relations are functions or not.relations are functions or not.

(a)(a)

(b)(b)

(c)(c)

(d)(d)

(e)(e)

(f)(f)

(g)(g)

(h)(h)

MappingMapping and functional notationand functional notation

Since functions are special relations, Since functions are special relations, we reserve certain notation strictly for we reserve certain notation strictly for use when dealing with functions. use when dealing with functions.

Consider the function Consider the function f = {(x; y)/ y = 3x),f = {(x; y)/ y = 3x), This function may be represented by This function may be represented by

means of mapping notation or means of mapping notation or functional notation.functional notation.

Mapping notationMapping notation

This is read as “f maps This is read as “f maps xx onto onto 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 the then the corresponding element in the range is range is 3(2) = 6.3(2) = 6.

We say that We say that 66 is the image of is the image of 22 in the in the mapping of mapping 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 the The 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 x -In 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, if For example, if x = 4,x = 4, then the corresponding y-value then the corresponding y-value

is obtained by substituting is obtained by substituting x = 4x = 4 into into 3x.3x. For For x = 4,x = 4, the y - value is f the y - value is f (4) = 3(4) = 12.(4) = 3(4) = 12. The brackets in the symbol f (4) do not mean f times The brackets in the symbol f (4) do not mean f times

4, but rather the y-value at 4, 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 by Suppose 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

If If Determine the value of:Determine the value of:(a)(a) f f (2)(2)

(b)(b) f f (-3)(-3)

(c)(c) f f

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

21

3

11

9

131

3

13)

3

1(

2

f

13)( 2 xxf

EXERCISEEXERCISE

1. Determine the domain and range of the 1. Determine the domain and range of the following relations. following relations.

State whether the relation is a function State whether the relation is a function or not.or not.

(a) (b)(a) (b)

(c) (d)(c) (d)

(e) (e)

(f)(f)

2. 2. Consider the function Consider the function Determine and then represent Determine 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

2f

3. If 3. 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

1f

2

1f

12)( 2 xxxf

THE LINEAR FUNCTIONTHE LINEAR FUNCTION

The graph of a linear function is a The graph of a linear function is a straight line. straight line.

The equation of a linear function takes The equation of a linear function takes the form the form y = m x + c.y = m x + c.

The Table MethodThe Table Method

Example 1Example 1

Sketch the graph of Sketch the graph of y = x - 2y = x - 2 by using the table by 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 by The y-values were found by

substituting the substituting the x -x - values into the values into the equation equation y = x - 2.y = x - 2.

Note:Note: The line cuts the x-axis at the point The line cuts the x-axis at the point (2; 0).(2; 0). This point represents the coordinates of the This point represents the coordinates of the

x-intercept.x-intercept. For the x - intercept of any line, it is clear that For the x - intercept of any line, it is clear that

the y-value is always the y-value is always 0.0. The line cuts the y-axis at the point The line cuts the y-axis at the point (0; - 2).(0; - 2). This point represents the coordinates of the This point represents the coordinates of the

y-intercept. y-intercept. For the y - intercept of any line, it is clear that For the y - intercept of any line, it is clear that

the x- value is always the x- value is always 0.0.

The Dual-Intercept MethodThe Dual-Intercept Method Example 2Example 2 Sketch the graph of Sketch 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 intercepts This 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) = 62x = 62x = 6 x = 3x = 3(3; 0)(3; 0)

y - intercept:y - intercept: let let x = 0x = 0- 3y = - 2x + 6 - 3y = - 2x + 6 y =y =

y =y =

(0; -2)(0; -2)

23

2x

223

)0(2

The x-intercept is the point The x-intercept is the point (3; 0)(3; 0) The y-intercept is the point The y-intercept is the point (0; - 2)(0; - 2)

Note:Note: The dual-intercept method will not work with The dual-intercept method will not work with

lines that cut the axes at the origin, i.e. at the lines that cut the axes at the origin, i.e. at the pointpoint

(0; 0).(0; 0). In these cases, use the table method to In these cases, use the table method to

sketch the graph.sketch the graph.

Example 3Example 3 Sketch the graph of Sketch 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 = 0y – 2x = 0y – 2x = 0y = 2xy = 2xy = 2(0)y = 2(0)y = 0y = 0

The x-intercept is the point The x-intercept is the point (0; 0).(0; 0). The y-intercept is the point The y-intercept is the point (0; 0)(0; 0) This line cuts the axes at the origin. We will This 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 as We 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 same Sketch the graphs of the following linear functions on the same

set of axes using either the dual-intercept method or the table set 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: xxh2: xxj

Now answer the following questions based on Now answer the following questions based on your graphs:your graphs:

(a) (a) What do you notice about the slope of each What do you notice about the slope of each line drawn?line drawn?

(b) What do you notice about the coefficient of x (b) What do you notice about the coefficient of x in each equation?in each equation?

(c) (c) So what can you conclude about the So what can you conclude about the coefficient of coefficient of x in each equation?x in each equation?

(d) What is the y - intercept of each line and how (d) What is the y - intercept of each line and how do these y - intercepts relate to the equations?do these y - intercepts relate to the equations?

(e) (e) In terms of translations, how can the graphs In 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?

Discovery Exercise 2Discovery Exercise 2

Sketch the graphs of the following Sketch the graphs of the following linear functions on the same set of linear functions on the same set of axes using either the dual - intercept axes using either the dual - intercept method or the table method where the method 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 based Now answer the following questions based on your graphs:on your graphs:

(a) (a) What do you notice about the slope What do you notice about the slope of each line drawn?of each line drawn?

(b) (b) What do you notice about the What 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 the So 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 line What is the y - intercept of each line and how do these y - intercepts and how do these y - intercepts relate to the equations?relate to the equations?

(e) (e) In terms of translations, how can the In terms of translations, how can the graphs of g, h and j be drawn using graphs of g, h and j be drawn using

the graph of f ?the graph of f ?

ConclusionConclusion For any linear function written in the form For 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 gradient The coefficient of x, namely in, represents the slope or gradient

of the line. of the line. If If m > 0,m > 0, the lines slope to the right. the lines slope to the right. If If 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 of The graph of y = mx + c is the translation of the graph of

y = m x. y = m x. If If c > 0c > 0, the graph of y = , the graph of y = m x + cm x + c shifts by shifts by c c

units upwards units upwards If If c < 0,c < 0, the graph of y = the graph of y = m x + cm x + c shifts by shifts by cc units downwardsunits downwards

Example 4Example 4 Rewrite the following equations in the form y = m x + c and Rewrite 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 slope the 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 point The 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

13

4x

(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 point The y-intercept is the point (0; 2).(0; 2). The gradient is The gradient is m =m = which is which is

negative. negative. The line slopes to the left.The line slopes to the left.

22)0(4

32

4

3

x

4

3

24

3 x

Example 5Example 5(a) (a) Sketch the graph of Sketch 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 is Now draw the graph of the line formed if the graph off is

translated 2 units upwards. Indicate the coordinates of the translated 2 units upwards. Indicate the coordinates of the intercepts with the axes as well as the equation of the newly intercepts 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 is Now draw the graph of the line formed if the graph off is translated 3 units downwards. Indicate the coordinates of the translated 3 units downwards. Indicate the coordinates of the intercepts with the axes as well as the equation of the newly intercepts with the axes as well as the equation of the newly formed line.formed line.

EXERCISE 1EXERCISE 11.1. Draw neat sketch graphs of the following Draw neat sketch graphs of the following

linear functions on separate axes. linear functions on separate axes. Use the dual-intercept method or where Use 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 Without actually sketching the graphs of of the the following linear functions, following linear functions, determine, by rewriting the equations in determine, by rewriting the equations in the form the form y = m x + c,y = m x + c, the gradient and y - the gradient and y - intercept of each line and state the intercept of each line and state the direction of the slope of each line.direction 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 of Sketch the graph of f (x) = x - 1f (x) = x - 1 on a on a set of axes.set of axes.

(b)(b) Now draw the graph of the line Now draw the graph of the line formed if the graph of f isformed if the graph of f istranslated 2 units upwards. Indicate translated 2 units upwards. Indicate the coordinates of the intercepts with the coordinates of the intercepts with

the the axes as well as the equation of the axes as well as the equation of the newlynewly

formed line.formed line. (c)(c) Now draw the graph of the line Now draw the graph of the line

formed if the graph of f isformed if the graph of f istranslated 3 units downwards. translated 3 units downwards. Indicate the coordinates of theIndicate the coordinates of theintercepts with the axes as well asintercepts with the axes as well asthe equation of the newlythe equation of the newlyformed line.formed line.

4.4. (a)(a) Sketch the graph of Sketch 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 line Now draw the graph of the line formed if the graph of f isformed if the graph of f is

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 line Now draw the graph of the line formed if the graph of f isformed if the graph of f is

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 function Consider the linear function y = 3x.y = 3x. Consider the points Consider the points (0; 0)(0; 0) and and (1; 3)(1; 3) change in y – values change in y – values change in x - valueschange in x - values Consider the points Consider the points (0; 0)(0; 0) and and (2; 6)(2; 6) change in y – values change in y – values change in x – valueschange in x – values Consider the points Consider the points (1; 3)(1; 3) and and (2; 6)(2; 6) change in y – values change in y – values change in x - valueschange in x - values Consider the points Consider the points (-1; -3)(-1; -3) and and (2; 6)(2; 6) change in y – values change in y – values change in x – valueschange in x – values

What can you concludeWhat can you conclude about about change in y – values change in y – values change in x – valueschange in x – values (called the gradient) between (called the gradient) between any two points on the line any two points on the line y = 3x?y = 3x?

The Gradient-Intercept MethodThe Gradient-Intercept Method

We can easily sketch lines using the We can easily sketch lines using the concept of gradient and the y-intercept concept of gradient and the y-intercept of the line.of the line.

Example 6Example 6

Sketch the following lines using the Sketch the following lines using the gradient-intercept methodgradient-intercept method

andand Consider:Consider:

clearlyclearly : :

xxf2

1)( xxg

2

1)(

2

1m

xy2

1

The positive sign indicates that the line The 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 unit The numerator tells us to rise up 1 unit

from the y - intercept.from the y - intercept. The denominator tells us to run 2 units The denominator tells us to run 2 units

to the right.to the right.

Clearly m =.Clearly m =.

xy2

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 form rewrite the equation in the form

y = m x + c y = 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

x3

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 can In 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 the It is clear from the above graph that the line is horizontal.line is horizontal.

Lines which cut the y - axis and are Lines which cut the y - axis and are parallel to the x - axis have equations of parallel to the x - axis have equations of the form:the form:

y = numbery = number

Discovery Exercise 4Discovery Exercise 4 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 vary In 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 and and y – 3 = 0y – 3 = 0 x + l = 0x + l = 0 y = 3y = 3 x = -1x = -1

Remember:Remember:

The gradient of a horizontal line is The gradient of a horizontal line is always zero.always zero.

The gradient of a vertical line is always The gradient of a vertical line is always undefined.undefined.

EXERCISE 2EXERCISE 2

1. Sketch the graphs of the following linear 1. Sketch the graphs of the following linear functions by using the gradient - intercept functions by using the gradient - intercept method:method: (a)(a) (d)(d) x – 5y = 0 x – 5y = 0 (b)(b) (e)(e) y – x = 0 y – x = 0 (c)(c) (f) (f)

2. Sketch the graphs of the following on the 2. 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

xy4

3

yx3

1

xy4

3 1

2

1 xy

3. Draw neat sketch graphs of the following 3. 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 = 0x4

1

13

2x

Finding the equation of a lineFinding the equation of a line Example 1Example 1 Determine the equation of the Determine the equation of the

following line in the form following line in the form y = m x + c :y = m x + c : The y-intercept is 3.The y-intercept is 3. Therefore Therefore c = 3.c = 3. y = m x + 3y = m x + 3 Substitute the point Substitute 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

Example 2Example 2 Determine the equation of the following line Determine the equation of the following line

in the form y = mx + c:in the form y = mx + c: Method 1Method 1 The y-intercept is 4.The y-intercept is 4. Therefore Therefore c = 4.c = 4. y = mx + 4y = mx + 4 Substitute the point Substitute the point (x; y) = (- 2 ; 0)(x; y) = (- 2 ; 0) into the equation into 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 is Therefore the equation is y = 2x + 4y = 2x + 4

Method 2Method 2 The y-intercept is 4.The y-intercept is 4. Therefore Therefore 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 is Therefore the equation is y = 2x + 4y = 2x + 4

22

4

Method 3Method 3 Use the formula for gradient from Use the formula for gradient from

Analytical Geometry.Analytical Geometry. (- 2; 0)(- 2; 0) and and (0; 4)(0; 4) Gradient = Gradient = The y – intercept is 4.The y – intercept is 4. So, So, y = 2x + 4y = 2x + 4

22

4

)2(0

04

EXERCISE 3EXERCISE 3 Determine the equations of the Determine 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 line Determine 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 a Determine the equation of the line with a gradient of - 2 and passing through the gradient of - 2 and passing through the

point point (2; 3).(2; 3). (c)(c) Determine the equation of the line which Determine 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 which Determine 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 linear If you have the equations of two linear

functions and you sketch the graphs of the functions and you sketch the graphs of the two lines, it is possible to determine the two lines, it is possible to determine the coordinates of the point of intersection of coordinates of the point of intersection of these lines by either:these lines by either:

reading off the solution graphicallyreading off the solution graphicallyoror

using simultaneous equations to determine using simultaneous equations to determine the solution algebraically.the solution algebraically.

The following exercise will illustrate this for The following exercise will illustrate this for you.you.

EXERCISE 4EXERCISE 4 1.1. (a)(a) Draw neat sketch graphs of the Draw neat sketch graphs of the

following lines on the same set offollowing lines on the same set ofaxes: axes: x + y = 3x + y = 3 and and x - y = -1.x - y = -1.

Hence write down the coordinates Hence write down the coordinates of of the point of intersection of these the point of intersection of these

lines.lines. (b) (b) Solve Solve x + y = 3x + y = 3 and and x - y = - 1x - y = - 1 using using

the method of simultaneous the method of simultaneous equations. equations. What do you notice about the What do you notice about the solutions?solutions?

2. 2. Determine the coordinates of the Determine the coordinates of the point of intersection of the following point of intersection of the following pairs of lines:pairs of lines:

x + 2y = 5x + 2y = 5 and and x- y = - 1x- y = - 1

3. 3. The graphs of two linear functions The graphs of two linear functions are represented below. are represented below.

Determine the coordinates of the Determine the coordinates of the point of intersection.point of intersection.

2

2

2

2

THE QUADRATIC FUNCTIONTHE QUADRATIC FUNCTION

The graphs of these The graphs of these functions are called functions are called parabolas and have the parabolas and have the general equation general equation y = y =

ax + qax + q where a 0. where a 0. Investigation 1Investigation 1 Complete the following Complete the following

table and then draw the table and then draw the graphs of each function graphs of each function on the set of axes on the set of axes provided below.provided below.

XX -2-2 -1-1 00 11 22

X X

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 following Complete the following table and then draw the table and then draw the graphs of each graphs of each function on the set of function on the set of axes provided below.axes provided below.

xx -2-2 -1-1 00 11 22

x x

2x 2x

3x3x

2

2

ConclusionConclusion (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 equation what variable in the equation y = ax + qy = ax + q tells you tells 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 the of the parabolas.parabolas. (e) (e) As read from left to right, for which values of x will 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) 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 minimum value and what is this maximum or minimum value?value?

2

2

2

2

2

2

Investigation 3Investigation 3 Complete the Complete the

following table and following table and then draw the then draw the graphs of each graphs of each function on the set function on the set of axes provided of axes provided below.below.

xx -1-1 00 11

xx

x + 2 x + 2

x - 1x - 1

x - 4x - 4

ConclusionConclusion(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 equation what variable in the equation y = ax+ qy = ax+ q tells you what tells 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 value value 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 and and y = x- 4y = x- 4 relate to the “mother” graph y = x in terms of relate 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 graphs intercepts of the graphs y = x- 1y = x- 1 and and 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 Write down the equation of the graph formed if

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

is is reflected about the x - axis. reflected about the x - axis. Draw this newly formed graph on the same set of axes Draw this newly formed graph on the same set of axes

asas

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 thus The coordinates of the y - intercept are thus (0; -9).(0; -9). (b) (b) For the x - intercepts, For the x - intercepts, let let 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 are The coordinates of the x-intercepts are

(- 3; 0)(- 3; 0) and and (3; 0)(3; 0) (c) (c) The “mother” graph in this example is and The “mother” graph in this example is and

the graph of the graph of is the translation of by 9 units downwards.is the translation of by 9 units downwards. Therefore sketch the graph of first Therefore 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

2xy 2xy

2xy

(d)(d) (1)(1) Turning point is Turning 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 = 0 The axis of symmetry is the y - axis which has an equation x = 0 (5)(5) The graph of f increases for The graph of f increases for x > 0x > 0 (6)(6) The graph of f decreases for The graph of f decreases for x < 0x < 0 (e) (e) If the graph of is shifted upward by 10 units, the new If the graph of is shifted upward by 10 units, the new

graph formed will have an equation (you added 10 graph formed will have an equation (you added 10 units units to the y intercept)to the y intercept)

(f) (f) If the graph of is reflected about the x- axis, then the If the graph of is reflected about the x- axis, then the y coordinates of the points will change sign as was done y coordinates of the points will change sign as was done

in transformation geometry.in transformation geometry.The equation of the reflected graph is The equation of the reflected graph is

obtained by 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 xy92 xy