Free Maths Methods Cheatsheets Year 11 Unit 1 2
Free Maths Methods Cheatsheets Year 11 Unit 1 2
Free Maths Methods Cheatsheets Year 11 Unit 1 2
Year 11
Unit 1 & 2
FREE V1.98
1
Purpose of this book
Hello!
This is a brief overview of Units 1 & 2 Mathematical Methods to help you learn and revise more efficiently. It is
essentially a cut down version of the Units 3 & 4 Overview.
It was originally designed as a reference book for students who use the online video tutorials on
MathsMethods.com.au but has since been used by many as their Bound Reference. Each page has a clickable link to
direct you to the relevant video tutorial if you have access and there’s plenty of other free resources if you don’t!
Please note, like many of our resources, this overview is designed to reinforce understanding and may not use the
exact notation you need to use when doing tests and exams.
Do well and I hope this overview makes the year a little less stressful for you :)
Kind regards
6
Covered in detail in video tutorials, see LINEAR EQUATIONS Perpendicular
Parallel means the −𝟏𝟏
Gradient-Intercept Form same gradient means 𝒎𝒎 =
𝒎𝒎
y = mx + c (0, c)
m means gradient
c means y-intercept y = 2x + 3 y = 2x - 2 y = 2x y = -½x
Linear Equations
𝜽𝜽 = 𝐭𝐭𝐭𝐭𝐧𝐧−𝟏𝟏 𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠
(x1, y1) is any point on the line
𝑦𝑦 −𝑦𝑦 rise
(x2, y2) is any different point on the line (x2, y2) gradient = 𝑥𝑥2−𝑥𝑥1 =
(x1, y1) 2 1 run
Want FREE RESOURCES on this topic? See LINEAR EQUATIONS (FREE VIDEO SERIES) For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see PARABOLAS & QUADRATICS
Intercept Form 𝒚𝒚 = 𝒅𝒅(𝒙𝒙 − 𝒂𝒂)(𝒙𝒙 − 𝒃𝒃)
1. See if positive or negative
2. Draw in x intercepts (which are a and b) 𝒂𝒂 𝒃𝒃
How to draw Parabolas
General Form
1. See if positive or negative
𝒚𝒚 = 𝒂𝒂𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒙𝒙 + 𝒄𝒄 𝒄𝒄
2. Draw in y-intercept
−𝒃𝒃 ± 𝒃𝒃𝟐𝟐 − 𝟒𝟒𝒂𝒂𝒂𝒂
3. Find x-intercepts if there are any 𝑥𝑥 intercepts = 𝒃𝒃
𝟐𝟐𝒂𝒂 −
4. Find turning point 𝟐𝟐𝟐𝟐
Want FREE RESOURCES on this topic? See PARABOLAS For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see HOW TO SKETCH ANY FUNCTION
y = sin(x)
y=x y= x2 y = x3
List of Main Functions
y = cos(x)
y = x½ y = x-1 y = x-2
y = tan(x)
y = ax y = 2x y = log2(x)
Want FREE RESOURCES on this topic? See SKETCHING FUNCTIONS (FREE VIDEO) For more resources, see MathsMethods.com.au
5 STEPS: 1. Change form 2. Factorise inside 3. Turning Point 4. Shape 5. Reflections
𝟒𝟒
Covered in detail in video tutorials, see TRANSLATION – MOVING FUNCTIONS and STRETCHING AND REFLECTING 𝒚𝒚 = + 𝟏𝟏
𝟒𝟒−𝒙𝒙
𝒚𝒚 = 𝟑𝟑 𝟔𝟔 − 𝟐𝟐𝟐𝟐 + 𝟏𝟏 1. 𝒚𝒚 = 𝟒𝟒 𝟒𝟒 − 𝒙𝒙 −𝟏𝟏
+ 𝟏𝟏
How to Sketch Any Function
𝟏𝟏
1. 𝒚𝒚 = 𝟑𝟑 𝟔𝟔 − 𝟐𝟐𝟐𝟐 𝟐𝟐 + 𝟏𝟏 −𝟏𝟏
𝟏𝟏
2. 𝒚𝒚 = 𝟒𝟒 −(𝒙𝒙 − 𝟒𝟒) + 𝟏𝟏
2. 𝒚𝒚 = 𝟑𝟑 −𝟐𝟐(𝒙𝒙 − 𝟑𝟑 𝟐𝟐 + 𝟏𝟏
(𝟑𝟑, 𝟏𝟏) (𝟒𝟒, 𝟏𝟏) 4.
4.
5. Negative in front of 𝒙𝒙, 5. Negative in front of 𝒙𝒙,
flip around the 𝒚𝒚-axis flip around the 𝒚𝒚-axis
𝟒𝟒
𝒚𝒚 = − 𝟐𝟐𝟐𝟐 + 𝟒𝟒 + 𝟑𝟑 𝒚𝒚 = + 𝟏𝟏
𝟒𝟒−𝒙𝒙 𝟐𝟐
−𝟐𝟐, 𝟑𝟑 𝟏𝟏
1. 𝒚𝒚 = − 𝟐𝟐𝟐𝟐 + 𝟒𝟒 𝟐𝟐 + 𝟑𝟑 5. Negative in front of 𝒙𝒙,
𝟏𝟏 flip around the 𝒚𝒚-axis
𝒚𝒚 = − 𝟐𝟐(𝒙𝒙 + 𝟐𝟐) 𝟐𝟐 + 𝟑𝟑 (𝟒𝟒, 𝟏𝟏)
4.
4.
1. 𝒚𝒚 = 𝟒𝟒 𝟒𝟒 − 𝒙𝒙 −𝟐𝟐 + 𝟏𝟏
5. Negative in front of 𝒚𝒚,
flip around the 𝒙𝒙-axis 2. 𝒚𝒚 = 𝟒𝟒 −(𝒙𝒙 − 𝟒𝟒) −𝟐𝟐 + 𝟏𝟏
Want FREE RESOURCES on this topic? See HOW TO SKETCH ANY FUNCTION For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see FUNCTIONS AND POINTS, USING MATRICES and SERIES OF TRANSFORMATIONS
(𝟒𝟒, 𝟗𝟗)
𝒙𝒙𝟐𝟐 → − 𝟐𝟐(𝒙𝒙 − 𝟒𝟒 )𝟐𝟐 + 𝟗𝟗
−𝒇𝒇 𝒙𝒙 = −𝒙𝒙𝟐𝟐 1. Reflection in the x-axis
1 2 3
Transformations
2. Followed by a dilation of
𝒇𝒇(𝟐𝟐𝒙𝒙) = −(𝟐𝟐𝒙𝒙)𝟐𝟐
factor ½ from the y-axis
𝟐𝟐 3. Then a translation of 4 units
𝒇𝒇 𝒙𝒙 − 𝟒𝟒 + 𝟗𝟗 = − 𝟐𝟐 𝒙𝒙 − 𝟒𝟒 + 𝟗𝟗 in positive x-direction and 9
units in the positive y-direction
𝟏𝟏
𝒇𝒇 𝒙𝒙 is a dilation of factor 𝒂𝒂 from the y-axis 𝒇𝒇 −𝒙𝒙 is a reflection in the y-axis
𝒂𝒂
(in the x-direction) −𝒇𝒇 𝒙𝒙 is a reflection in the x-axis
𝒃𝒃𝒇𝒇 𝒙𝒙 is a dilation of factor 𝒃𝒃 from the x-axis 𝒇𝒇 𝒙𝒙 + 𝒌𝒌 is a translation along the y-axis
(in the y-direction) 𝒇𝒇 𝒙𝒙 − 𝒉𝒉 is a translation along the x-axis
𝒙𝒙𝒎𝒎 𝟏𝟏 𝟏𝟏
Exponential Laws
Negative
Power = 𝒙𝒙𝒎𝒎−𝒏𝒏 𝒙𝒙−𝟏𝟏 = 𝒙𝒙−𝒏𝒏 = 𝒏𝒏
𝒙𝒙𝒏𝒏 𝒙𝒙 𝒙𝒙
𝟏𝟏 𝟏𝟏 𝒎𝒎
Fraction 𝒎𝒎 𝒏𝒏
Power 𝒙𝒙𝟐𝟐 = 𝒙𝒙 𝒙𝒙𝒎𝒎 = 𝒙𝒙 𝒙𝒙 𝒏𝒏 = 𝒙𝒙𝒎𝒎
𝒏𝒏
𝒙𝒙 𝒙𝒙𝒏𝒏
𝒙𝒙𝒎𝒎 𝒙𝒙𝒏𝒏 = 𝒙𝒙𝒎𝒎+𝒏𝒏 (𝒙𝒙𝒎𝒎 )𝒏𝒏 = 𝒙𝒙𝒎𝒎𝒏𝒏 = 𝒏𝒏
𝒚𝒚 𝒚𝒚
Want FREE RESOURCES on this topic? See POWER LAWS (FREE VIDEO) For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see WHAT ARE LOGARITHMS?
log28 = 3 to get 8
How many 2s are
multiplied together
Arithmos means number
23 =8 2
Logarithm originally means
how many numbers
Want FREE RESOURCES on this topic? See LOGARITHMS (MUSIC VIDEO) For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see SKETCHING LOGS AND EXPONENTIALS
y = -3e(2x + 1) – 2 y = loge(-2x + 4) – 3
Sketching Logs and Exponentials
1) Find any reflections 1) Find any reflections
reflected in x-axis reflected in y-axis
Want FREE RESOURCES on this topic? See SKETCHING LOGS AND EXPONENTIALS
Covered in detail in video tutorials, see DEFINITIONS OF SIN AND COS and THE UNIT CIRCLE
5 y y
Sin, Cos and Tan Definitions
3
θ
sin(θ)
4 θ
x
θ
x
cos(θ)
Length of 𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎 𝟑𝟑
𝐬𝐬𝐬𝐬𝐬𝐬 𝛉𝛉 = =
Length of 𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇 𝟓𝟓 Tangent is a line which touches a tan(θ) is the length of the tangent,
circle only at one point. cut off by the x axis and the radius.
Length of 𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀 𝟒𝟒 y
𝐜𝐜𝐜𝐜𝐜𝐜 𝛉𝛉 = =
Length of 𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇𝐇 𝟓𝟓
tan(θ)
Length of 𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎𝐎 𝟑𝟑 tangent θ
𝐭𝐭𝐭𝐭𝐭𝐭 𝛉𝛉 = =
Length of 𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀 𝟒𝟒 x
Want FREE RESOURCES on this topic? See SIN, COS AND TAN (FREE VIDEO) For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see PROVING EXACT VALUES
Angle sin(θ) cos(θ) tan(θ)
𝟎𝟎 𝟎𝟎 𝟏𝟏 𝟎𝟎
𝟏𝟏
𝝅𝝅 𝟏𝟏 𝟑𝟑 𝟏𝟏 𝐬𝐬𝐬𝐬𝐬𝐬(𝟎𝟎) = 𝟎𝟎 𝐜𝐜𝐜𝐜𝐜𝐜(𝟎𝟎) = 𝟏𝟏 𝐭𝐭𝐭𝐭𝐭𝐭(𝟎𝟎) = 𝟎𝟎
𝟔𝟔 𝟑𝟑𝟑𝟑 𝟐𝟐 𝟐𝟐 𝟑𝟑
Exact Values
𝟏𝟏 𝟏𝟏
𝟏𝟏
𝟐𝟐
𝝅𝝅 𝟐𝟐 𝟐𝟐
𝟒𝟒𝟒𝟒 𝟏𝟏 30° 60° 45°
𝟒𝟒 𝟐𝟐 𝟐𝟐 𝟏𝟏
𝟏𝟏 𝟏𝟏 𝟏𝟏
𝐭𝐭𝐭𝐭𝐭𝐭(𝟒𝟒𝟒𝟒) = 𝟏𝟏
𝟐𝟐
𝐬𝐬𝐬𝐬𝐬𝐬(𝟑𝟑𝟑𝟑) = 𝐜𝐜𝐜𝐜𝐜𝐜(𝟔𝟔𝟔𝟔) =
𝟐𝟐 𝟐𝟐
𝝅𝝅 𝟑𝟑 𝟏𝟏
𝟑𝟑
𝟔𝟔𝟔𝟔
𝟐𝟐 𝟐𝟐
𝟑𝟑 𝟏𝟏 𝟏𝟏
1
0
𝝅𝝅 𝟐𝟐𝝅𝝅 𝝅𝝅 𝝅𝝅 𝟑𝟑𝝅𝝅 𝝅𝝅
𝟒𝟒 𝟐𝟐 𝟒𝟒
-3 -2
Want FREE RESOURCES on this topic? See SKETCHING CIRCULAR FUNCTIONS For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see FINDING THE DERIVATIVE
𝒅𝒅𝒅𝒅 𝒅𝒅𝒅𝒅 𝒅𝒅𝒅𝒅
Derivative of 𝒙𝒙 = ×
𝒅𝒅𝒅𝒅 𝒅𝒅𝒅𝒅 𝒅𝒅𝒅𝒅
is exactly the same as
𝒅𝒅𝒅𝒅
𝒇𝒇 𝒙𝒙 = 𝟓𝟓𝒙𝒙𝟒𝟒 𝒇𝒇′ 𝒙𝒙 = 𝟒𝟒 × 𝟓𝟓𝒙𝒙𝟑𝟑 = 𝒇𝒇′(𝒈𝒈 𝒙𝒙 ) × 𝒈𝒈′ 𝒙𝒙
𝒅𝒅𝒅𝒅
1. Multiply the 𝒙𝒙 by the power
2. Minus one from the power Chain Rule (short version)
Finding 𝒇𝒇𝒇(𝒙𝒙)
𝟐𝟐
𝒇𝒇 𝒙𝒙 = 𝟔𝟔𝒙𝒙𝟓𝟓 − 𝟑𝟑𝒙𝒙𝟑𝟑 + 𝟐𝟐𝒙𝒙−𝟏𝟏 − 𝟒𝟒 𝒅𝒅𝒅𝒅 𝟒𝟒
2. = 𝟓𝟓 × 𝟐𝟐 𝒙𝒙𝟑𝟑 − 𝟓𝟓 × 𝟑𝟑𝟑𝟑𝟐𝟐
𝟐𝟐 −
𝟏𝟏 𝒅𝒅𝒅𝒅
𝒇𝒇′ 𝒙𝒙 = 𝟓𝟓 × 𝟔𝟔𝒙𝒙𝟒𝟒 − × 𝟑𝟑𝒙𝒙 𝟑𝟑 + −𝟏𝟏 × 𝟐𝟐𝒙𝒙−𝟐𝟐 + 𝟎𝟎
𝟑𝟑
Want FREE RESOURCES on this topic? See CALCULUS BASICS (FREE VIDEO) For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see TYPES OF STATIONARY POINTS and HOW TO FIND STATIONARY POINTS
Stationary point means where How to find stationary points 𝒇𝒇 𝒙𝒙 = 𝟐𝟐𝒙𝒙𝟑𝟑 + 𝟏𝟏
the gradient of the curve is zero. 1. Find 𝒇𝒇𝒇(𝒙𝒙) = 𝟎𝟎 and solve for x
𝒇𝒇′ 𝒙𝒙 = 𝟔𝟔𝒙𝒙𝟐𝟐 𝟔𝟔𝒙𝒙𝟐𝟐 = 𝟎𝟎 𝒙𝒙 = 𝟎𝟎
𝒇𝒇′ 𝒙𝒙 = 𝟎𝟎 3. To find type: Sub in two 𝒙𝒙 values (before and after the S.P.)
𝒇𝒇′ −𝟏𝟏 = 𝟔𝟔(−𝟏𝟏)𝟐𝟐 = 𝟔𝟔 𝒇𝒇′ 𝟏𝟏 = 𝟔𝟔(𝟏𝟏)𝟐𝟐 = 𝟔𝟔
Types of S.P positive positive
Minimum Point of
Maximum inflexion
𝒇𝒇′ 𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃 = 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝒇𝒇′ 𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃 = 𝒏𝒏𝒏𝒏𝒏𝒏 𝒇𝒇𝒇 𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃𝒃 = 𝒑𝒑𝒑𝒑𝒑𝒑
𝒇𝒇𝒇 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 = 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝒇𝒇𝒇 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 = 𝒑𝒑𝒑𝒑𝒑𝒑 𝒇𝒇𝒇 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 = 𝒑𝒑𝒑𝒑𝒑𝒑
Want FREE RESOURCES on this topic? See STATIONARY POINTS (FREE VIDEO) For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see KINEMATICS – VELOCITY, ACCELERATION AND STUFF EQUATIONS
Instantaneous means gradient
Kinematics is the subject Distance means how far one point on the curve
about how objects move something has moved 𝒙𝒙
𝒙𝒙 = displacement
𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓
𝒅𝒅𝒙𝒙
= velocity Displacement means how 𝒓𝒓𝒓𝒓𝒓𝒓 𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅
far away something is
Kinematics
𝒅𝒅𝒅𝒅
= acceleration 𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅
𝒙𝒙 𝒕𝒕
2 𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓
Average means
𝒓𝒓𝒓𝒓𝒓𝒓
two points on the curve
𝒕𝒕
Differentiate 1 2 3
-1
instantaneous 𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅 𝒅𝒅𝒅𝒅 velocity
=
𝒙𝒙 time 0 1 2 3 𝒅𝒅𝒅𝒅
𝒅𝒅𝒅𝒅 𝒅𝒅𝒅𝒅 Distance 0 2 4 5 average 𝒓𝒓𝒓𝒓𝒓𝒓𝒓𝒓
=
Antidifferentiate Displacement 0 2 0 -1 velocity 𝒓𝒓𝒓𝒓𝒓𝒓
Want FREE RESOURCES on this topic? See RATES OF CHANGE For more resources, see MathsMethods.com.au
Covered in detail in video tutorials, see RANDOM VARIABLES and DISCRETE RANDOM VARIABLES
Discrete Random Variable is a letter that represents an Example: 10 balls in a bag: 4 blue and 6 orange
outcome in terms of countable numbers
If picking 3 balls at a time (with
Discrete Random Variables
If you add all the 𝐏𝐏𝐏𝐏 𝐗𝐗 , it will = 1 OOO 0 0.6 × 0.6 × 0.6 = 0.216
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