FUNCTIONS

A function is a mathematical relationship between two variables, where every input variable
has one output variable. The input is the number or value put into the function. The output is
the number or value the function gives. In functions, the 𝑥-variable is known as the input or
independent variable, because its value can be chosen freely. The calculated 𝑦-variable is
known as the output or dependent variable, because its value depends on the chosen input
value. Functions can be written in terms of 𝑦 or 𝑓(𝑥). We can also use other letter such as
𝑔(𝑥) or ℎ(𝑥) to show that each 𝑦-value is a function of an 𝑥-value.

The Linear or Straight Line Graph

It is given by the equation: 𝑦 = 𝑚𝑥 + 𝑐

where: 𝑚 is the gradient or slope of the graph 𝑐: 𝑦-intercept

In order to draw a straight line graph, you must determine the 𝑥-intercept when 𝑦 = 0 and
𝑦-intercept when 𝑥 = 0. The gradient is found from the equation:

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Rules for sketching a straight line graph

1. Determine the 𝑥-intercept (let 𝑦 = 0)

2. Determine the 𝑦-intercept (let 𝑥 = 0)

3. Plot these two points and draw a straight line through them.

Rules for finding the equation of a straight line graph

1.Find the gradient 𝑚 using any two points on the graph.

2.Find the equation by using the gradient and any point on the graph in the equation:

𝑦 = 𝑚𝑥 + 𝑐 or 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )

The Hyperbola

It is given by the general equation:

𝑎
𝑦= + 𝑞
𝑥– 𝑝

where 𝑎 ≠ 0; 𝑝 ≠ 0 𝑎𝑛𝑑 𝑞 ≠ 0

𝑎: the value that indicates the shape of the graph

𝑝: 𝑥-asymptote 𝑞: 𝑦-asymptote

The asymptote is an imaginary line that a graph approaches but never touches. It indicates the
values of 𝑥 for which the function does not exist. These values are excluded from the domain
and the range. The horizontal asymptote is the line 𝒚 = 𝒒 and the vertical asymptote is the
line 𝒙 = 𝒑.

Rules for sketching the hyperbola graph:

1. Determine the shape of the graph using the value of 𝑎.

2. Draw the asymptotes on the set of axes as dotted lines.
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 3. Determine the 𝑥-intercept(s) let 𝑦 = 0.
4. Determine the 𝑦- intercept(s) let 𝑥 = 0.
5. Plot the points and then draw the graph.

Rules for finding the equation of a hyperbola

1. Identify the 𝑥 and 𝑦 asymptotes.

2. Use a random point on the graph to determine the value 𝑎.
3. If equation is in the form 𝑥𝑦 = 𝑘, then substitute a point on the graph to determine 𝑘.

Equation of line of symmetry for a hyperbola

A line of symmetry is an imaginary line that divides a graph into two mirror images of each
other.
There are two lines about which a hyperbola is symmetrical :
𝑦 = 𝑥 + 𝑘 (increasing function) and 𝑦 = −𝑥 + 𝑘 (decreasing function).

For example a hyperbola with the 𝑥-asymptote at 𝑥 = −1 and the 𝑦-asymptote at

𝑦 = −3 would have the equation of symmetry for the decreasing function as
𝑦 = −𝑥 + 𝑘

−3 = − (−1) + 𝑘

𝑘 = −4

Therefore, the equation is : 𝑦 = −𝑥 − 4

Note: the two line of the axis of symmetry intersect at the points ( 𝑝; 𝑞)

Exponential graph

It is given by the general equation:

𝑦 = 𝑎. 𝑏 𝑥− 𝑝 + 𝑞 where 𝑏 > 0 𝑎𝑛𝑑 𝑏 ≠ 1

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𝑎 and 𝑏 are constants and 𝑞 is the 𝑦-asymptote.

The value of 𝑏 affects the direction of the graph:

• If 𝑏 > 1, 𝑓(𝑥) is an increasing function.

• If 0 < 𝑏 < 1, 𝑓(𝑥) is a decreasing function.

• If 𝑏 ≤ 0 , 𝑓(𝑥) is not defined.

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Rules for drawing an exponential graph

1. Write down the 𝑦-asymptote (𝑞) and draw it using a dotted line.
2. Find the 𝑥 and 𝑦 intercepts and plot them on the set of axes.
3. Draw the new formed graph.

Rules for finding the equation of an exponential graph

1. When given two random points; to find the value of 𝑎 use the point where 𝑥 = 0
first i.e. the 𝑦-intercept.
2. After finding 𝑎, substitute the value of 𝑎 into the equation and find 𝑏.

The Parabola or Quadratic Graph

The general equation of a parabola can be written in one of three formats

• 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 [standard form]

• 𝑦 = 𝑎(𝑥 − 𝑝)2 + 𝑞 [turning point form]
• 𝑦 = 𝑎(𝑥 − 𝑥1 )(𝑥 − 𝑥2 ) [root form]

Properties

1. Shape

−𝑏
2. The graph has an axis of symmetry given by 𝑥 = 2𝑎

The value of 𝑥 in the axis of symmetry equation is also the 𝑥 coordinate of the turning
point of the graph.

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 −𝑏 −𝑏
3. The function has one turning point given by ( ) ; 𝑓 ( 2𝑎 )
2𝑎

4. The function may have either a maximum or a minimum value but never both.

Rules for sketching a parabola graph

To sketch any quadratic function, follow the following steps:

1. Write down the 𝑦-intercept (let 𝑥 = 0)
2. To calculate the 𝑥-intercepts, write the equation in the form∶
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 and factorise
3. Determine the axis of symmetry.
4. Substitute the 𝑥-value of the axis of symmetry into the original equation of the function
to calculate the co-ordinates of the turning point.
5. Plot the points and then draw the function using free hand.

Note: By completing the square for the standard form of the equation of a parabola, it can be
re-written as:

𝑦 = 𝑎(𝑥 − 𝑝)2 + 𝑞

𝑝: 𝑥-coordinate of turning point or line of symmetry

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𝑞: 𝑦-coordinate of turning point. In other words (𝑝; 𝑞) are just the turning points.

Rules for finding the equation of a parabola

Given the 𝒙-intercepts and one point:

Use the formula: 𝑦 = 𝑎(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )

• Substitute the values of the 𝑥-intercepts.

• Substitute the given point which is not the 𝑥-intercept so as to find 𝑎.
• Write the equation in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐

Given the turning point and one point:

Use the formula: 𝑦 = 𝑎(𝑥 − 𝑝)2 + 𝑞

• Substitute the co-ordinates of the turning point (𝑝; 𝑞).

• Substitute the given or random point on the graph and solve for 𝑎.
• Write the equation in the form 𝑦 = 𝑎(𝑥 − 𝑝)2 + 𝑞 or 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
depending on the instruction in the question.

Given the co-ordinates of three points on the parabola

Use the formula: 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐

• One of the given point is the 𝑦-intercept, therefore 𝑐 is given, so substitute its value.
• Substitute the co-ordinates of the other two points into 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
• Solve the two equations simultaneously for 𝑎 and 𝑏.

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Nature of roots and the quadratic graph

The discriminant(∆) of a quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, is 𝑏 2 − 4𝑎𝑐. The discriminant is

found inside the square root of the quadratic formula. The discriminant indicates the number
and nature of the roots of the quadratic as shown below:

Note:

Domain: the set of possible 𝑥-values

Range: the set of possible 𝑦-values

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Summary of Transformations of Function Graphs

Transformation Description
𝑦 = 𝑓(𝑥 − 𝑐) horizontal transformation to the right by 𝑐 units
𝑦 = 𝑓(𝑥 + 𝑐) horizontal transformation to the left by 𝑐 units

𝑦 = 𝑓(𝑥) + 𝑑 vertical transformation upwards by a factor 𝑑 units

𝑦 = 𝑓(𝑥) − 𝑑 vertical transformation downwards by a factor 𝑑 units
𝑦 = 𝑓(−𝑥) reflection in the 𝑦-axis
𝑦 = −𝑓(𝑥) reflection in the 𝑥-axis
1
𝑦 = 𝑓(𝑏𝑥) Horizontal expansion or compression by a factor of 𝑏

𝑦 = 𝑎𝑓(𝑥) Vertical expansion or compression by a factor of 𝑎

Examples
Given 𝑦 = 𝑓(𝑥) describe the following transformations

1. 𝑦 = −2𝑓(𝑥 + 3) − 5
• Reflection in the 𝑥-axis
• Vertical expansion by a factor of 2
• Horizontal translation 3 units to the left
• Vertical translation 5 units downwards

2. 𝑦 = 5𝑓(4𝑥) + 1
• Vertical expansion by a factor of 5
1
• Horizontal compression by a factor of 4

• Vertical transformation 1 unit up

3. 𝑦 = 𝑓(2𝑥 + 8)
• Horizontal compression by a factor of 2

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 • Horizontal translation 4 units to the left
Note: Remember to factorise 2𝑥 + 8 to 2(𝑥 + 4)

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PAST EXAM PAPERS PRACTISE QUESTIONS
2013 Gauteng June Paper 1 Q 6

𝑎
The diagram represents the graphs of 𝑓(𝑥) = 𝑥−𝑝 + 𝑞 and 𝑔(𝑥) = 𝑚𝑥 + 𝑐 .The graph of 𝑔

cuts the 𝑥-axis at 2 and the 𝑦-axis at 2. The 𝑦- intercept of 𝑓 is 2.

6.1 Determine the values 𝑎, 𝑝, 𝑞. (4)

6.2 Write down the domain and range of 𝑓. (4)

6.3 Write down the equations of axes of symmetry of 𝑓(𝑥) + 1. (4)

6.4 Determine the equation of g in the form 𝑦 = 𝑚𝑥 + 𝑐. (2)

6.5 Calculate the points of intersection of 𝑓 and 𝑔. (5)

2015 Gauteng Preliminary Paper 1 Q 7

The figure below represents the sketches of 𝑓 and 𝑔 defined by:

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 𝑓(𝑥) = −2𝑥 2 − 4𝑥 + 30 and 𝑔(𝑥) = −2𝑥 + 10

A and B are the 𝑥-intercept of 𝑓. The graphs of 𝑔 passes through A. C are the point of
intersection of 𝑓 and 𝑔. The graph of 𝑔 intersects the 𝑦-axis at D.

Referring to the sketch above

7.1 Determine the coordinates of A and B. (4)

7.2 Write down the function of 𝑓 in the form 𝑓(𝑥) = 𝑎(𝑥 − 𝑝)2 + 𝑞 and hence write

down the coordinates of the turning points. (5)

7.3 Determine whether (1; 12) are the coordinates of C. Show all the working. (2)

7.4 The straight line with the equation 𝑦 = 𝑚𝑥 + 32 is a tangent to the graph of 𝑓.

Calculate the possible values of 𝑚. (6)

7.5 Determine the values of 𝑥 for which 𝑓(𝑥). 𝑔(𝑥) > 0. (2)

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2015 North West Preliminary Paper 1 Q 4

The graphs of ℎ(𝑥) = 3−𝑥 ; 𝑓(𝑥) = −(𝑥 + 1)2 + 9 and 𝑔(𝑥) = 𝑎. 2𝑥 + 𝑞 are represented in
the sketch below. D, the turning point of 𝑓, is also a point of intersection of 𝑔 and 𝑓. The
asymptote of 𝑔 passes through C, the 𝑦-intercept of 𝑓.

4.1 Write down the coordinates of C. (2)

4.2 Calculate the values of 𝑎 and 𝑞. (3)

4.3 Write down the range of 𝑔. (2)

4.4 Write down the coordinates of D′, if D is reflected about the line 𝑦 = 8. (1)

4.5 If 𝑘(𝑥) = (𝑥 + 2)2 + 9 , describe the transformation from 𝑓 to 𝑘. (3)

4.6 Write down the equation of ℎ−1 (𝑥) in the form 𝑦 = … (1)

4.7 Determine the minimum value of :

1 𝑓(𝑥) − 5
𝑦=( )
3

(2)

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2010 November Paper 1 Q 5

Consider the function 𝑓(𝑥) = 4−𝑥 − 2

5.1 Calculate the coordinates of the intercepts of 𝑓 with the axes. (4)

5.2 Write down the equation of the asymptote of 𝑓. (1)

5.3 Sketch the graph of 𝑓. (3)
5.4 Write down the equation of 𝑔 if 𝑔 is the graph of 𝑓 shifted 2 units upwards. (1)
5.5 Solve for 𝑥 if 𝑓(𝑥) = 3 . (You need not simplify your answer.) (3)

2015 Western Cape Preliminary Paper 1 Q 4

Sketched below are the graphs of 𝑓(𝑥) = −(𝑥 + 2)2 + 4 and 𝑔(𝑥) = 𝑎𝑥 + 𝑞. R is the turning
point of 𝑓.

4.1 Give the coordinates of R . (2)

4.2 Calculate the length of AB. (2)

4.3 Determine the equation of 𝑔. (2)

4.4 For which values of 𝑥 is 𝑔(𝑥) > 𝑓(𝑥)? (2)

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4.5 Write down the equation of the axis of symmetry of ℎ if ℎ(𝑥) = 𝑓(−𝑥). (2)

4.6 Give the range of 𝑝 if 𝑝(𝑥) = −𝑓(𝑥). (2)

2013 Gauteng June Paper 1 Q 5

In the diagram below, the graphs of the following functions are represented:
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and 𝑔(𝑥) = 𝑎𝑏 𝑥 . A (2; 4) is a point on 𝑔. The graphs cut the axes as
given below.

5.1 Determine the equation of 𝑔. (4)

5.2 Write down the equation of the asymptote of 𝑔 . (1)

5.3 Determine the equation of 𝑓 in the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. (4)

5.4 Find the equation of the axis of symmetry of 𝑓. (1)

5.5 Write down the equation of ℎ, the reflection of the graph of 𝑔 about the 𝑥-axis. (1)

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2015 November Paper 1 Q 4

Given 𝑓(𝑥) = 2𝑥+1 − 8

4.1 Write down the equation of the asymptote of 𝑓 . (1)

4.2 Sketch the graph of 𝑓. Clearly indicate ALL intercepts with the axes as well as

the asymptote. (4)

4.3 The graph of 𝑔 is obtained by reflecting the graph of 𝑓 in the 𝑦-axis Write down the
equation of 𝑔. (1)

2009 March Paper 1 Q 6

𝑎
Sketched below are the graphs of 𝑓(𝑥) = (𝑥 − 𝑝)2 + 𝑞 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥−𝑝 + 𝑞
1
A (2 2 ; 0)is a point on the graph of 𝑓. P is the turning point of 𝑓. The asymptotes of 𝑔 are

represented by the dotted lines. The graph of 𝑔 passes through the origin.

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6.1 Determine the equation of 𝑔. (4)
6.2 Determine the coordinates of P, the turning point of 𝑓. (4)
6.3 Write down the equations of the asymptotes of 𝑔(𝑥 − 1). (2)
6.4 Write down the equation of ℎ, if ℎ is the image of 𝑓 reflected in the 𝑥-axis. (1)

2014 Exemplar Paper 1 Q 4

2
4.1 Given: 𝑓(𝑥) = 𝑥+1 − 3

4.1.1 Calculate the coordinates of the 𝑦-intercept of 𝑓. (2)

4.1.2 Calculate the coordinates of the 𝑥-intercept of 𝑓. (2)
4.1.3 Sketch the graph of 𝑓 in your ANSWER BOOK, showing clearly the asymptotes and
the intercepts with the axes. (3)
4.1.4 One of the axes of symmetry of 𝑓 is a decreasing function. Write down the equation
of this axis of symmetry. (2)

4.2 The graph of an increasing exponential function with equation 𝑓(𝑥) = 𝑎. 𝑏 𝑥 + 𝑞

• Range: 𝑦 > −3
• The points (0; −2) and (1; −1) lie on the graph of 𝑓.
4.2.1 Determine the equation that defines 𝑓. (4)
4.2.1 Describe the transformation from 𝑓(𝑥) to ℎ(𝑥) = 2.2𝑥 + 1. (2)

2015 Mpumalanga Preliminary Paper 1 Q 6

Given 𝑓(𝑥) = −√4𝑥 ; 𝑥 ≥ 0 and 𝑔(𝑥) = 3𝑥 − 3
6.1 Write down the equation of the asymptote of 𝑔. (1)
6.2 Draw a neat sketch of 𝑓 and 𝑔 on the same set of axes. Cleary indicate the intercepts
with the axis. (5)
6.3 Consider the graph of 𝑓 −1 (𝑥) . Determine the equation ℎ the reflection of 𝑓 −1 (𝑥)
in the 𝑥-axis. Leave your answer in the form ℎ(𝑥) = ……. (4)

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2010 March Paper 1 Q 6
The graphs of 𝑓(𝑥) = −𝑥 2 + 7𝑥 + 8 and 𝑔(𝑥) = −3𝑥 + 24 are sketched below. 𝑓 and 𝑔
intersect in D and B. A and B are the 𝑥-intercepts of 𝑓.

6.1 Determine the coordinates of A and B. (4)

6.2 Calculate 𝑎, the 𝑥-coordinate of D. (4)
6.3 S(𝑥; 𝑦)is a point on the graph of 𝑓; where 𝑎 ≤ 𝑥 ≤ 8. ST is drawn parallel to the 𝑦-axis
with T on the graph of 𝑔. Determine ST in terms of 𝑥. (2)
6.4 Calculate the maximum length of ST. (2)

2017 Johannesburg South Cluster June Paper 1 Q 4

Sketched are the functions 𝑓(𝑥) = (𝑥 + 𝑝)2 + 𝑞 and 𝑔(𝑥): (𝑥 + 2)(𝑦 + 3) = 𝑡 . If
5
𝑔(0) = − 2 and 𝑔 is a rectangular hyperbola with one of its asymptotes an axis of

symmetry for 𝑓 as shown. Answer the following:

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4.1.1 Write down the equations of the asymptotes of 𝑔. (2)
4.1.2 Determine the values of:
4.1.2.1 𝑡 (2)
4.1.2.2 𝑝 and 𝑞 (3)
𝑎
4.2 Write 𝑔 in the form: 𝑦 = 𝑥+ 𝑝 + 𝑞 (3)

4.3 If ℎ(𝑥) = 𝑥 − 1 is the line of symmetry to 𝑔. Determine the coordinates of the

points of intersection of ℎ and 𝑔. (6)
4.4 If 𝑘 = 𝑥 2 + 4𝑥 + 3 Determine the values of 𝑘 if its roots are
4.4.1 non-real (2)
4.4.2 negative and unequal (2)
4.5 Write down the:
4.5.1 domain of 𝑔 (2)
4.5.2 range of 𝑓 (1)

2014 Standard Grade Paper 1 Q 3.1.1 – Q 3.1.4

The diagram below (not drawn to scale) represents the graphs of:
• A parabola f which cuts the 𝑦-axis at −2 passes through points P (−1; 1), and
Q(2; −2) and has turning point R.
• A straight line g that passes through Q and R.

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 • A hyperbola ℎ(𝑥) < 0, , that passes through point P.

3.1.1 Show that the equation of the parabola is 𝑦 = 𝑥 2 − 2𝑥 − 2 . (6)

3.1.2 Show that R is the point (1; −3). (3)
3.1.3 Determine the equation of 𝑔 in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 . (3)
3.1.4 Write down the equation of the hyperbola ℎ . (2)

2017 IEB March Paper 1 Q 6 (a) − (d)

If 𝑓(𝑥) = 2𝑥 2 − 20𝑥 + 47 and ℎ(𝑥) = 𝑏 𝑥 + 𝑞
a) Rewrite the equation of in the form 𝑓(𝑥) = 𝑎(𝑥 + 𝑝)2 + 𝑞 (4)
b) Given that ℎ(𝑥) = 𝑏 𝑥 + 𝑞 as a y-intercept of (0; −2)and passes through, (1; −1),
determine 𝑏 and 𝑞. (3)
c) Sketch the graphs of 𝑓 and ℎ on the same set of axes. Indicate on your graphs the
coordinates of the turning point of 𝑓 and the 𝑦-intercept of ℎ. Also indicate
asymptote(s), if any, by means of dotted lines. (5)
d) On your sketch, indicate by using the letter A, where 𝑓(𝑥) − ℎ(𝑥) = 0 (1)

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2015 IEB November Paper 1 Q 2
1
a) Given 𝑓(𝑥) = 𝑥+ 1 + 2

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

2) Determine the 𝑥 and 𝑦-intercepts of the graph of 𝑓. (3)
3) Sketch the graph of 𝑓 . Show all asymptotes and intercepts with axes. (3)
b) Given 𝑔(𝑥) = 2.3𝑥 − 1
1) Determine the intercepts with axes, correct to 2 decimal digits, if necessary. (4)
2) Sketch the graph of 𝑔 . Label clearly all asymptotes and intercepts with axes. (3)

2016 Overberg Preliminary Paper 1 Q 4

The sketch shows 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and 𝑔(𝑥) = 𝑚𝑥 + 𝑐. The turning point of the
parabola is (1; 8). Both 𝑓(𝑥) and 𝑔(𝑥) pass through the point (0 ; 6).

4.1 Show that 𝑎 = −2 ; 𝑏 = 4 𝑎𝑛𝑑 𝑐 = 6 (5)

4.2Determine the equation of 𝑔(𝑥). (5)
4.3 Determine the coordinates of the point of contact of the tangent to 𝑓(𝑥)
that is parallel to 𝑔(𝑥). (4)

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4.4 If PQ is a line parallel to the 𝑦-axis with P on 𝑓(𝑥) and Q on 𝑔(𝑥),
determine the length of PQ in terms of 𝑥. (2)
4.5 For what values of 𝑥 is 𝑓(𝑥). 𝑔(𝑥) < 0? (2)
4.6 For what values of 𝑥 is 𝑓(𝑥) ≥ 𝑔(𝑥)? (2)

2017 IEB March Paper 1 Q 7

−2𝑥+2
Given 𝑓(𝑥) = 𝑥+1
𝑎
Express 𝑓(𝑥) in the form 𝑓(𝑥) = + 𝑞 and hence write down the equation of the
𝑥+𝑝

asymptotes of 𝑓(𝑥). (6)

2016 IEB November Paper 1 Q 1 (b) and (c)

(b) Given: 𝑓(𝑥) = 2(𝑥 + 2)2 − 8
Sketch the graph of 𝑓. Show the turning point and intercept with the axes (5)
4
(c) Given 𝑔(𝑥) = 𝑥+1 + 2

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

2015 IEB November Paper 1 Q 6 (a)

a) The graph of 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 is sketched below where 𝑎, ; 𝑏; 𝑐 ∈ 𝑅

For each of the equations given, choose the statement (i), (ii), or (iii)) that applies.

22
 (1) 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 (1)
(2) 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = −2 (1)
(3) 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 4 (1)
(i) Roots are non-real
(ii) Roots are real and unequal
(iii) Roots are real and equal
(2) Determine the points of intersection of the graphs of 𝑔(𝑥) and 𝑦 = 𝑥. (4)

2014 IEB Exemplar Paper 1 Q 6

−4
Sketched are the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) where 𝑓(𝑥) = 𝑥+1 − 3

Determine:
(a) The equations of the asymptotes of 𝑓. (2)
(b) The length of AB, if A and B are the 𝑥 and 𝑦 intercepts of 𝑓. Leave answer in surd
form. (6)
(c) The equation of 𝑔, if 𝑔 is an axis of symmetry of 𝑓. (2)

2008 Higher Grade June Paper 1 Q 1.3

The graphs of 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3 and 𝑔(𝑥) = 2𝑥 + 𝑎 do not intersect. Find all the
possible values of 𝑎. (6)

23
2010 Higher Grade June Paper 1 Q 1.3
Determine the points of intersection of the graphs of 𝑥 2 + 𝑦 2 − 25 = 0 and
𝑥 − 2𝑦 = 5. (5)

2012 Higher Grade June Paper 1 Q 2.1.1

4
Given: 𝑓(𝑥) = 3 𝑥 2 − 4𝑥 and 𝑔(𝑥) = 3 − 𝑥

Sketch the graphs of 𝑓 and 𝑔 on the same system of axes, clearly indicating the
intercepts with the axes as well as the turning point of 𝑓 on your graph. (7)

2012 IEB November Paper 1 Q 3 (a) – (c)

(a) Refer to the figure showing a sketch of 𝑦 = 𝑓(𝑥)

(1) Give the domain and range for the graph. (3)
(2) Use the graph to determine the values of 𝑥 for which 𝑓(𝑥) > 0 (2)
(3) Use the graph to determine the values of 𝑥 for which𝑓´(𝑥) < 0 . (1)
(4) Determine 𝑓(𝑓(3)). (2)
(b) Refer to the figure showing a hyperbola.
Determine the equation of this graph. (4)

24
(c) Given: 𝑔(𝑥) = 3𝑥 2 − 7
The graph of 𝑦 = 𝑔(𝑥) is shifted 3 units down and 2 units to the left, giving
𝑦 = ℎ(𝑥).Determine an expression for ℎ(𝑥) in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. (4)

2008 Standard Grade Paper 1 Q 2

The graph below, which is not drawn to scale, shows a parabola with 𝑥-intercepts at
A (−3; 0) and B. The axis of symmetry has equation 𝑥 = −1 and the 𝑦-intercept is
C (0; −6).

25
Determine:
2.1.1 The coordinates of B. (2)
2.1.2 The equation of the parabola. (5)
2.1.3 The coordinates of the turning point D. (4)
2.1.4 The equation of the straight line passing through A and C. (2)
2.1.5 The values of 𝑥 for which 𝑓(𝑥) ≥ 0. (2)

2009 Standard Grade Paper 1 Q 3

The sketch shows the graphs of 𝑓(𝑥) = −2𝑥 2 − 4𝑥 + 6 and 𝑔(𝑥). A, B and C are the
intercepts of 𝑓 with the coordinate axes. T is the turning point of the graph of 𝑓 .The graph
of 𝑔(𝑥) = 𝑚𝑥 + 𝑐 is a straight line parallel to AC, and is a tangent to the graph of 𝑓(𝑥) at D.

3.1.1 Determine the lengths of OC and AB. (5)

3.1.2 Determine the equation of the axis of symmetry of the graph of 𝑓. (2)
3.1.3 Show that the length of ST = 8 units. (3)
3.1.4 If the graph of 𝑔(𝑥) = 𝑚𝑥 + 𝑐 is a straight line through D parallel to AC, calculate:
a) The gradient of AC. (3)
b) The value of 𝑚. (1)
c) The coordinates of D. (5)

26
 12
3.2 Given ℎ(𝑥) = − 𝑥

3.2.1 Draw the graph of ℎ. Show the coordinates of any one point on the graph. (2)
2
3.2.2 Determine: ℎ(𝑥 2 ) − (ℎ(𝑥)) . (2)

2014 March Paper 1 Q 6

6
Given 𝑓(𝑥) = +3
𝑥 −2

6.1 Write down the equations of the asymptotes of the graph of 𝑓. (2)
6.2 Write down the domain of 𝑓. (1)
6.3 Draw a sketch graph of 𝑓 in your ANSWER BOOK, indicating the intercept(s) with the
axes and the asymptotes. (4)
6.4 The graph of 𝑓 is translated to 𝑔. Describe the transformation in the form∶ (𝑥 ; 𝑦)
if the axes of symmetry of 𝑔 are 𝑦 = 𝑥 + 3 and 𝑦 = −𝑥 + 1. (4)

2014 March Paper 1 Q 7

The graph of where 𝑓(𝑥) = 𝑎(𝑥 − 𝑝)2 + 𝑞 where 𝑎, 𝑝 and 𝑞 are constants, is given
below. Points E, F (1; 0) and C are its intercepts with the coordinates axes. A (−4; 5) is
the reflection of 𝐶 across the axis of symmetry of 𝑓. D is a point on the graph such that
the straight line through A and D has equation 𝑔(𝑥) = −2𝑥 − 3.

27
7.1 Write down the coordinates of C. (1)
7.2 Write down the equation of the axis of symmetry of 𝑓. (1)
7.3 Calculate the values of 𝑎, 𝑝 and 𝑞. (6)
7.4 If 𝑓(𝑥) = −𝑥 2 − 4𝑥 + 5 calculate the 𝑥-coordinate of D. (4)
7.5 The graph of 𝑓 is reflected about the 𝑥-axis.
Write down the coordinates of the turning point of the new parabola. (2)

2014 November Paper 1 Q 4

2
The diagram below shows the hyperbola 𝑔 defined by 𝑔(𝑥) = 𝑥 + 𝑝 + 𝑞 with asymptotes

𝑦 = 1 and 𝑥 = −1. The graph of 𝑔 intersects the 𝑥-axis at T and the 𝑦-axis at (0; 3). The
line 𝑦 = 𝑥 intersects the hyperbola in the first quadrant at S.

4.1 Write down the values of 𝑝 and 𝑞. (2)

4.3 Calculate the 𝑥-coordinate of T. (2)
4.3 Write down the equation of the vertical asymptote of the graph of ℎ if
ℎ(𝑥) = 𝑔(𝑥 + 5). (1)
4.4 Calculate the length of OS. (5)
4.5 For which values of 𝑘 will the equation 𝑔(𝑥) = 𝑥 + 𝑘 have two real roots that are
of opposite signs. (1)

28
2015 Metro East June Paper 1 Q 3
The graphs of 𝑓(𝑥) and 𝑔(𝑥)are shown in the diagram below. The turning point of 𝑓(𝑥) is
A (2; 9) and the graphs 𝑓 and 𝑔 intersects at B (3; 8). C is a point on 𝑔(𝑥) and is on the
axis of symmetry of 𝑓.

3.1 Show that the function 𝑓 can be defined by the equation:

𝑓(𝑥) = −𝑥 2 + 4𝑥 + 5 (4)
3.2 Write down the equation of the axis of symmetry of 𝑓. (1)
3.3 The graph (𝑥), has the equation 𝑦 = 𝑎 𝑥 . Determine the value of 𝑎. (2)
3.4 If it is given that (−1; 0) is one root of 𝑓, write down the coordinates of the other
root. (1)
3.5 For which value(s) of 𝑥 will 𝑓(𝑥) < 0? (2)
3.6 Determine the length of AC.
3.7 Discuss the nature of the roots of ℎ(𝑥) if ℎ(𝑥) = 𝑓(𝑥) − 9. (2)

2015 Metro East June Paper 1 Q 4

𝑎
In the diagram below are the sketches of : 𝑓(𝑥) = −2𝑥 + 2 and 𝑔(𝑥) = 𝑥 + 𝑃 + 𝑞.

The graph of 𝑔 cuts the 𝑦-axis at (0; 4).

29
4.1 Calculate the values of 𝑝 and 𝑞. (3)
4.2 Show that 𝑎 = −2. (2)
4.3 Determine the coordinates of point A the 𝑥-intercept of 𝑔(𝑥). (2)
4.4 The value of 𝑞 is increased by 1 unit. What effect will it have on the graph of 𝑔? (1)

2015 Senior Certificate Paper 1 Q 4

𝑎
The diagram below shows the graphs of 𝑔(𝑥) = 𝑥 −1 − 2. The point (0; −5) lies on 𝑔.

4.1 Write down the range of 𝑔. (2)

30
4.2 Determine the value of 𝑎. (2)
4.3 If another function ℎ defined as ℎ(𝑥) = 𝑔(𝑥 − 3) + 7, determine the coordinates
of the point intersection of the asymptotes of ℎ. (3)

2015 Senior Certificate Paper 1 Q 6

The sketch below shows the graphs of 𝑔(𝑥) = −12𝑥 + 12 and 𝑓(𝑥) = −3𝑥 2 − 9𝑥 + 30.
A and B are the 𝑥-intercepts of 𝑓 and C is a point on 𝑓. D is a point on 𝑔 such that CD is
parallel to the 𝑦-axis. H and K are the points of intersection of 𝑓 and 𝑔.

6.1 Determine the length of AB. (4)

6.2 Determine the coordinates of K. (5)
6.3 Determine the values of 𝑥 for which 𝑓(𝑥) − 𝑔(𝑥) ≤ 0. (3)
6.4 Determine the maximum length of CD for −2 ≤ 𝑥 ≤ 3. (5)

2015 March Paper 1 Q 4

6
Given 𝑓(𝑥) = 𝑥 −1
+2

4.1 Write down the equation of the asymptote of 𝑔 . (2)

31
4.2 Calculate:
4.2.1 The 𝑦-intercept of 𝑔. (1)
4.2.2 The 𝑥-intercept of 𝑔. (2)
4.3 Draw the graph of 𝑔, showing clearly the asymptotes and the intercepts with
the axes. (3)
4.4 Determine the equation of the line of symmetry that has a negative gradient, in
the form 𝑦 = …… (3)
6
4.5 Determine the value(s) of 𝑥 for which − 1 ≥ −𝑥 − 3. (2)
𝑥+2

2015 March Paper 1 Q 6

The graphs of 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐; 𝑎 ≠ 0 and 𝑔(𝑥) = 𝑚𝑥 + 𝑘 are drawn below.
D (1; −8) is a common point on 𝑓 and 𝑔.
• 𝑓 intersects the 𝑥-axis at (−3; 0) and (2; 0).
• 𝑔 is a tangent to 𝑓 at D.

6.1 For which value(s) of 𝑥 is 𝑓(𝑥) ≤ 0? (2)

6.2 Determine the values of 𝑎, 𝑏 and 𝑐. (5)
6.3 Determine the coordinate of the turning point of 𝑓. (3)
6.4 Write down the equation of the axis of symmetry of ℎ if ℎ(𝑥) = 𝑓(𝑥 − 7) + 2. (2)

32
6.5 Calculate the gradient of 𝑔. (3)

2015 Eastern Cape Preliminary Paper 1 Q 6

Given 𝑓(𝑥) = 2𝑥 2 − 10𝑥 − 28 and 𝑔(𝑥) = 𝑚𝑥 + 𝑘.
6.1 Write down the 𝑦-intercept of 𝑓. (1)
6.2 Determine the 𝑥-intercept of 𝑓. (3)
6.3 Determine the coordinates of the turning point of 𝑓. (2)
6.4 Sketch the graph of 𝑓.Clearly show the intercepts with both axes as well as the
coordinates of the turning point. (2)
6.5 Determine the coordinates of point P, a point on 𝑓, where the gradient of the
tangent to 𝑓 at P is equal to 6. (4)
6.6 Determine the equation of 𝑔, the straight line passing through the points (−2; 0)
and (4; −36). (3)
6.7 Write down the equation of ℎ in the form ℎ(𝑥) = (𝑥 + 𝑝)2 + 𝑞 if
ℎ(𝑥) = 𝑓(𝑥 + 2) − 3. (3)

2015 Kwazulu Natal Prelimary Paper 1 Q 6

𝑎
The diagram below shows the graph of 𝑓(𝑥) = 𝑥 + 𝑃 + 𝑞. The lines 𝑥 = −1 and 𝑦 = 1 are

the asymptoes of 𝑓. P (−2; 4) is a point on 𝑓 and T is the 𝑥-intercept of 𝑓.

33
6.1 Determine the values of 𝑎, 𝑝 and 𝑞. (4)
6.2 Calculate the coordinates of T, the 𝑥-intercept of 𝑓. (3)
6.3 If the graph of 𝑓 is symmetrical with respect to the line 𝑦 = 𝑥 + 𝑐, determine the
value of 𝑐. (2)

2015 Mpumalanga Preliminary Paper 1 Q 4

Sketched below are graphs of : 𝑔(𝑥) = 𝑥 − 2
and 𝑓(𝑥) = 𝑦 = 𝑎(𝑥 + 𝑝)2 + 𝑞
P (3; 2) is the turning point of 𝑓. A and B are the 𝑥-intercept of 𝑓, D is the point of intersection
of 𝑓 and 𝑔. C is the 𝑦-intercept of 𝑔.

34
4.1 Determine the equation of 𝑓 in the form 𝑦 = 𝑎( 𝑥 + 𝑝)2 + 𝑞. (4)
4.2 Calculate the coordinates of A and D. (5)
4.3 Write down the values of 𝑥 for which:
4.3.1 𝑓 ′ (𝑥) > 0. (2)
4.3.2 𝑓(𝑥). 𝑔(𝑥) ≥ 0. (2)
4.4 Describe the transfromation from 𝑓 to 𝑝 if 𝑝(𝑥) = −2𝑥 2 . (2)

2016 March Paper 1 Q 4

Given: 𝑓(𝑥) = 2−𝑥 + 1
4.1 Determine the coordinates of the 𝑦-intercept of 𝑓. (1)
4.2 Sketch the graph of 𝑓, clearly indicating ALL intercepts with the axes as well as any
asymptotes. (3)
4.3 Calculate the average gradient of 𝑓 between the points on the graph where
𝑥 = −2 and 𝑥 = 1. (3)
4.4 If ℎ(𝑥) = 3𝑓(𝑥), write down an equation of the asymptote of ℎ. (1)

2016 Eastern Cape June Paper 1 Q 3

3
Given: 𝑓(𝑥) = 𝑥 −1 − 2

3.1 Write down the equation of the:

35
3.1.1 horizontal asymptote of 𝑓. (1)
3.1.2 vertical asymptote of 𝑓. (1)
3.2 Determine the 𝑥 and 𝑦-intercepts of 𝑓. (3)
3.3 Sketch the graph of 𝑓,showing clearly the asymptotes and the intercepts with the
axes. (3)
3.4 If another function 𝑔 is defined as 𝑔(𝑥) = 𝑓(𝑥 − 3) + 7 , determine the coordinates
of the point of intersection of the asymptotes of 𝑔. (2)

2016 Eastern Cape June Paper 1 Q 4

The functions 𝑓(𝑥) = −𝑥 2 − 2𝑥 + 3and 𝑔(𝑥) = 𝑚𝑥 + 𝑐 are drawn below, with 𝑔 passing
through E, C and A. A and B are the 𝑥-intercepts of , and CD is the axis of symmetry of 𝑓 .
E is the 𝑦 -intercept of 𝑔.

4.1 Determine the coordinates of C, the turning point of the graph of 𝑓 . (3)
4.2 Determine the coordinates of A and B. (3)
4.3 Determine the values of 𝑚 and 𝑐. (2)
4.4 Calculate the length of CE. (leave your answer in surd form) (3)
4.5 Determine the values of 𝑥, for which 𝑓(𝑥). 𝑔(𝑥) < 0. (2)

36
2016 North West Preliminary Paper 1 Q 4
−3
Given 𝑓(𝑥) = 𝑥 +1
− 2

3.1 Calculate the coordinates of the 𝑦-intercept of 𝑓. (2)

3.2 Calculate the coordinates of the 𝑥-intercept of 𝑓. (2)
3.3 Sketch the graph of 𝑓 in your ANSWER BOOK, clearly showing the asymptotes and
the intercept with the axes. (3)
3.4 Write down the range of 𝑓. (2)
3.5 Another function ℎ, is formed by translating 𝑓 3 units to the right and 4 units down.
Write down the equation of ℎ. (2)
3.6 For which value(s) of 𝑥 is ℎ(𝑥) ≤ −4? (3)
3𝑥−5
3.7 Determine the equation of the asymptotes of 𝑘(𝑥) = . (3)
𝑥 −1

2016 Mpumalanga Preliminary Paper 1 Q 3

Sketched below are the graphs of:
𝑔(𝑥) = −2𝑥 + 8
𝑓(𝑥) = 𝑥 2 + 𝑐
6
and ℎ(𝑥) = 𝑥 − 2 + 1

A and B are the 𝑥 and 𝑦-intercepts of ℎ respectively, C (−6; 20) and E are the points of
intersection of 𝑓 and 𝑔.

37
3.1 Calculate the coordinates of A, B and E. (4)
3.2 Show that the value of 𝑐 = −16. (2)
3.3 Write down the values of 𝑥 for which 𝑔(𝑥) − 𝑓(𝑥) ≥ 0. (2)
3.4 Determine the equation of the symmetry axis of ℎ if the gradient is negative. (2)
3.5 Show that the length of BE= 2√5. (2)
3.6 Write down the range of 𝑠, if 𝑠(𝑥) = ℎ(𝑥) + 2. (2)
3.7 If 𝑡 is a tangent to 𝑓 and parallel to 𝑔, determine the equation of the tangent, 𝑡 in
the form 𝑦 = 𝑚𝑥 + 𝑐. (5)

2016 Kwazulu Natal Preliminary Paper 1 Q 6

𝑎
The diagram below shows the graph of ℎ(𝑥) = + 𝑞 . The lines 𝑥 = 3 and 𝑦 = −2
𝑥+𝑝

are asymptotes of ℎ. P (−4; 4) is a point on ℎ.

38
6.1 Write down the values of 𝑝 and 𝑞. (2)
6.2 Calculate the value of 𝑎. (2)
6.3 Calculate the coordinates of the 𝑦-intercept of ℎ. (2)
6.4 If 𝑔(𝑥) = ℎ(𝑥 + 2) , write down the equation of the vertical asymptote of 𝑔. (2)
6.5 If the graph of ℎ is symmetrical about the line 𝑦 = −𝑥 + 𝑐 , determine the value
of 𝑐. (2)

2016 Kwazulu Natal Preliminary Paper 1 Q 7

The sketch below shows the graphs of 𝑔(𝑥) = 𝑥 2 − 3𝑥 − 10 and ℎ(𝑥) = 𝑎𝑥 + 𝑞 .
The graphs intersect at B and D. The graph of 𝑔 intersects the 𝑥-axis at A and B and has a
turning point at C. The graph of ℎ intersects the 𝑦-axis at D and the 𝑥-axis at B.

39
7.1 Write down the coordinates of D. (1)
7.2 Determine the coordinates of A and B. (4)
7.3 Write down the values of 𝑎 and 𝑞. (2)
7.4 Calculate the coordinates of C, the turning point of 𝑔. (3)
7.5 Write down the turning point of 𝑡, if 𝑡(𝑥) = 𝑔(−𝑥) + 3 (2)
7.6 For which values of 𝑥 will 𝑔′ (𝑥). ℎ′ (𝑥) ≥ 0 ? (2)

2016 Eastern Cape Preliminary Paper 1 Q 7

𝑎
The sketch below shows the graph of 𝑓(𝑥) = 𝑥 + 𝑝 + 𝑞 and 𝑔(𝑥) = 𝑏 𝑥 + 𝑐. The 𝑥-intercept

of 𝑓 is at (−6; 0) and the 𝑦 -intercept of 𝑓 and 𝑔 is at (0; −3). The point (2; 5) lies on the
graph of 𝑔.

40
7.1.1 For which value(s) of 𝑥 is 𝑓(𝑥) = 𝑔(𝑥)? (1)
7.1.2 For which values of 𝑥 is 𝑓(𝑥) < 𝑔(𝑥)? (2)
7.1.3 Write down the equation of the asymptote of 𝑔 . (1)
7.1.4 Determine the equation of 𝑔 . (4)
7.1.5 Write down the equations of the asymptotes of 𝑓. (2)
7.1.6 Determine the equation of 𝑓. (3)
7.1.7 Determine the equations of the axes of symmetry of 𝑓. (3)

2016 Gauteng Preliminary Paper 1 Q 4

𝑎
The graph of 𝑓(𝑥) = 𝑥 + 𝑝 + 𝑞 is sketched below with asymptotes 𝑥 = 4 and 𝑦 = 2.

T (5; 3) is a point on 𝑓 and C is the point of intersection of the asymptotes.

41
4.1 Determine the values of 𝑎, 𝑝 and 𝑞. (3)
4.2 Give the equation of ℎ, the reflection of 𝑓 in the 𝑦-axis. (1)
4.3 If the graph of 𝑓 is symmetrical about the line 𝑦 = −𝑥 + 𝑐, determine the
value of 𝑐. (2)

2016 Eastern Cape Preliminary Paper 1 Q 8

The sketch below, which is not drawn to scale, shows the graph of 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and a
straight line 𝑔, passing through the origin. The 𝑦-intercept of 𝑓 is (0; 4). Point A (−3; −5) is
the turning point of 𝑓.

42
8.1 Write down the equation of the axis of symmetry of 𝑓. (1)
8.2 Show by calculation that 𝑎 = 1 and 𝑏 = 6. (3)
8.3 Discuss the nature of the roots of 𝑓. (3)
8.4 𝑔 is a tangent to 𝑓 and the gradient of line 𝑔 is 2. Determine the coordinates of
the point of contact. (4)

2017 March Paper 1 Q 5

The sketch below shows the graphs of 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 3 and 𝑔(𝑥) = 𝑥 − 3.
• A and B are the 𝑥-intercepts of 𝑓
• The graphs of 𝑓 and 𝑔 intersect at C and B
D is the turning point of 𝑓.

5.1.1 Determine the coordinates of C. (1)

5.1.2 Calculate the length of AB. (4)
5.1.3 Calculate the coordinates of D. (2)
5.1.4 Calculate the average gradient of 𝑓 between C and D. (2)
5.1.5 Calculate the size of ∠OCB. (2)
5.1.6 Determine the values of 𝑘 for which 𝑓(𝑥) = 𝑘 will have unequal positive real
roots. (3)

43
5.17 For which values of 𝑥 will 𝑓 ′ (𝑥). 𝑓 ′′ (𝑥) > 0 ? (3)
5.2 The graphs of a parabola 𝑓 has an 𝑥-intercepts at 𝑥 = 1 and 𝑥 = 5. 𝑔(𝑥) = 4 is
a tangent to 𝑓 at P, the turning point of 𝑓.Sketch the graph of 𝑓, clearly showing the
intercepts with the axes and the coordinates of the turning point. (5)

2017 Eastern Cape June Paper 1 Q 4

𝑎
Given the graph of 𝑓, a hyperbola of the form 𝑦 = 𝑥 + 𝑞 , answer the questions that
+ 𝑝

follow.

4.1 Write down the values of 𝑝 and 𝑞. (2)

4.2 Determine the value of 𝑎, and write down the equation of 𝑓 in the form 𝑦 = .. (3)
4.3 The axes of symmetry of 𝑓 are 𝑦 = 𝑥 + 3 and 𝑦 = −𝑥 + 1. The graph of 𝑓 is
transformed to 𝑔 such that the axes of symmetry of g are given by 𝑦 = 𝑥 − 3 and
𝑦 = −𝑥 + 1. Describe the transformation. Show all calculations to support
your answer. (5)

44
2017 Eastern Cape June Paper 1 Q 3
3.1 Given a function 𝑓: 𝑦 + 4 = (𝑥 − 5)2
3.1.1 Write down the equation of the axis of symmetry of 𝑓. (1)
3.1.2 Determine the 𝑥-intercepts of 𝑓. (3)
3.1.3 Sketch the graph of 𝑓, clearly showing the intercepts with the axes and the
turning point. (4)
3.1.4 Write down the range of 𝑓. (1)
3.1.5 𝑓(𝑥) is transformed to 𝑔(𝑥) where the 𝑥-intercepts of 𝑔(𝑥) is the same as that of
𝑓(𝑥) and the turning point of 𝑔(𝑥) is (5; 4).Describe the transformation and write
down the equation of 𝑔(𝑥). (2)
3.2 Given:
𝑓(𝑥) = 𝑥 2 + 3 and 𝑔(𝑥) = 𝑘𝑥 − 1, determine the value(s) of 𝑘 if 𝑔 a tangent to the
graph of 𝑓. (5)

2017 Senior Certificate Paper 1 Q 5

Given 𝑓(𝑥) = 𝑥 2 − 5𝑥 − 14 and 𝑔(𝑥) = 2𝑥 − 14
5.1 On the same set of axes, sketch the graphs of 𝑓 and 𝑔. Clearly indicate all intercepts
with the axes and turning points. (6)
1
5.2 Determine the equation of the tangent to 𝑓 at 𝑥 = 2 2 . (2)

5.3 Determine the value(s) of 𝑘 for which 𝑓(𝑥) = 𝑘 will have two unequal positive real
roots. (2)
5.4 A new graph ℎ is obtained by first reflecting 𝑔 in the 𝑥-axis and then translating it
7 units to the left. Write down the equation of ℎ in the form ℎ(𝑥) = 𝑚𝑥 + 𝑐. (2)

2017 Eastern Cape Preliminary Paper 1 Q 5

1
The sketch shows the graph of 𝑓(𝑥) = 𝑥(𝑥 + 3) and 𝑔(𝑥) = − 2 𝑥 + 2

45
5.1 Determine the coordinates of A. (1)
5.2 Calculate the coordinates of P, the turning point of 𝑓 . (3)
5.3 Determine the average gradient of 𝑓 between 𝑥 = −5 and = −3 . (3)
5.4 Determine the value(s) of 𝑥 for which 𝑓(𝑥) > 0. (2)
5.5 Determine the coordinates of the turning point of ℎ if ℎ(𝑥) = 𝑓 (𝑥 − 2). (2)
5.6 L is a point on the straight line and M is a point on the parabola. LM is
perpendicular to the 𝑥-axis. Show that the length LM can be written as:
7 2 81
LM = − (𝑥 + ) +
4 6
(4)

2017 Eastern Cape Preliminary Paper 1 Q 7

𝑑−𝑥
A sketch of the hyperbola 𝑓(𝑥) = 𝑥 − 𝑝 where 𝒑 and 𝒅 are constants, is given below.

The dotted lines are the asymptotes. The point 𝐴 (5; 0) is given on the graph of 𝑓.

46
7.1 Determine the values of 𝒅 and 𝒑 . (2)
3
7.2 Show that the equation can be written as 𝑦 = 𝑥 − 2 − 1 (2)

7.3 Write down the image of 𝐴 if 𝐴 is reflected about the axis of symmetry
𝑦 = 𝑥 − 3. (2)

2018 March Paper 1 Q 4

Below are the graphs of 𝑓(𝑥) = (𝑥 − 4)2 − 9 and a straight line 𝑔.
• A and B are the 𝑥-intercepts of 𝑓 and E is the turning point of 𝑓.
• C is the 𝑦-intercept of both 𝑓 and 𝑔.
• The 𝑥-intercept of 𝑔 is D. DE is parallel to the 𝑦-axis.

47
4.1 Write down the coordinates of E. (2)
4.2 Calculate the coordinates of A. (3)
4.3 M is the reflection of C in the axis of symmetry of 𝑓. Write down the coordinates
of M. (3)
4.4 Determine the equation of 𝑔 in the form 𝑦 = 𝑚𝑥 + 𝑐 (3)
4.5 Write down the equation of 𝑔−1 in the form 𝑦 =.. (3)
4.6 For which values of 𝑥 will 𝑥(𝑓(𝑥)) ≤ 0? (4)

2018 Eastern Cape June Paper 1 Q 3

The diagram below represents the graphs of 𝑓(𝑥) = 𝑎(𝑥 − 2)2 + 4 and 𝑔(𝑥) = 𝑏 𝑥 . The
graphs intersect at P, the turning point of 𝑓 and at Q , the 𝑦-intercept of both 𝑓 and 𝑔.

48
3.1.1 Write down the coordinates of P and Q. (2)
3.1.2 Determine the values of 𝑎 and 𝑏. (4)
3.1.3 How can the domain of 𝑓 be restricted such that 𝑓 −1 may be a function? (2)
3.1.4 Determine the maximum value of ℎ(𝑥) = 𝑔[𝑓(𝑥)]. (2)
Consider the following two functions: 𝑝(𝑥) = 𝑥 2 + 1 + and 𝑟(𝑥) = 𝑥 2 + 2𝑥
3.2.1 Write down the range of 𝑝. (1)
3.2.2 Describe the transformation from 𝑝 to 𝑟. (3)

2018 Eastern Cape June Paper 1 Q 4

−3
Given the equation of 𝑓, a hyperbola, 𝑓(𝑥) = 𝑥 + 1 + 5, answer the questions that follow.

4.1 Calculate the 𝑦-intercept of 𝑓. (1)

4.2 Calculate the 𝑥-intercept of 𝑓. (2)
4.3 Sketch the graph of 𝑓, clearly indicating the asymptotes and intercepts with axes. (3)
4.4 Write down the equation of the graph formed if the graph 𝑓 is shifted 3 units to
the right and then reflected across the 𝑥-axis. (3)

49
2018 Senior Certificate Paper 1 Q 5
2
The graphs of 𝑓(𝑥) = 𝑥 + 1 + 4 and parabola 𝑔 are drawn below.

• C, the turning point of 𝑔, lies on the horizontal asymptote of 𝑓.

• The graph of 𝑔 passes through the origin.
14
• B (𝑘 ; )is a point on f such that BC is parallel to the 𝑦-axis.
3

5.1 Write down the domain of 𝑓. (2)

5.2 Determine the 𝑥-intercept of 𝑓. (2)
5.3 Calculate the value of 𝑘. (3)
5.4 Write down the coordinates of C. (2)
5.5 Determine the equation of 𝑔 in the form 𝑦 = 𝑎(𝑥 + 𝑝)2 + 𝑞. (3)
𝑓(𝑥)
5.6 For which value(s) of 𝑥 will ≤ 0? (4)
𝑔(𝑥)

5.7 Use the graph of 𝑓 and 𝑔 to determine the number of real roots of
2
− 5 = −(𝑥 − 3)2 − 5. Give reasons for your answer. (4)
𝑥

2018 Winelands Preliminary Paper 1 Q 4

𝑘
In the sketch below, the graphs of 𝑔(𝑥) = 𝑥 + 𝑃 + 4 and 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 are given.

The asymptotes of 𝑔 intersect at B, the turning point of 𝑓. The graphs of 𝑓 and 𝑔 intersect

50
at C. The axis of symmetry of 𝑔 that has a negative gradient, is the line ℎ(𝑥) that intersects
the graph of 𝑓 at A (2; 3) and B.

4.1 Determine the equation of ℎ. (2)

4.2 Show that the coordinates of B are (1; 4) . Clearly show all calculations. (2)
4.3 Show that the equation of 𝑓(𝑥) is given by 𝑓(𝑥) = −𝑥 2 + 2𝑥 + 3
Clearly show all calculations. (4)
4.4 Determine the equation of 𝑔. (3)
4.5 Determine the equations of the asymptotes of 𝑔(𝑥 + 1). (2)
4.6 Determine the value(s) of 𝑥 for which 𝑔′ (𝑥). 𝑓 ′ (𝑥) ≥ 0? (2)

2018 Winelands Preliminary Paper 1 Q 6

𝑘
Sketch the graph of 𝑓(𝑥) = 𝑥 + 𝑝 + 𝑞 if:

• the domain is given as : 𝑥 ∈ 𝑅; 𝑥 ≠ −1.

• the range is given as: 𝑦 ∈ 𝑅; 𝑦 ≠ 2.
• 𝑘<0
1
• the 𝑥-intercepts is : (− 2 ; 0 ).

• 𝑓(0) = 1 (5)

51
2018 North West Preliminary Paper 1 Q 5
2
The graphs of 𝑦 = 𝑎(𝑥 + 𝑝)2 + 𝑞 and 𝑔(𝑥) = 𝑥 + 1 − 3 are sketched below. P is the

𝑦-intercept of 𝑓 and 𝑔. The horizontal asymptote of 𝑔 is also a tangent to 𝑓 at the turning

point of 𝑓.

5.1 Write down the equation of the vertical asymptote of 𝑔. (1)

5.2 Determine the coordinates of P. (2)
5.3 Determine the equation of 𝑓. (3)
5.4 One of the axes of symmetry of 𝑔 is a decreasing function. Write down the equation
of this axis of symmetry, ℎ(𝑥). (2)
5.5 For which values of 𝑘 will 𝑔(𝑥) = ℎ(𝑥) + 𝑘 have TWO real roots that are of
opposite signs? (2)
5.6 Give the domain of 𝑚(𝑥) if 𝑚(𝑥) = 𝑔(2𝑥) + 5. (3)

2018 North West Preliminary Paper 1 Q 4

Given:
𝑘
𝑓(𝑥) = +𝑞
𝑥 + 𝑝
• The point B (−1; 0) is an 𝑥-intercept of 𝑓.

52
 • The domain of 𝑓 is real numbers, but 𝑥 ≠ 2.
• The range of 𝑓 is real numbers, but 𝑦 ≠ 3.
• 𝑓 is a decreasing function.
4.1 Determine the equation of 𝑓. (3)
4.2 Determine the coordinates of the 𝑦-intercept of 𝑓. (2)
4.3 Sketch the graph of 𝑓 in your ANSWER BOOK, clearly showing the asymptotes and
the intercepts with axes. (3)

2018 Mpumalanga Preliminary Paper 1 Q 4

In the diagram below,𝑓(𝑥) = 𝑥 2 + 2𝑥 + 3 and 𝑔(𝑥) = −𝑥 + 3 are drawn. C is the
𝑦-intercept of 𝑓 and 𝑔. A and B are the 𝑥-intercept of 𝑓,and B is the 𝑥-intercept of 𝑔.
D is the turning point of 𝑓. EGF is a straight line parallel to the 𝑦-axis with E on 𝑓 and F
on 𝑔.

3.1 Determine the coordinates of D, the turning point of 𝑓. (4)

3.2 Calculate the length AB. (3)
3.3 Calculate the value of 𝑥 for which EF has a maximum length. (4)
3.4 Determine the range of 𝑝 if 𝑝(𝑥) = 𝑓(𝑥) − 2 (2)

53
3.5 If ℎ(𝑥) = 𝑥 2
3.5.1 Describe the transformation from 𝑓 to ℎ. (3)
3.5.2 Restrict the domain of ℎ for ℎ−1 to be a function. (2)
3.6 Determine the values of 𝑥 for which:
3.6.1 𝑓(𝑥) − 𝑔(𝑥) > 0 (2)
3.6.2 𝑓 ′ (𝑥) < 0 (2)

2018 Mpumalanga Preliminary Paper 1 Q 4

𝑘 1
In the diagram below, the graphs of 𝑓(𝑥) = 𝑎 𝑥 and 𝑔(𝑥) = 𝑥 + 𝑝 + 𝑞 are drawn. P (1; 3)

is the point of intersection of 𝑓 and 𝑔. Q (−2; 9) is the point of 𝑓 and the vertical asymptote
of 𝑔.

4.1 Determine:
4.1.1 the value of 𝑎. (2)
4.1.2 the equation of 𝑔. (4)
4.2 Write down the equation of 𝑓 −1 (𝑥), in the form 𝑓 −1 (𝑥) =… (1)
4.3 If ℎ(𝑥) = 𝑥 + 𝑐 is the axis of symmetry of 𝑔, determine the value of 𝑐. (2)
4.4 Use the graph to find the solution of 𝑙𝑜𝑔𝑎 𝑥 > 0. (2)

54
2018 Limpopo Preliminary Paper 1 Q 4
Sketched below are the graphs of 𝑓(𝑥) = 2𝑥 and 𝑔(𝑥) = −(𝑥 − 1)2 + 𝑞, where 𝑞 is a constant.
The graphs of 𝑓 and 𝑔 at C and D. C is the 𝑦-intercept of both 𝑓 and 𝑔. D is the turning point
of 𝑔.

4.1 Show that 𝑞 = 2. (2)

4.2 Write down the coordinates of the turning point of 𝑔. (2)
Determine the value(s) of 𝑡 for 𝑔(𝑥) = 𝑡 if the roots are equal. (1)
4.4 Write down the equation of 𝑓 −1 (𝑥) in the form 𝑦 = …. (2)
4.5 Sketch the graph of 𝑓 −1 on a system of axes. Indicate the 𝑥-intercept and the
coordinates of one other point on your graph. (3)
4.6 Write down equation of ℎ if ℎ(𝑥) = 𝑔(𝑥 + 1) − 2 (2)
4.7 How can the domain of ℎ be restricted so that ℎ−1 will be a function? (1)

2018 Limpopo Preliminary Paper 1 Q 5

4
Sketched below are the graphs of 𝑦 = 𝑓(𝑥) = 𝑥 + 2 + 1 and 𝑔(𝑥) = 2𝑥 + 3.A, the 𝑦-intercept

of both 𝑓 and 𝑔, and B are the points of intersection of 𝑓 and 𝑔. D is the 𝑥-intercept of 𝑓.

55
5.1 Write down the equations of the asymptotes of 𝑓. (2)
5.2 Determine the coordinates of:
5.2.1 A (1)
5.2.2 B (3)
5.2.3 B (4)
5.3 Calculate the average gradient of 𝑓 between A and D. (2)
5.4 For which value(s) of 𝑥 is 𝑓(𝑥). 𝑔′ (𝑥) ≤ 0 (3)

2018 Kwazulu Natal Preliminary Paper 1 Q 5

Given:
𝑥 − 3
𝑓(𝑥) =
𝑥 +2
5
5.1 Show that 𝑓(𝑥) = 1 − 𝑥 +2 (1)

5.2 Write down the equations of the vertical and horizontal asymptotes of 𝑓. (2)
5.3 Determine the intercepts of the graph of 𝑓 with the 𝑥-axes and 𝑦-axes. (2)
5.4 Write down the value of 𝑐 if 𝑦 = 𝑥 + 𝑐 is a line of symmetry to the graph of 𝑓. (2)

56
2018 Kwazulu Natal Preliminary Paper 1 Q 6
𝑓(𝑥) = 𝑙𝑜𝑔𝑝 𝑥 and 𝑔(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 are sketched below. A is the turning point of f and B is the
common 𝑥-intercept of f and g. The point C (2; −1) lies on the graph of 𝑓

6.1 Calculate the value of 𝑝. (2)

6.2 Write down the coordinates of B. (1)
1
6.3 If 𝑝 = 2 determine the coordinates of A. (3)

6.4 Determine the values of 𝑎 and 𝑏. (4)

6.5 Write down the equation of 𝑓 −1 inverse of 𝑓, in the form 𝑦 = … (2)
6.6 Determine the values of 𝑥 for which 𝑓(𝑥) ≥ −1 (2)
6.7 Determine the values of 𝑥 for which 𝑓(𝑥). 𝑔′ (𝑥) ≤ 0 (2)

2018 Gauteng Preliminary Paper 1 Q 4

The graphs of 𝑓(𝑥) = 2𝑥 − 8 and 𝑔(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 are sketched below. Point Q (0 ; 4,5)
and point D are the 𝑦-intercepts of graph 𝑔 and 𝑓 respectively. The graphs intersect at P, which
is the turning point of graph 𝑔 and the common 𝑥-intercept of 𝑓 and 𝑔.

57
4.1 Write down the equation of the asymptotes of graph 𝑓. (1)
4.2 Determine the coordinates of point P and point D. (4)
4.3 Determine the equation of ℎ if ℎ(𝑥) = 𝑓(2𝑥) + 8. (2)
4.4 Determine the equation of ℎ−1 in the form 𝑦 =…. (2)
4.5 Write down the range of ℎ−1 . (1)
4.6 Determine the equation of 𝑔. (3)
4.7 Calculate:
3 5

∑ 𝑔(𝑘) − ∑ 𝑔(𝑘)
𝑘=0 𝑘=4

(4)
4.8 Describe the transformation that should be applied to graph 𝑔 so that the new
graph obtained will have non-real roots? (1)

2018 Gauteng Preliminary Paper 1 Q 5

1
The graphs of 𝑓(𝑥) = − 2 𝑥 2 + 2𝑥 + 6 and 𝑔(𝑥) = 𝑥 + 2 are sketched below. The graphs

intersect at (−2,0) and (4; 6).

58
Use the graphs to determine the values of 𝑥 for which:
5.1 𝑓(𝑥) = 𝑔(𝑥) (2)
𝑓(𝑥)
5.2 ≥0 (2)
𝑔(𝑥)

5.3 𝑓 ′ (𝑥). 𝑔(𝑥) ≥ 0 (2)

2018 Gauteng Preliminary Paper 1 Q 6

1
Given 𝑓(𝑥) = 4 𝑥 2

6.1 Write down the equation of 𝑔 if 𝑔 is the reflection of 𝑓 about the 𝑦-axis. (1)
6.2 Write down the equation of ℎ if 𝑓 is translated TWO units down to obtain ℎ. (1)
6.3 Write down the range of ℎ. (1)

59
2018 Gauteng Preliminary Paper 1 Q 7
3
The graphs of 𝑓(𝑥) = 𝑥 − 2 − 3 and 𝑔, an axis of symmetry of 𝑓, are sketched below. The

vertical asymptote cuts the 𝑥-axis at C.

7.1 Write down the equation of the vertical asymptote of 𝑓. (1)

3
7.2 Describe how the graph of ℎ(𝑥) = 𝑥 was transformed to obtain 𝑓. (2)

7.3 Write down the domain of 𝑓(𝑥 − 1). (1)

7.4 Determine the equation of the line, parallel to 𝑔 (an axis of symmetry of 𝑓) passing
through point C. (3)

2018 Free State Preliminary Paper 1 Q 4

In the sketch below, the graphs of ℎ(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and 𝑔(𝑥) = 4𝑥 + 8 are drawn. P and R
are the 𝑥-intercepts of ℎ and D is the turning point of ℎ. C and F are the 𝑦-intercepts of ℎ
and 𝑔 respectively. The two graphs intersect at P and E (6; 32).

60
4.1 Write down the coordinates of point P. (2)
4.2 If the equation of the axis of symmetry of ℎ is 𝑥 = 4, write down the coordinates
of point R. (2)
4.3 Determine the equation ℎ of in the form ℎ(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. (4)
4.4 Calculate the length of FC. (2)
4.5 Determine the range of ℎ. (2)
4.6 For which value(s) of 𝑥 is ℎ(𝑥). 𝑔(𝑥) < 0? (2)
4.7 Determine the value of 𝑘 if 𝑦 = 4𝑥 + 𝑘 is a tangent to ℎ(𝑥) = −𝑥 2 + 8𝑥 + 20. (4)

2018 Free State Preliminary Paper 1 Q 5

2+𝑥
The sketch below shows the graph of 𝑓(𝑥) = 𝑥 − 1. A and B are the 𝑥-intercepts and the 𝑦-

intercepts of 𝑓.

61
 𝑎
5.1 Write down the equation of 𝑓 in the form 𝑓(𝑥) = 𝑥 + 𝑝 + 𝑞. (3)

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

5.3 Write down the coordinates of A, the 𝑥-intercepts of 𝑓. (2)
5.4 The graph of 𝑘(𝑥) is obtained by shifting the graph of 𝑓(𝑥) horizontally such that
point A is at the origin, O. Write down the equation of the vertical asymptote of 𝑘(𝑥). (1)

2018 Eastern Cape Preliminary Paper 1 Q 5

Sketched below are the graphs of 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and 𝑔(𝑥) = −𝑥 + 5
3 49
A and B are the 𝑥-intercept of 𝑓. T ( 2 ; ) is the turning point of 𝑓 . B and S are the
4

points of intersection of 𝑓 and 𝑔.

62
5.1 Calculate the coordinates of B. (2)
5.2 Determine the equation of 𝑓 in the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 (4)
5.3 If 𝑓(𝑥) = −𝑥 2 + 3𝑥 + 10, calculate the coordinates of S. (4)
5.4 Use the graphs to solve for 𝑓 where:
5.4.1 𝑓(𝑥) ≥ 𝑔(𝑥) (2)
1
5.4.2 −𝑥 2 + 3𝑥 + 2 4 < 0 (3)

2018 Eastern Cape Preliminary Paper 1 Q 6

2
Given: 𝑓(𝑥) = 𝑥 − 1

6.1 Draw a neat sketch of 𝑓 indicating all intercepts and asymptotes. (4)
6.2 Determine 𝑓 ′ (𝑥) . (2)
6.3 Determine the equation of ℎ , the axis of symmetry of 𝑓 that has a negative
gradient. (2)
6.4 A constant value of 𝑘 is added to ℎ so that the straight line becomes a tangent to
the graph of 𝑓 with 𝑥 > 0. Determine the value of 𝑘. (5)

63