KZN DEPARTMENT OF EDUCATION MATHEMATICS JUST IN …

1

KZN DEPARTMENT OF EDUCATION

MATHEMATICS

JUST IN TIME MATERIAL

GRADE 12

TERM 2 – 2020

TABLE OF CONTENTS

TOPIC PAGE

NO.

FUNCTIONS AND INVERSES

2 - 21

ANALYTICAL GEOMETRY

22 - 37

APPLICATION OF CALCULUS

38 – 46

This document has been compiled by the FET Mathematics Subject Advisors together with Lead

Teachers. It seeks to unpack the contents and to give more guidance to teachers.

2

FUNCTIONS AND INVERSES

Weighting in final

examinations 35 ±3 marks out of 150 marks in Paper 1

Methodology

HINTS TO TEACHERS: FOCUS ON THE FOLLOWING

CHARACTERISTICS:

Definition of a function, vertical line test

Domain and the Range

General concept of the inverse of a function

Restricting domain of MANY TO ONE function to make its INVERSE

a function

Intercepts with the axes

Turning points (Minima and Maxima)

Asymptotes (horizontal and vertical) and line of symmetry

Shape of the graph

Intervals on which the function INCREASES/DECREASES

How f(x) has been transformed to generate f(-x), -f(x), f(x+a), a.f(x) and

x = f(y) where a R

Errors/Misconceptions Learners confuse domain and range

Reflection about the x- axis, y-axis and the line y=x.

FROM ATP:

DATES CURRICULUM STATEMENT

09/4

(1 day)

1. Definition of a function.

2. General concept of the inverse of a function.

3. Determine and sketch graphs of the inverse of the function defined by

𝑦 = 𝑎𝑥 + 𝑞

4. Focus on the following characteristics:

domain and range, intercepts with the axes, shape and symmetry, gradient, whether the function

increases/decreases.

14/4 – 15/4

(2 days)


𝑦 = 𝑎𝑥2

6. Determine how the domain of the function may need to be restricted (in order to obtain a one-to-one

function) to ensure that the inverse is a function.


domain and range, intercepts with the axes, turning points, minima, maxima, shape and symmetry,

average gradient (average rate of change), intervals on which the function increases/decreases.

16/4 – 20/4

(3 days)


𝑦 = 𝑏𝑥 for 𝑏 > 0, 𝑏 ≠ 1.


domain and range, intercepts with the axes, asymptotes (horizontal and vertical), shape and symmetry,

average gradient (average rate of change), intervals on which the function increases/decreases.

10. Understand the definition of a logarithm:

𝑦 = 𝑙𝑜𝑔𝑏𝑥 ⟺ 𝑥 = 𝑏𝑦, where 𝑏 > 0 and 𝑏 ≠ 1.

The graph of the function defined by 𝑦 = 𝑙𝑜𝑔𝑏𝑥 for both the cases 0 < 𝑏 < 1 and 𝑏 > 1.

21/4 – 24/4

(4 days) Further sketching and interpretation of graphs of functions and their inverses.

3

FURTHER HINTS (OBJECTIVES) TO TEACHERS:

Be able to recognise the graph as being:

Linear ( qaxy ),

Quadratic(y = a(𝑥 + 𝑝)2 + q),

Hyperbolic (y = 𝑎

𝑥+𝑝 + q),

Exponential (y = a𝑏𝑥+𝑝 + q)

Understand the effects of the parameters a, p and q

Sketch the graphs, indicating all intercepts with the axes and turning points or asymptotes

(it is important to be able use and interpret functional notation)

Understand the definition of a FUNCTION

Understand the INVERSE of a function(swop x and y)

Know the inverse of xayaxyqaxy ,, 2 where a > 0

Be able to write the inverse in the form y = …

Understand the definition of a logarithm: y = log𝑏 𝑥 ↔ x = 𝑏𝑦 where b > 1

and b ≠ 1

Understand reflection about the y-axis (f(x) = f(-x)) and about the x-axis (f(x) = -f(x))

Understand reflection about the point (use mid-point formula)

Understand vertical translation (f(x) = f(x) + q) and horizontal translation

(f(x) = f(x + p))

If given a sketch of a function [f(x)], be able to determine the values of x for which:

1. f(x) < 0 OR f(x) > 0

2. f’(x) ≥ 0 OR f’(x) ≤ 0

3. 𝑓−1(x) < 0 OR 𝑓−1(x) ≥ 0

If given two functions (f(x) and g(x)) on the same set of axes, be able to determine the values of x for

which:

1. )()( xgxf 2. )()( xgxf OR 0)()( xgxf

3. )()( xgxf OR 0)()( xgxf 4. F(x).g(x)>0 OR f(x).g(x)<0

5. 𝑓(𝑥)

𝑔(𝑥)>0 6. G(x).𝑓1(x)>0

7. 𝑓−1(x) = 𝑔−1(x) 8. 𝑔′(x)>𝑓′(𝑥)

4

THE PARABOLA

In Grade 11 you were introduced to the parabola in the forms shown in the table below. The table below

summarises the properties of the parabola which you learnt in Grade 11.

FO

RM

S

y = a(x + p)2 + q y = ax2+bx+c y = a(x – x1)(x – x2)

In this form – p is the x-

value of the turning

point and q is the y-

value of the turning

point. The parabola is

symmetrical about its

turning point.

In this form c is the y-

intercept. The parabola is

symmetrical about its

turning point. The x-value

of the turning point and

axis of symmetry can be

found using equation

bx = -

2a.

In this form x1 and x2 are x-

intercepts of the parabola.

Remember the x-value of the

turning point lies halfway

between the x-intercepts

(i.e 1 2x + x

x =2

). This is

due to the symmetry of the

parabola about its turning

point

SH

AP

E

(a)

a > 0

concave up (happy face)

a < 0

concave down (sad face)

As the value of a increases from 0

the graph is compressed and gets

narrower.

Example:

x

y

f(x)=½x2

As the value of a decreases from 0 the graph

is compressed and gets narrower.

Example:

x

y

f(x)=-2x2

g(x)=-x2

h(x)=-½x2

VERTICAL

SHIFT

(q or c)

The value of q or c results in a vertical shift. If the value of q or c is greater than 0 the

graph will be shifted up. If q or c is less than 0 the graph will be shifted down.

DOMAIN

and

RANGE

The domain of a parabola is:

x ϵ ℝ or x ϵ ( ; )

The range of the parabola is:

y ≥ q or y ϵ[q; ) if a > 0

y ≤ q or y ϵ( ; q] if a < 0

5

THE HYPERBOLA

In Grade 11 you were introduced to the hyperbola in the form of y =a

+ qx + p

.

It was shown that a determines the shape of a hyperbola as well as quadrants which the hyperbola lies in.

The value of q resulted in a vertical shift (up or down) and the line y = q is the vertical asymptote. The value

of p resulted in a horizontal shift (left or right) and the line x = p is the vertical asymptote. The properties of

a hyperbola that you learnt in Grade 11 are summarised below:

a > 0 a < 0

SHAPE

(a)

axis of

symmetry

The hyperbola is symmetrical about the lines:

y = x + c and y = - x + c (use coordinates of the point of intersection of the

asymptotes, (-p ;q) to calculate value of c)

OR

y = x + p + q and y = - x – p + q

DOMAIN

And

RANGE

The domain of the hyperbola is:

x ϵ ℝ; x - p or x ϵ ( ; - p) (-p; )

The range of the hyperbola is:

y ϵ ℝ; y q or x ϵ ( ; q) (q; )

THE EXPONENTIAL FUNCTION

In Grade 11 you were introduced to exponential function in the form of:

y = a.bx+p + q where b > 0 and b 0

It was shown that the value of a determine the shape of the graph as well as on which side of the asymptote

the exponential function lie. Together with a, the value of b determines whether the function is increasing or

decreasing.

The value of q resulted in a vertical translation (up or down) and the line y = q is the horizontal asymptote.

The value of p resulted in a horizontal translation. The properties of the exponential function that you learnt

in Grade 11 are summarised below:

SH

AP

E (

a a

nd b

)

a > 0 and b>1 a < 0 and b > 1

The graph lies above the horizontal asymptote

and is an increasing function.

The graph lies below the horizontal

asymptote and is a decreasing function.

6

A > 0 and 0 < b < 1 a < 0 and 0 < b < 1

The graph lies above the horizontal asymptote

and is a decreasing function.

The graph lies below the horizontal

asymptote and is an increasing function

DO

MA

IN A

ND

RA

NG

E

The domain of the exponential function is:

x ϵ ℝ or ( ; )

The range of the exponential function is:

y > q or y ϵ (q ; ) if a > 0

y < q or y ϵ ( ;q) if a < 0

DEFINITION OF A FUNCTION

A function is defined as a relationship between values, where each input value maps to one output value. In

other words, for an equation to be called a function, there can only be one y-value for a particular x-value.

There are two types of functions: 1. One-to-one functions

A function where there is single y-value for a particular x-value.

One-to-one function

2. Many-to-one functions

A function cannot have more than one y-value to each x-value (One-to-many). However, a

function can have more than one x-value for a particular y-value. These are known as

many-to-one functions.

Many-to-one function One-to-many (not a function)

NB: A: Use a vertical line test to check if the relation is a function or not.

B: Use horizontal line test to check if the function is one-to-one or many-to-one.

7

INVERSE OF A FUNCTION

You obtain the inverse of a function by swopping the x and y values.

The domain becomes the range and the range becomes the domain.

A one-to-one function becomes a one-to-one inverse function, but a many-to-one function becomes a

one-to-many relation which is not a function.

For a many-to-one function (quadratic function/ parabola), you must restrict domain if you want to

make its inverse a function.

By swopping x and y values we get (x ; y) (y ; x). This means the graph is reflected about the line

y = x. A graph and its inverse are therefore always symmetrical about the line y = x.

You use f -1 to represent the inverse of f(x). Be careful not to mistake the -1 in f -1 for an exponent.

f -1(x) does not mean the reciprocal 1

f(x)

The inverse of an exponential function is a logarithmic function and a visa versa.

8

BASELINE ASSESSMENT

QUESTION 1 KZN JUNE 2019 GR11

The hyperbola, f, is defined by 3

( ) 12

f xx

.

1.1 Write down the equations of the asymptotes of f. (2)

1.2 Write down the y-intercept of f. (1)

1.3 Determine the x-intercept of f. (2)

1.4 Sketch the graph of f , showing all intercepts with the axes and any asymptotes. (3)

1.5 Write down the domain of h, if ( ) ( 2).h x f x (2)

[10]

QUESTION 2 KZN JUNE 2019 GR11

2.1 The following are graphs defined by 2 .y ax bx c

A B C

Match each of the following statements below with associated graph(s) above.

You must only write down the letter of the corresponding graph(s).

2.1.1 042 acb (1)

2.1.2 042 acb (1)

2.1.3 042 acb (1)

2.1.4 2 equal roots (1)

2.1.5 Non real roots (1)

2.1.6 2 unequal roots (1)

2.1.7 Concave down (1)

2.1.8 Concave up (1)

2.1.9 a > 0; b < 0 & c > 0 (1)

O

x

x

y

O

x

x

y

O

x x

y

9

2.2 The sketch below shows the graphs of xy k t and 2 .y ax bx c

The x-intercepts of h are (0 ; 0) and (0 ; 4). Q(1 ; –3) is a point on h.

The horizontal line through P, the turning point of h, is the asymptote of g.

The x-intercept of g lies on the axis of symmetry of h.

2.2.1 Show that 1a and 4.b

(3)

2.2.2 Determine the coordinates of P, the turning point of h.

(2)

2.2.3 Write down the range of h.

(1)

2.2.4 Write down the value of t.

(1)

2.2.5 Hence, calculate the value of .k

(2)

2.2.6 For which value(s) of r will the roots of 2 4x x r be non-real?

(2)

[11]

10

QUESTIONS FROM PAST EXAMINATION PAPERS

QUESTION 1 KZN SEPT 2019

Given 12

4)(

xxf

1.1 Write down the equations of the vertical and horizontal asymptotes of f. (2)

1.2 Determine the intercepts of the graph of f with the axes. (3)

1.3 Draw the graph of f. Show all intercepts with the axes as well as the asymptotes of the

graph.

(4)

[9]


In the diagram, the graphs of 65)( 2 xxxf and 1)( xxg are drawn below.

The graph of f intersects the x – axis at B and C and the y – axis at A.

The graph of g intersects the graph of f at B and S. PQR is perpendicular to the x – axis

with points P and Q on f and g respectively. M is the turning point of f.

2.1 Write down the co-ordinates of A. (1)

2.2 S is the reflection of A about the axis of symmetry of f. Calculate the coordinates of S. (2)

2.3 Calculate the coordinates of B and C. (3)

2.4 If PQ = 5 units, calculate the length of OR. (4)

2.5 Calculate the:

2.5.1 Coordinates of M. (4)

2.5.2 Maximum length of PQ between B and S. (4)

[18]

P

f

A

y

g

S

Q

x C B

R O

M

11

TEACHER ACTIVITIES


In the diagram, the graph of xxg 5log)( is drawn.

3.1 Write down the equation of 𝑔−1, the inverse of g, in the form y = … (2)

3.2 Write down the range of 𝑔−1. (1)

3.3 Calculate the value(s) of x for which .4)( xg (4)

[7]

QUESTION 4

Given 12

4)(

xxf


4.2 Determine the intercepts of the graph of f with the axes. (3)

4.3 Determine the coordinates of the image of the x intercept if it is reflected about the point

of intersection of the asymptotes .

(3)

[8]

12

QUESTION 5 FREE STATE 2019

In the diagram, the graph of xxf alog)( is drawn. B

2;

9

25is a point on f.

5.1 Determine the value of a. (2)

5.2 Determine the value(s) of x for which .0)( xf (2)

5.3 Write down the equation of 1f , the inverse of f , in the form y = … (2)

5.4 B// is the reflection of B on the graph .5

3)(

x

xg

Write down the coordinates of B//.

(2)

5.5 Determine for which value(s) of x will 9

25)(1 xf .

(2)

[10]

QUESTION 6

Given 2

6)(

x

xxf

6.1 Express )(xf in the form of q

px

axf

)(

(2)

6.2 Determine the equations of the axis of symmetry. (4)

6.3 Draw the graph of f. Show all intercepts with the axes as well as the asymptotes of the

graph.

(4)

[10]

B

y

x O

f

13


In the diagram below, the graph of 34

2)(

xxg is drawn. The graph f passes through A, the point of

intersection of the asymptotes of g, and cuts the x-axis and the y-axis at L and R respectively. K is the

y-intercept of g.

7.1 Determine the equation of f in the form cmxy . (3)

7.2 Write down the equation of the asymptotes of 2xg + 1. (2)

7.3 Calculate the length of KR. (3)

7.4 The graph of h, where h is the reflection of f in the line y = –7, passes through the point

S(– 4 ; p). Calculate the area of ARS. (4)

[12]

QUESTION 8

Given qpx

axf

)(

a > 0

Range; yϵR; y ≠ -2

Domain; xϵR; x ≠ -3

f(0) < 0

Use the given information above to sketch the graph of f [4]

y

A

L O

f

g

𝑥

R

K

14


In the diagram below, the graphs of 16)( 2 bxaxxf and 2412 xxg are drawn. The graph of g

is a tangent to the graph of f at B. A and B are the x-intercepts of f and C, the turning point.

9.1 Calculate the coordinates of B. (2)

9.2 Determine the values of a and b. (6)

9.3 If it is given that ,1642)( 2 xxxf determine:

9.3.1 The range of f (5)

9.3.2 The value(s) of x for which f / (x). g(x) > 0 (2)

[15]

QUESTION 10

Given qpx

axf

)(

a < 0

p < 0

q > 0

Use the given information above to sketch the graph of f. [4]

y

A B

C

f

g

x O

15

QUESTION 11 NORTH WEST 2019

The graphs of 21

( ) 2 9 and ( )2

ag x x h x q

x p

are sketched below.

The axis of symmetry of graphg is the vertical asymptote of graph h. The line f is an axis

of symmetry of graph h. B is the y-intercept of h, g and f .

y h

g

x

B

f

11.1 Write down the coordinates of C, the turning point of g.

(2)

11.2 Determine the coordinates of B.

(2)

11.3 Write down the equation of f.

(2)

11.4 Determine the equation of h.

(5)

11.5 Write down the equations of the vertical and horizontal asymptotes of

( ) 3 ( ) 2k x h x .

(2)

11.6 Determine the x-intercept of h.

(3)

11.7 For which values of x will:

4.7.1

/ ( )0

( )

g x

h x

(3)

4.7.2 1 ( 1) 2f x

(4)

11.8 Calculate the value(s) of k for which g(x) = f (x) +k has two unequal positive roots.

(6)

[29]

h

C

16

QUESTION 12 NORTH WEST 2019

QUESTION 13

Given .42.2)( 1 xxg

13.1 Determine the x and y intercepts of g (2)

13.2 Determine the domain and range of g (2)

13.3 Draw the graph of g Show all intercepts with the axes as well as the asymptotes of the

graph.

(4)

[8]

12.1 Consider the function 5

( )6

x

f x

12.1.1

Write down the equation of h, the reflection of f in the y-axis.

(1)

12.1.2 Write down the equation of 1 ( )f x

in the form y = ...

(2)

12.1.3 For which value(s) of x will1 ( ) 0f x ?

(2)

12.2 The function defined as 2( )f x ax bx c has the following properties:

/ ( 2,5) 0f

(1) 0f

2 4 0b ac

( 2,5) 6f

Draw a neat sketch graph of f. Clearly show all x-intercepts and turning point.

(4)

[9]

17

QUESTION 14 DBE NOV 2019

Below are the graphs of 3)( 2 bxxxf and px

axg

)( .

f has a turning point at C and passes through the x-axis at (1 ; 0).

D is the y-intercept of both f and g. The graphs f and g also intersect each other at E

and J.

The vertical asymptote of g passes through the x-intercept of f.

14.1 Write down the value of p. (1)

14.2 Show that a = 3 and b = 2. (3)

14.3 Calculate the coordinates of C. (4)

14.4 Write down the range of f. (2)

14.5 Determine the equation of the line through C that makes an angle of 45 with the

positive x-axis. Write your answer in the form y = …

(3)

14.6 Is the straight line, determined in QUESTION 4.5, a tangent to f? Explain your answer.

(2)

14.7 The function h(x) = f (m – x) + q has only one x-intercept at x = 0. Determine the

values of m and q.

(4)

[19]

E

J f

g

y

x O 1

D

C .

.

. g

18


Sketched below is the graph of xkxf )( ; k > 0. The point (4 ; 16) lies on f.

15.1 Determine the value of k. (2)

15.2 The graph of g is obtained by reflecting graph f about the line y = x. Determine the

equation of g in the form y = …

(2)

15.3 Sketch the graph g. Indicate on your graph the coordinates of two points on g. (4)

15.4 Use your graph to determine the value(s) of x for which:

15.4.1 f (x) × g(x) > 0 (2)

15.4.2 g(x) 1 (2)

15.5 If h(x) = f (–x), calculate the value of x for which 4

15h(x)f(x)

(4)

[16]

x

y

f

O

(4 ; 16)

1

.

.

19

QUESTION 16 DBE MAY/JUNE 2019

16.1 Given: 32

1

xxf

.

16.1.1 Determine the equations of the asymptotes of f. (2)

16.1.2 Write down the y-intercept of f. (1)

16.1.3 Calculate the x-intercept of f. (2)

16.1.4 Sketch the graph of f . Clearly label ALL intercepts with the axes and any

asymptotes.

(3)

16.2 Sketched below are the graphs of cbxaxxk 2 and 42 xxh .

Graph k has a turning point at (–1 ; 18). S is the x-intercept of h and k.

Graphs h and k also intersect at T.

16.2.1 Calculate the coordinates of S. (2)

16.2.2 Determine the equation of k in the form qpxay 2

(3)

16.2.3 If 1642 2 xxxk , determine the coordinates of T. (5)

16.2.4 Determine the value(s) of x for which )()( xhxk . (2)

16.2.5 It is further given that k is the graph of xg /.

(a) For which values of x will the graph of g be concave up? (2)

(b) Sketch the graph of g, showing clearly the x-values of the turning

points and the point of inflection.

(3)

[25]

(–1 ; 18)

T k

h

y

x O S

.

20

QUESTION 17 KZN JUNE 2019

2

1;0A;)(Given c

bx

axf is the y – intercept of the graph. The asymptotes to the graph intersects

the x – axis at 1 and the y – axis at 2 respectively.


17.2 Calculate the value of a. (3)

17.3 Determine the coordinates of A/, the image of A, if it is reflected about ).2;1( (4)

17.4 Determine the equation of g if ).3()( xfxg (2)

[11]

x

y

A

1

2

f

O

21


In the diagram below, the graph of 2)( axxf is drawn in the interval .x 0

The graph of 1f is also drawn. P(–6 ; –12) is a point on f and R is a point on .1f

18.1 Is 1f a function? Motivate your answer. (2)

18.2 If R is the reflection of P in the line xy , write down the coordinates of R. (1)

18.3 Calculate the value of .a (2)

18.4 Write down the equation of 1f in the form ...y (3)

[8]

x

y

f

R .

P(–6 ; –12)

O

.

22

ANALYTICAL GEOMETRY

Weighting 37 – 43 marks (±40)

Sub-topics/Clarification 1. EQUATION OF A CIRCLE: (𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 = 𝑟2

Use the same formula no matter the centre is at the origin

A point lie inside, outside or on the circle

Completing the square to find centre

2. INTERSECTION OF CIRCLES

Intersect at one point: the distance between the centres is equal to

the sum of their radii

Intersection of two circles: the distance between the centres is

less than the sum of their radii

Never intersect. The distance between the centres is greater than

the sum of their radii

3. STRAIGHT LINES AND TANGENTS TO CIRCLES

Radius/diameter is perpendicular to the tangent.

In a tangent the product of the gradient and radius/diameter is

negative one.

Point or points of intersection of straight lines and the circle.

The two tangents from the common point outside the circle are

equal.

Related

concepts/terms/vocabulary Gradient of a line: 𝑚 =

𝑦2−𝑦1

𝑥2−𝑥1

Inclination : tanα = m

Midpoint: When given two points 𝑚 = 𝑥1

𝑤ℎ𝑒𝑛 𝑔𝑖𝑣𝑒𝑛 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑎𝑛𝑑 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑢𝑠𝑒 ∶ 𝑥 =𝑥1 + 𝑥2

2; 𝑦

=𝑦1 + 𝑦2

2

Equation of a straight line : 𝑦 = 𝑚𝑥 + 𝑐 𝑜𝑟 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Parallel lines: equal gradients

Perpendicular lines: 𝑚1 × 𝑚2 = −1

Collinear points gradients between the points are equal.

Equation of a tangent: 𝑦 = 𝑚𝑥 + 𝑐 𝑜𝑟 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Prior-knowledge/

Background knowledge Revision of grades 10 and 11:

Grade 10 and 11 Analytical Geometry.

Properties and definitions of geometric shapes.

Different types of triangles.

4. Definition: circle, Radius, Tangent, Centre

5. Distance formula

6. Interpretation of circles (centre theorem; rad-tan theorem; tan-chord

theorem)

7. Finding equations of the straight lines and tangents/normal and tangent

8. Intersecting circles

Resources Chalkboard Instruments

Mathematical instruments

Scientific Calculator

Activities

23

Exercises on different maths textbooks can be used

Maths Handbook and Study Guide by Kevin Smith (page 221- 224).

See the activities below.

Give leaners past examination papers

Methodology Revise grades 10 and 11 analytical Geometry. Then give learners the

revision exercises below

Use the distance formula to derive the standard form of equation of a

circle:

(x – a)2 + (y – b)2 = r2 , where (a ; b) is the centre point of

circle and r is the length of the radius. Then do a couple of examples from any CAPS approved textbooks

Integration with Trigonometry and Euclidean Geometry

Assessment

o The practice exercises below can be taken from any relevant

textbooks.

Classwork

Homework

Assignment

Test

Errors/misconceptions Incorrect use of gradient formula

FROM ATP:

HINTS:

Should be able to determine the:


31/02 – 03/04

(4 days)

1. The equation 222 )()( rbyax defines a circle with radius r and

centre ).;( ba

NOTE: Include circles that touch internally and externally

06/04 – 08/04

(3 days) 2. Determination of the equation of the tangent to given circle.

Distance formula: 2

12

2

12 yyxxd

Gradient formula: 12

12

xx

yym

Relationship between the Parallel lines: 21 mm

Relationship between the Perpendicular lines 121 mm

Midpoint of a line segment:

2;

2

1212 yyxxM

Equation of a straight line: cmxy OR 11 xxmyy

Angle of inclination: tanm

Completing a square Writing cbxax 2 in the form of qpxa 2)(

The relationship between tangent and a radius Tangent is perpendicular to the radius at point of contact.

24

N(4; 1)

K(−1; 4)

L(3; 6)

𝜃 O

𝑥

𝑦

R

P

M

BASELINE ASSESSMENT

QUESTION 1

In the diagram, K (–1 ; 4) ; L(3 ; 6); M and N(4 ; 1) are vertices of a parallelogram. R is the

midpoint of LN. P is the x -intercept of the line MN produced.

1.1 Calculate:

1.1.1 gradient of KL . (2)

1.1.2 coordinates of R. (3)

1.1.3 coordinates of M. (4)

1.2 Determine the equation of NM in the form y = mx + c. (3)

1.3 Calculate the:

1.3.1 coordinates of P. (2)

1.3.2 size of Ɵ , the inclination of PM. (4)

[20]

Properties of polygons Quadrilaterals and triangles

Centre and the radius of a circle Centre: 0;0 OR Centre: );ba and radiusr

The equation of a circle:

with centre at the origin 0;0 : 222 ryx

AND

with centre ba; : 222rbyax

The equation of the tangent to a circle: Using 1 rt mm and a given point.

25

THE CIRCLE

Equation of a circle

222 )()( rbyax

Where: a is the x -coordinate of the circle centre

b is the y -coordinate of the circle centre

r is the length of the circle’s radius (by distance formula)

If the centre is at the origin (0;0), the equation reduces to: 222 ryx

EXAMPLES

1. Write down the equation of the circle with centre (-2;-1) and radius 4

2. Determine the equation of the circle with centre M(-3;1) and A(2;-2) a point on the circle.

3. Given the equation of the circle with centre O is 02024 22 yyxx .

Determine the coordinates of O and the radius of the circle

SOLUTIONS

1. 222 )()( rbyax

Substitute the centre (-2;-1) and radius 4

222 4))1(())2(( yx

16)1()2( 22 yx 222 )()( rbyax

2. Find the radius of the circle 222 ))2(1()23( r

9252 r

342 r

Equation of the circle

34)1())3(( 22 yx

34)1()3( 22 yx

3. 2024 22 yyxx

1420)12()44( 222 yyxx

25)1()2( 22 yx

The circle has a centre at O (2 ; -1) and a radius of 5 units

26

ACTIVITY

Write down the equation of a circle with

a) centre at (0;0) and radius 3

b) centre at );( qp and radius a

c) centre at (3 ; -2) and passes through (3 ; 4)

Determine the centre and the radius of the circles defined by:

a) 082422 yxyx

b) 03622 xyx

c) 024222 yxyx

INTERSECTION OF CIRCLES

Two circles can intersect one another at two points, one point or not at all

1. Circles intersecting at two points

2. Circles intersecting at one point(touching)

21 rrd

3. Circles that do not intersect

21 rrd

r

RA

B

27

4. Circles that touch internally

rRd

EXAMPLE

Given two circles 422 yx and 9)3()2( 22 yx , Determine:

a) The circle centres and lengths of the radii of each circle

b) Whether or not the circles intersect each other

SOLUTION

a) Centres (0;0) and (2;-3)

21 r and 32 r

220302 d

13

21 rrd the two circles do intersect (at two points)

ACTIVITY

Given two circles with equations 0161012 22 yyxx and 02422 yxyx ,

prove that circles touch each other.

r

R

28

TANGENTS TO CIRCLES

A radius is any line that connects the centre

of a circle and any point on the

circumference.

A diameter is any line that joins two points

on the circumference and passes through the

centre. (Diameter=2radii)

A tangent is any line that touches a curve (in

this case a circle) externally at only one

point.

The radius is perpendicular to the tangent at

the point of contact. (tan rad)

1tan gentradius mm

THE EQUATION OF A TANGENT

EXAMPLES

1. Determine the equation of a tangent which touches the circle 17)3()2( 22 yx at the

point (1;1)

2. Write down the equations of the tangents to the circle 36)3( 22 yx which are parallel

to the x -axis

SOLUTIONS

1. 4

12

13

radiusm

4

1

4

11tan

radius

gentm

m

Equation of a tangent: )( 11 xxmyy

)1(4

11 xy

4

3

4

1 xy

2.

Centre(0 ; 3)

Radius = 6 units

9y and 3y

(0 ;3)

y

x

29

ACTIVITY

1.Determine the equation of the tangent which touches the circle with centre at the origin, at

the point (4 ; 2)

2.Determine the length of a tangent from the point R(-2 ; 2) to the point of contact P on the

circle 0128 22 yxx

DBE/2017

In the diagram, N is the centre of the circle. M(-3 ; -2) and P(1 ; 4) are points on the circle.

MNP is the diameter of the circle. Tangents drawn to circle N from point R, outside the

circle meet the circle at S and M respectively.

1. Determine the coordinates of N.

N is the Midpoint of the diameter MP

N

2

24;

2

)3(1

N )1;1(

2. Determine the equation of the circle in the form 222 )()( rbyax

2212)1(3 radius

= 13 132 r

Centre = (-1 ; 1) equation 13)1()1( 22 yx

3. Determine the equation of the tangent RM in the form cmxy

2

3

)3(1

)2(1

NMm

3

2RMm

Equation of RM ))3((3

2)2(

xy

43

2 xy

30

4. If it is given that the line joining the S to M is perpendicular to the x -axis, determine the

coordinates of S.

RSNM is a kite (adjacent sides are equal)/ tangents from same point

RN bisects SM

Vertical distance from -2 to 1 is 3 to get to S from 1 the distance will also be 3 (diagonals of

a kite)

S is (-3 ; 4)

5. Determine the coordinates of R, the common external point from which both tangents to

the circle are drawn

2

3

)3(1

41

NSm

3

2RSm

))3((3

24 xy

63

2 xy

63

24

3

2

xx

304 x

2

15x

2

7y S

2

7;

2

15

31

N(4; 1)

K(−1; 4)

L(3; 6)

𝜃 O

𝑥

𝑦

R

P

M

PRACTICE EXERCISES


In the diagram, K (–1 ; 4) ; L(3 ; 6); M and N(4 ; 1) are vertices of a parallelogram. R is the

midpoint of LN. P is the x -intercept of the line MN produced.

1.1 Calculate:

1.1.1 distance of KL . (2)

1.1.2 coordinates of R. (3)

1.1.3 coordinates of M. (4)

1.2 Determine the equation of NM in the form y = mx + c. (3)

1.3 Calculate the:

1.3.1 coordinates of P. (2)

1.3.2 size of Ɵ , the inclination of PM. (4)

[20]

32


In the diagram below, ),4;1(A )4;3( B , );2( C k and );( D yx are the vertices of a

rectangle. AB and DC cuts the x axis at G and H respectively. GD is drawn. CHG .

2.1 Calculate the gradient of BG. (2)

2.2 Determine the equation of AB in the form .cmxy (2)

2.3 Calculate the:

2.3.1 Value of k (𝑦-coordinate of C). (4)

2.3.2 Coordinates of D. (3)

2.3.3 Size of . (3)

2.3.4 Area of DHG. (7)

[21]

A (1; 4)

D (x; y)

O G H

B (–3; –4)

C (2; k)

33


In the diagram below, the circle centred at 1;3E passes through point 5;5P .

3.1 Determine the equation of:

3.1.1 The circle in the form 0CBA22 yxyx . (4)

3.1.2 The tangent to the circle at 5-;5P in the form cxy m . (5)

3.2 A smaller circle is drawn inside the circle. Line EP is a diameter of the small circle.

Determine the:

3.2.1 Coordinates of the centre of the smaller circle. (3)

3.2.2 Length of the radius. (3)

3.3 Hence, or otherwise, determine whether point 3;9C lies inside or outside the

circle centre at E. (3)

[18]

●

O

●

34

QUESTION 4 SCE: DBE/May-June 2018

In the diagram, ABCD is a quadrilateral having vertices A(7; 1), B(-2; 9), C(-3; -4) and D(8;-11). M

is the midpoint of BD.

4.1 Calculate the gradient of AC. (2)

4.2 Determine :

4.2.1 The equation of AC in the form cmxy (2)

4.2.2 Whether M lies on AC. (4)

4.3 Prove that BDAC. (3)

4.4 Calculate:

4.4.1 , the inclination of BD (2)

4.4.2 The size of angle of DBC ˆ (3)

4.4.3 The length of AC (2)

4.4.4 The area of ABCD (5)

[23]

35

QUESTION 5 SCE: DBE/ May-June 2018

In the diagram, a circle having centre at the origin passes through P (4; -6). PO is the diameter of a

smaller circle having centre at M. The diameter RS of the larger circle is a tangent to the smaller

circle at O.

5.1 Calculate the coordinate of M. (2)

5.2 Determine the equation of:

5.2.1 The large circle (2)

5.2.2 The small circle in the form 022 EDyCxyx (3)

5.2.3 The equation of RS in the form cmxy (3)

5.3 Determine the length of cord NR, where N is the reflection of R in the y-axis. (4)

5.4 The circle with centre at M is reflected about the x-axis to form another circle centred at K.

Calculate the length of the common chord of these two circles

(3)

[17]

36

QUESTION 6 NSC: DEB/Feb.-Mar. 2018

In the diagram, P, Q (-7; -2), R and S (3; 6) are vertices of quadrilateral. R is a point on the x-axis.

QR is produced to N such that QR = 2RN. SN is drawn. 057,71ˆ OTP and NRS ˆ .

Determine:

6.1 The equation of SR (1)

6.2 The gradient of QP to the nearest integer (2)

6.3 The equation of QP in the form cmxy (2)

6.4 The length of QR. Leave your answer in surd form (2)

6.5 090tan (3)

6.6 The area of RSN , without using a calculator (6)

[16]

37

QUESTION 7 NSC: DEB/Feb.-Mar. 2018

In the diagram, PKT is a common tangent to both circles at K (a; b). the centres of both circles lie

on the line xy2

1 . The equation of the circle centred at O is 18022 yx . The radius of the

circle is three times that of the circle centred at M.

7.1 Write down the length of OK in surd form (1)

7.2 Show that K is the point (-12; -6) (4)

7.3 Determine:

7.3.1 The equation of the common tangent, PKT, in the form cmxy (3)

7.3.2 The coordinate of M (6)

7.3.3 The equation of the smaller circle in the form 222rbyax (2)

7.4 For which value(s) of r will another circle, with equation 222 ryx , intersect the circle

centred at M at two distinct points?

(3)

7.5 Another circle 0240163222 yxyx , is drawn. Prove by calculation that this

circle does not cut the circle with centre M(-16;-8).

(5)

[24]

38

APPLICATION OF CALCULUS

Sub-topics/Clarification Optimization and rate of change

Related

concepts/terms/vocabulary

Maxima and minima.

Rate of change /stationary points.

Prior-knowledge/

Background knowledge

Measurements (perimeter, area, Total Surface Area, volumes etc).

Rules of differentiation.

Resources Worksheet, textbooks, study guide, tin, cans, chalk box.

Methodology Revise with learners prior knowledge done in grade 10 and 11

(measurement)

Remind them formulae (Perimeter, Area, TSA, and Volume)

Look at the terminology that will be used in the sub-topic

Drill learners on the concept of maxima and minima integrating with

graphs (stationery points)

Drill learners on the concept of rate of change and other terms they

will be using in the sub-topic for example stationery point is a

turning point; gradient of a tangent is a derivative

Practical activity can be performed by learners in terms of

optimization, e.g.: using can, boxes and their nets etc.

Theory on the calculus of motion

S(t) represents an equation of motion (height, distance,

displacement) at time t.

/s t represents velocity or speed at time t.

//s t represents acceleration at time t.

Assessment Class activities

Short test

Home work

Errors/Misconceptions/

Problem Areas Learners use equation of Area instead of Total Surface Area

FROM ATP:


28/4

(1 day)

1. An intuitive understanding of the limit concept.

2. Use limits to define the derivative of a function 𝑓 at any 𝑥:

𝑓′(𝑥) = limℎ→0

𝑓(𝑥+ℎ)−𝑓(𝑥)

ℎ.

Generalise to find the derivative of 𝑓 at any point 𝑥 in the domain of 𝑓, i.e., define the

derivative function 𝑓′(𝑥) of the function𝑓(𝑥).

Understand intuitively that 𝑓′(𝑎) is the gradient of the tangent to the graph of 𝑓 at the point

with 𝑥-co-ordinate 𝑎.

29/4 – 30/4

(2 days)

3. Using the definition (first principles), find the derivative, 𝑓′(𝑥), for

a. 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐;

b. 𝑓(𝑥) = 𝑎𝑥3;

c. 𝑓(𝑥) =𝑎

𝑥; and

d. 𝑓(𝑥) = 𝑐 (𝑎, 𝑏 and 𝑐 are constants).

39

HINTS TO TEACHERS: FOCUS ON THE FOLLOWING CHARACTERISTICS:

Definition of terms, e.g.

Displacement

120 10s t t

Velocity

/ 10s t v t

Acceleration

// / 0s t v t a t

Rate

Speed

Maximum

Minimum

Do a thorough revision of the rules of differentiation.

Emphasise sketching and interpretation of cubic functions in real life contexts.

Develop a strategic approach to problems on optimisation, e.g.

Highlight what the question asks and make an expression for what must be optimised.

Make a rough sketch of the situation if possible.

If there two variables in the expression, use simultaneous equations to eliminate one of them.

Write down the derivative of the expression in terms of one of the variables.

Set the derivative equal to zero and solve for one the variables.

The value/s obtained can then be used to determine the maxima or minima.

04/5 – 07/5

(4 days)

4. Use the formula 𝑑

𝑑𝑥(𝑎𝑥𝑛) = 𝑎𝑛𝑥𝑛−1, for any real number 𝑛, together with the rules

a. 𝑑

𝑑𝑥[𝑓(𝑥) ± 𝑔(𝑥)] =

𝑑

𝑑𝑥[𝑓(𝑥)] ±

𝑑

𝑑𝑥[𝑔(𝑥)]; and

b. 𝑑

𝑑𝑥[𝑘𝑓(𝑥)] = 𝑘

𝑑

𝑑𝑥[𝑓(𝑥)] (𝑘 a constant).

08/5

(1 day) 5. Find equations of tangents to graphs of functions.

11/5

(1 day) 6. Apply the Remainder and Factor Theorems to polynomials of degree at most 3.

7. Factorise third degree polynomials.

12/5 – 19/5

(6 days)

8. Introduce the second derivative 𝑓′′(𝑥) =𝑑

𝑑𝑥[𝑓′(𝑥)] of 𝑓(𝑥), and how it determines the

concavity of a function.

9. Sketch graphs of polynomial functions using differentiation to determine the coordinates of

stationary points, and points of inflection (where concavity changes). Also determine the

𝑥-intercepts of the graph, using the factor theorem and other techniques.

20/5 – 26/5

(5 days)

10. Solve practical problems concerning optimisation and rate of change, including calculus of

motion.

40

LEARNERS SHOULD:

Be able to use rules of differentiation to determine the first and second derivative.

Be able to sketch graphs of cubic functions and their related derivatives.

Be able to interpret sketched graphs of cubic functions in context.

Be able to relate the point of inflection to the concavity of a graph.

Understand the concept of stationary points of a graph and how it relates to maxima and minima.

Understand the relation between the rate of change and concept of optimisation.

Be able to determine the maximum or minimum values of given expressions.

SUMMARY OF CONCEPTS

CONCEPT CONTEXT

The derivative of a function, 𝑓/(𝑥). An expression relating a change in one variable

with respect to another variable.

When a derivative is equal to zero, 𝑓/(𝑥) = 0. It is at an instant where the change does not happen.

Mensuration formulae.

Only formulae for cones, spheres and pyramids will

be given during exams, but learners should be able

to identity them from an array of formulae.

Calculus of motion.

𝑠(𝑡) represents an equation of motion at time, t.

𝑠/(𝑡) represents speed or velocity at time, t.

𝑠//(𝑡) represents acceleration at time, t.

41

PRACTICE EXERCISES


The sketch below represents the curve of 𝑓(𝑥) = 𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 8. The solutions of the equation

𝑓(𝑥) = 0 are −2 ; 1 and 4.

1.1 Calculate the values of b, c and d. (4)

1.2 Calculate the x-coordinate of B, the maximum turning point of f. (4)

1.3 Determine an equation for the tangent to the graph of f at 𝑥 = −1. (4)

1.4 Sketch the graph of 𝑓//(x). Clearly indicate the x- and y-intercepts on your sketch. (3)

1.5 For which value/s of x is f(x) concave upwards? (2)

[17]

42


The sketch below shows the graph of 𝑝(𝑥) where 𝑝(𝑥) = 𝑥3 + 𝑏𝑥2 + 24𝑥 + 𝑐. 𝐴(2 ; 0) is an

x-intercept of both 𝑝(𝑥) and 𝑝/(𝑥). C is the other x-intercept of 𝑝/(𝑥).

2.1 Show that the numerical value of 𝑏 = −9. Clearly show all your calculations. (3)

2.2 Calculate the coordinates of C. (3)

2.3 For what value(s) of x will 𝑝(𝑥) be increasing? (3)

2.4 Calculate the value(s) of x for which the graph of p is concave up. (2)

2.5 Sketch the graph of 𝑝(𝑥). Clearly indicate the intercepts with the axes, turning points and

the point of inflection.

(4)

[15]

43


The function by 𝑓(𝑥) = 2𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is sketched below.

The turning points of the graph of f are 𝑇(2 ; −9) and 𝑆(5 ; 18).

3.1 Show that 𝑎 = 21, 𝑏 = −60 and 𝑐 = 43. (7)

3.2 Determine an equation of the tangent to the graph of f at 𝑥 = 1 (5)

3.3 Determine the x-value at which the graph of f has a point of inflection. (2)

[14]

QUESTION 4 NW SEPT 2018

A shopkeeper finds that a number of people visiting his shop at any moment during the ten (10)

hours that the shop is open, is represented by N(𝑡) = 𝑡3 − 12𝑡2 + 36𝑡 + 8, where 𝑁(𝑡) is the

number of people in the shop, t hours after the shop opened.

4.1 How many people are in the shop when it opens? (1)

4.2 At what stage is the number of people in the shop increasing? (5)

4.3 At which stage is it the best time for the shopkeeper to take a break and leave his assistant

alone in the shop?

(1)

[7]


A particle moves along a straight line. The distance, s (in metres) of the particle from a fixed point

on the line at time t seconds (𝑡 ≥ 0) is given by 𝑠(𝑡) = 2𝑡2 − 18𝑡 + 45.

5.1 Calculate the particle’s initial velocity. (Velocity is the rate of change of distance.) (3)

5.2 Determine the rate at which the velocity of the particle is changing at t seconds. (1)

5.3 After how many seconds will the particle be closest to the fixed point? (2)

[6]

44

𝐶 =1200

𝑥+ 600𝑥2


The depth of water (in metres) left in the dam, t hours after the sluice gate was opened to allow the

flow of water to drain from the dam is given by the equation 𝐷(𝑡) = 28 −1

9𝑡2 −

1

27𝑡3.

6.1 Calculate the average rate of change in the depth of the water after the first two hours. (4)

6.2 Determine the rate at which the level of the water is decreasing after 16 hours. (4)

[8]

QUESTION 7 DBE FEB/MARCH 2013

A rectangular box is constructed in such a way that the length (l) of the base is three times as

Long as it’s width. The material used to construct the top and the bottom of the box costs R100

per square metre. The material used to construct the sides of the box costs R50 per square metre.

The box must have a volume equal to 9 𝑚3. Let the width of the box be x metres.

7.1 Determine an expression for the height (h) of the box in terms of x. (3)

7.2

Show that the cost to construct the box can be expressed as

(3)

7.3 Calculate the width of the box (that is the value of x) if the cost is to be a minimum. (4)

[7]

45

r

h

QUESTION 8 LIMPOPO SEPT 2019

PQRS is a rectangle with P on the curve ℎ(𝑥) = 𝑥2 and with the x-axis and the line 𝑥 = 6 as

boundaries.

8.1 Show that the area of rectangle 𝑃𝑄𝑅𝑆 can be expressed as 𝐴 = 6𝑥2 − 𝑥3 (3)

8.2 Determine the largest possible area of rectangle 𝑃𝑄𝑅𝑆. Show all your calculations. (4)

[7]

QUESTION 9 DBE FEB/MARCH 2016

A soft drink can has a volume of 340𝑐𝑚3,

a height of h cm and a radius of r cm.

9.1 Express h in terms of r. (2)

9.2 Show that the surface area of the can is given by 𝐴(𝑟) = 2𝜋𝑟2 + 680𝑟−1. (2)

9.3 Determine the radius of the can that will ensure that the surface area is a minimum. (4)

[8]

46

QUESTION 10 MIND ACTION SERIES

The diagram represents a right circular cone of height, h cm, and a base radius of r cm. If the sum of

the height and base radius is 12 cm.

10.1 Express r in terms of h. (2)

10.2 Express the volume of the cone in terms of h. (4)

10.3 Determine the maximum volume of the cone. (4)

[10]

11 During an experiment, the volume of water in a dam is found to

be 260 8 3V t t t , where V t represents the volume of

water in

the dam, in kilolitres, t days after the experiment has begun.

There is a suspected leak in the dam wall.

a) Calculate the rate of change of volume of water after 3 days.

b) When did the volume of water start decreasing?

c) When will the dam be empty?

(3)

(3)

(4)

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