2023 BOT Winter School Grade 12 Learner Notes
2023 BOT Winter School Grade 12 Learner Notes
2023 BOT Winter School Grade 12 Learner Notes
MATHEMATICS
Grade 12
Functions
The Parabola
The Hyperbola
Inverse Functions
Consolidation
6.1 Algebra
6.2 Functions
• 𝑓𝑓 ′ (𝑥𝑥) = 0 • 𝑦𝑦 = 𝑞𝑞
EXAMPLE of the
SKETCH of the
graph
EXAMPLE 1:
1.4 Draw the graph of f. Clearly show the turning point and intercepts with the axes.
1.5.3 Determine the equation of ℎ where ℎ is the graph after 𝑓𝑓 moved 1 unit to
the right.
1.5.4 Write down the turning point for 𝑔𝑔 if 𝑔𝑔 is the reflection of 𝑓𝑓 in the 𝑥𝑥 −axis.
2.4 Draw the graph of f. Clearly show the turning point and intercepts with the axes.
2.5.4 Does the graph – 𝑓𝑓(𝑥𝑥) have a maximum or minimum value? Give this
value.
The graph of 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 and 𝑘𝑘(𝑥𝑥) = 2𝑥𝑥 + 10 is drawn below.
• The parabola cuts the 𝑥𝑥-axis at A and B and the 𝑦𝑦-axis at P.
• B and P are also the intercepts of 𝑘𝑘.
• T is the turning point of f.
3.5 Determine the values of 𝑘𝑘 for which 𝑓𝑓(𝑥𝑥) = 𝑘𝑘 will have no real roots.
The graph of 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 is drawn below. W, P and Q are the intercepts of 𝑓𝑓.
R(1 ; −8) is the turning point of f. S(2 ; −6) is a point on f.
4.1 Write down the coordinates of Q if S and Q are reflections in the axis of symmetry.
4.6 Write down the equation of the axis of symmetry for 𝑓𝑓(𝑥𝑥 + 3)
4.8 Determine the values of 𝑘𝑘 for which 𝑓𝑓(𝑥𝑥) = 𝑘𝑘 will have two unequal, positive real
roots.
𝑘𝑘
𝑦𝑦 = + 𝑞𝑞
𝑥𝑥 + 𝑝𝑝
6
Given: 𝑓𝑓(𝑥𝑥) = +4
𝑥𝑥−3
5.2 Determine:
5.2.1 the 𝑥𝑥-intercept of f.
5.2.2 the coordinates of the 𝑦𝑦-intercept of f.
5.3 Draw the graph of f. Clearly show ALL the asymptotes and intercepts with the axes.
5.5 Determine the equation of ℎ where ℎ is the graph after 𝑓𝑓 moved 1 unit to the right.
𝑘𝑘
In the diagram is the graph of: 𝑓𝑓(𝑥𝑥) = + 𝑞𝑞. C and D are the intercepts of 𝑓𝑓.
𝑥𝑥+𝑝𝑝
6.2 Determine:
6.2.1 the numerical values of 𝑘𝑘, 𝑝𝑝 𝑎𝑎nd 𝑞𝑞.
6.2.2 the coordinates of the 𝑦𝑦-intercept of f.
6.2.3 the coordinates of the 𝑥𝑥-intercept of f.
6.2.4 the equation of the axis of symmetry of 𝑓𝑓 that has a negative gradient.
7.2 Determine:
7.2.1 the 𝑥𝑥-intercept of f.
7.2.2 the coordinates of the 𝑦𝑦-intercept of f.
7.3 Draw the graph of f. Clearly show ALL the asymptote(s) and intercepts with the axes.
7.5 Determine the equation of ℎ where ℎ is the graph after 𝑓𝑓 moved 2 unit to the right and
1 unit up.
In the diagram is the graph of: 𝒇𝒇(𝒙𝒙) = 𝟐𝟐𝒙𝒙+𝒑𝒑 + 𝒒𝒒 . A(3;0) and B are the intercepts of 𝒇𝒇.
The asymptote of 𝑓𝑓 is at 𝑦𝑦 = −8
8.2 Determine:
8.2.1 the numerical values of 𝑝𝑝 and 𝑞𝑞.
8.2.2 the coordinates of B, the 𝑦𝑦-intercept of f.
1 𝑥𝑥
8.5 Describe the transformation from 𝑓𝑓 to ℎ(𝑥𝑥) = � �
2
FUNCTIONAL NOTATION
DEFINITION OF A FUNCTION
INVERSE
1. The inverse of a function takes the 𝑦𝑦-values (range) of the function to the corresponding 𝑥𝑥-
values (domain) and vice versa. Therefore, the 𝑥𝑥 and 𝑦𝑦 values are interchanged.
Example 1:
𝑦𝑦 = 2𝑥𝑥 − 4 𝑓𝑓
Interchange 𝑥𝑥 and 𝑦𝑦
𝑓𝑓 −1
Example 2:
If 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 + 6
We know that:
If 𝑦𝑦 = 𝑎𝑎 𝑥𝑥
21 = 2 then log 𝑎𝑎 𝑦𝑦 = 𝑥𝑥
22 = 4
Example 1 Example 2
52 = 25 34 = 81
If log 𝑎𝑎 𝑦𝑦 = 𝑥𝑥
then 𝑦𝑦 = 𝑎𝑎 𝑥𝑥
Example 1 Example 2
2 = log 5 25 4 = log 3 81
Example 3 Example 4
1
−2 = log 2 2 = log 100
4
Example 3:
𝑥𝑥 −2 −1 0 1 2
𝑓𝑓: 𝑦𝑦 = 2𝑥𝑥
1 1
𝑦𝑦 1 2 4
4 2
𝑥𝑥
𝑦𝑦
QUESTION 5
Sketched below is the graph of 𝑓𝑓(𝑥𝑥) = 𝑘𝑘 𝑥𝑥 ; 𝑘𝑘 > 0. The point (4; 16) lies on 𝑓𝑓.
5.3 Sketch the graph 𝑔𝑔. Indicate on your graph the coordinates of two points on 𝑔𝑔. (4)
QUESTION 6
6.3 Write down the equation of 𝑓𝑓 −1 , the inverse of 𝑓𝑓, in the form 𝑦𝑦 = … (2)
Example 4:
1
Given:𝑓𝑓(𝑥𝑥) = − 𝑥𝑥 2
2
4.1 Determine the equation of the inverse function in the form 𝑓𝑓 −1 (𝑥𝑥) = …
4.2 Sketch the graph of the function and its inverse on the same set of axes
4.3 Now restrict the domain of the original function in two different ways to form new
one-to-one functions.
5.1 Determine the inverse of 𝑓𝑓. Hence rewrite the equation of the inverse function in the
form 𝑓𝑓 −1 (𝑥𝑥) = …
5.2 Sketch the graphs of the new function and its inverse function and 𝑦𝑦 = 𝑥𝑥 on the same
set of axes.
QUESTION 4
In the diagram below, the graph of 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 is drawn in the interval 𝑥𝑥 ≤ 0. The graph 𝑓𝑓 −1 is
also drawn. P(−6; −12) is a point on 𝑓𝑓 and R is a point on 𝑓𝑓 −1 .
4.2 If R is the reflection of P in the line 𝑦𝑦 = 𝑥𝑥, write down the coordinates of R. (1)
6.1 Algebra
1.3 If 𝑛𝑛 is the largest integer for which 𝑛𝑛200 < 5300 , determine the value of 𝑛𝑛. (3)
[22]
2.3 Show that 2.5𝑛𝑛 − 5𝑛𝑛+1 + 5𝑛𝑛+2 is even for all positive integer vales of 𝑛𝑛. (3)
64
3.2 Given: 𝑎𝑎 + = 16
𝑎𝑎
−𝑛𝑛±�𝑛𝑛2 −4𝑚𝑚𝑚𝑚
4.3 The roots of an equation are 𝑥𝑥 = where 𝑚𝑚, 𝑛𝑛 and 𝑝𝑝 are positive real
2𝑚𝑚
numbers. The numbers 𝑚𝑚, 𝑛𝑛 and 𝑝𝑝 , in that order, form a geometric sequence.
(4)
Prove that 𝑥𝑥 is a non-real number.
[24]
5.3 If it is given that 2𝑥𝑥 × 3𝑦𝑦 = 246 , determine the numerical value of 𝑥𝑥 − 𝑦𝑦. (4)
[24]
2
Given: 𝑓𝑓(𝑥𝑥) = +1
𝑥𝑥−3
The graphs of 𝑓𝑓(𝑥𝑥) = −𝑥𝑥 2 + 7𝑥𝑥 + 8 and 𝑔𝑔(𝑥𝑥) = −3𝑥𝑥 + 24 are sketched below.
𝑓𝑓and 𝑔𝑔 intersect in D and B. A and B are the 𝑥𝑥 − intercepts of 𝑓𝑓.
Below are the graphs if 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 4)2 − 9 and a straight line 𝑔𝑔.
• A and B are the 𝑥𝑥 −intercepts of 𝑓𝑓 and E is the turning point of 𝑓𝑓.
• C is the 𝑦𝑦 −intercept of both 𝑓𝑓 and 𝑔𝑔.
• The 𝑥𝑥 −intercept of 𝑔𝑔 is D. DE is parallel to the 𝑦𝑦 −axis.
The graph of 𝑔𝑔(𝑥𝑥) = 𝑎𝑎 𝑥𝑥 is drawn in the sketch below. The point S(2; 9) lies on 𝑔𝑔.
T is the 𝑦𝑦 −intercept of 𝑔𝑔.
𝑎𝑎
The function 𝑓𝑓, defined by 𝑓𝑓(𝑥𝑥) = + 𝑞𝑞, has the following properties:
𝑥𝑥+𝑝𝑝
• The range of 𝑓𝑓 is 𝑦𝑦 ∈ ℝ, 𝑦𝑦 ≠ 1.
• The graph 𝑓𝑓 passes through the origin.
• P(√2 + 2; √2 + 1) lies on the graph 𝑓𝑓.
7.2 S(−2; 0) and T(6; 0) are the 𝑥𝑥 −intercepts of the graphs of 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐
and R is the 𝑦𝑦 −intercept. The straight line through R and T represents the graph of
𝑔𝑔(𝑥𝑥) = −2𝑥𝑥 + 𝑑𝑑.
8.1 Use the graph to determine the values of 𝑥𝑥 for which 𝑓𝑓(𝑥𝑥) ≥ −9. (2)
8.2 Write down the equation of 𝑓𝑓 −1 in the form 𝑦𝑦 =. .. Include ALL restrictions. (3)
8.3 Sketch 𝑓𝑓 −1 , the inverse of 𝑓𝑓, in your ANSWER BOOK.
Indicate the intercept(s) with the axes and the coordinates of ONE other point. (3)
8.4 Describe the transformation from 𝑓𝑓 to 𝑔𝑔 if 𝑔𝑔(𝑥𝑥) = √27𝑥𝑥, where 𝑥𝑥 ≥ 0. (1)
[9]
The graph of a hyperbola with the equation 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) has the following properties:
• Domain: 𝑥𝑥 ∈ ℝ, 𝑥𝑥 ≠ 5
• Range: 𝑦𝑦 ∈ ℝ, 𝑦𝑦 ≠ 1
• Passes through point (2; 0)
Determine 𝑓𝑓(𝑥𝑥).
[4]
The graphs of 𝑔𝑔(𝑥𝑥) = 2𝑥𝑥 + 6 and 𝑔𝑔−1 , the inverse of 𝑔𝑔, are shown in the diagram below.
• D and B are the 𝑥𝑥 − and 𝑦𝑦 − intercepts respectively of 𝑔𝑔.
• C is the 𝑥𝑥 − intercept of 𝑔𝑔−1 .
• The graphs of 𝑔𝑔 and 𝑔𝑔−1 intersect at A.