90°
MADE BY L.
TYHAWANA
CONTACT: 076 237 3731
90°
ππππ
INTRODUCTION
Welcome to your Grade 12 Technical Mathematics notes on functions, made by Mr.
Tyhawana from Living the Dream Academy! These notes are specifically designed for
Tech Math learners, though they might also be useful to some Pure Math learners.
These notes do not cover everything in exhaustive detail and use certain notations
for simplicity. For example, I used inequalities rather than brackets because I find
them easier to understand and that's how I taught my learners. While everything in
this document is correct, not everything is fully explained, so it's important to
continue attending classes. You can also visit our YouTube channel (Living the Dream
Academy) for more information or contact me at 076 237 3731.
In these notes, you will explore:
β Definitions and Notations: Basic terminology and symbols used in functions.
β Domain and Range: Determining the set of possible inputs (domain) and
outputs (range) for different functions.
β Graphing Functions: Visualizing functions by plotting their graphs and
interpreting these graphs to understand function behavior.
β Types of Functions: Various functions such as linear, quadratic, polynomial,
exponential, and trigonometric functions.
By the end of these notes, you will have a solid foundation in the concept of
functions, enabling you to excel in your studies and apply these principles in
technical and everyday contexts. Let's dive into the fascinating world of functions
and discover the power and beauty of mathematics together!
FUNCTIONS AND GRAPHS
Total marks in Functions and graphs ≈ ππ
What learners should know???
β’Draw sketch graphs of functions by dual
intercept method.
β’Identify area in the graph where:
βIs increasing
βIs decreasing
βIs above the x-axis
βIs below the y-axis
β’Write down the domain and range of graphs
β’Read and Plot the graph sketch by determining:
β’Equations
β’Intercepts
β’Lengths
β’Turning points
β’Asymptotes
β’However, it’s okay to plot the graph using the
table.
FUNCTIONS TO LEARN ABOUT
STRAIGHT LINE
π = ππ + π π¨π«
π² − ππ = π(π − ππ )
PARABOLA
π = πππ + ππ + π
or
π=π π−π π+π
or
π = π(π − ππ )(π − ππ )
HYPERBOLA
π
π=
+π
π−π
(p;q)
(p;q)
(π; π)
π=
−π
−π
π+π
π =N/A
πΎπππ π < π
π=π
(-p;-q)
FUNCTIONS TO LEARN ABOUT
EXPONENTIAL
π = π. ππ + π
CIRCLE
SEMI-CIRCLE
ππ − ππ
π=
ππ = ππ + ππ
r
π
π = −π. ππ − π
r
π = − ππ − ππ
r
-π
Notation and Definitions
β’Intercepts: the points where the graph cuts the axes.
β’Turning point: is a point where the graph changes direction.
Turning point can either be local minima or local maxima,
minimum or maximum.
β’Point of intersection: the point where the two functions are
equal.
β’Asymptotes: a line that a function approaches as it heads
towards positive or negative infinity.
β’Axis of symmetry: a line that divides the graph into two equal
parts
β’π π > π: Area where the function is above the x-axis.
β’π π < π: Area where the function is below the x-axis.
β’π′ π > π: area where the function is increasing/gradient is
positive.
β’π′ π < π: area where the function is decreasing/gradient is
negative.
Advanced Notation and Definitions
NOTATION INTERPRETATION/DEFINITION
π π <π
π(π) ≤ π
π π >π
π π ≥π
π π > π(π)
π π ≥ π(π)
π π .π π > π
π π .π π ≥ π
π π .π π < π
π π .π π ≤ π
π π −π π =π
β’ The graph is below the x-axis
β’ Value(s) is/are excluded
β’ The graph is below the x-axis
β’ Value(s) is/are included
β’ The graph is above the x-axis
β’ Value(s) is/are excluded
β’ The graph is above the x-axis
β’ Value(s) is/are included.
β’ Graph of f is above the graph of g
β’ Value(s) is/are excluded.
β’ Graph of f is above the graph of g
β’ Value(s) is/are included.
β’ Both graphs of f and g are either above or
below the x-axis
β’ Value(s) are excluded
β’ Both graphs of f and g are either above or
below the x-axis
β’ Value(s) are included
β’ One of the graphs of f or g is above the xaxis and the other is below the x-axis
β’ Value(s) is/are excluded.
β’ One of the graphs of f or g is above the xaxis and the other is below the x-axis.
β’ Value(s) is/are included
β’ π π =π π
β’ Point of intersection
RANGE AND DOMAIN
FUNCTION
DOMAIN
RANGE
π∈β
π∈β
π = πππ + ππ + π
π∈β
π ≥ ππ»π· ; π ∈ β
π = −πππ + ππ + π
π∈β
π ≤ ππ»π· ; π ∈ β
π ≠ π; π ∈ β
π ≠ π; π ∈ β
π∈β
π ≠ π; π ∈ β
ππ − ππ ≥ π; π ∈ β
−π ≤ π ≤ π
π ≤ π ≤ π; π ∈ β
STRAIGHT LINE
π = ππ + π
PARABOLA
HYPERBOLA
π
π=
+π
π−π
EXPONENTIAL
π = π. ππ + π
;π ≠ π
SEMI-CIRCLE
π=
ππ − ππ
π − ππ ≥ π; π ∈ β
π
−π ≤ π ≤ π; π ∈ β
π = − ππ − ππ −π ≤ π ≤ π
CIRCLE
ππ = ππ + ππ
−π ≤ π ≤ π; π ∈ β −π ≤ π ≤ π; π ∈ β
STRAIGHT LINE AND PARABOLA
FUNCTION
WHAT TO FIND
Straight Line
y-intercept
x-intercept
β For y-int, let x=0
β For x-int, let y=0
y-intercept
x-intercept
β For y-int, let x=0
β For x-int, let y=0
β Solve for x by
quad. Formula or
factorization
Turning Point
π
πππ − ππ
πΏπ»π· = −
;π =
ππ π»π·
ππ
π = ππ + π
Parabola/ Quadratic
π = πππ + ππ + π
π= π π−π π+π
π = π(π − ππ )(π − ππ )
HOW
or
ππ°ππ π + ππ°ππ π
πΏπ»π· =
π
1.
2.
3.
4.
or
πΏπ»π· = πΏπΊπππππππ
or
Find the first
derivative
Equate to Zero
Solve for x
Subst. the value of x
into the original
equation
STRAIGHT LINE AND PARABOLA
STRAIGHT LINE AND PARABOLA
STRAIGHT LINE AND PARABOLA
STRAIGHT LINE AND PARABOLA
Practice Problems – Straight line and
parabola
Practice Problems – Straight line and
parabola
Practice Problems – Straight line and
parabola
HYPERBOLA
FUNCTION
WHAT TO FIND?
HYPERBOLA
y-intercept
x-intercept
π=
π
+π
π−π
HOW?
β For y-int, let x=0
β For x-int, let y=0
Asymptotes
Vertical
Asymptote, but
Horizontal Shift
Horizontal
Asymptote, but
Vertical Shift
π = π; π ∈ β
π = π; π ∈ β
Practice Problems – Hyperbola
Practice Problems – Hyperbola
Worked Examples – Hyperbola
Practice Problems – Hyperbola
Practice Problems – Hyperbola
EXPONENTIAL
FUNCTION
WHAT TO FIND
EXPONENTIAL FUNCTION
y-intercept
x-intercept
π = π. ππ + π;
π > π πππ
π ≠ π
HOW
β For y-int, let x=0
β For x-int, let y=0
Asymptotes
Horizontal
Asymptote, but
Vertical Shift
π = π; π ∈ β
Practice Problems – Exponential
Practice Problems – Exponential
Practice Problems – Exponential
Practice Problems – Exponential
Practice Problems – Exponential
SEMI-CIRCLE
FUNCTION
WHAT TO FIND
SEMI-CIRCLE
y-intercept
x-intercept
β For y-int, let x=0
β For x-int, let y=0
Radius
Use Pythagorean
Theorem.
π=
ππ − ππ
Whether it’s
above or below
the x-axis
HOW
π = + ππ − ππ
π = − ππ − ππ
Semi-circle
Semi-circle
Semi-circle
MY MATH PHILOSOPHY
Many mathematicians and math experts believe that you have to understand math in
order to master it. This belief has made many learners struggle with math. However, I
believe that math is the same as any other subject. You can memorize it at first, and
then the understanding part will follow later. Here are few tips to help you along the
way.
1. Memorize First, Understand Later: Start by memorizing key formulas and
methods. As you practice more, the understanding of how and why they
work will come naturally.
2. Apply Math to Real-Life Situations: Try to relate mathematical concepts to
real-life scenarios. This will help you see the practical use of what you are
learning and deepen your understanding.
3. Develop a Mathematical Perspective: Look at everything with the eye of a
mathematician. When you start seeing patterns and relationships that are
not obvious to others, you know you're on the right track.
4. Embrace the Crazy: If people think you're crazy because of your unique
way of seeing things, it means the spirit of math is penetrating. Embrace it
and keep exploring.
5. Be "Lazy" in a Smart Way: To be a great mathematician, be "lazy" in the
sense that you avoid writing long paragraphs and focus on finding as many
tricks and shortcuts as possible to get to the answer.
6. Practice, Practice, Practice: Math is a skill that gets better with practice.
Work on different types of problems regularly to improve your proficiency.
7. Stay Curious: Always ask questions and be curious about why things work
the way they do. This mindset will help you dig deeper and understand
concepts more thoroughly.
8. Use Resources: Don't hesitate to use additional resources like our YouTube
channel (Living the Dream Academy) for more explanations and examples.
You can also contact me at 076 237 3731 if you have any questions.
9. Don’t rush yourself: Remember, learning math is a journey. Stay
motivated, keep practicing, and you will see improvement.
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