LINEAR EQUATIONS
(It helps to download Desmos on phone
to check graphs)
y = mx + c

m is the gradient. m is the coefficient of x when equation is in terms of y, NOT in terms
5
of y2, 2y, 0.5y, 𝑦, etc.




Positive gradient means graph is increasing(going from bottom left to up right),
Negative gradient means graph is decreasing(going from up left to bottom right)
c is the y-intercept(when x=0). Keep x=0 and solve for y to find y-intercept/
roots/solution to equation or zeros of equation means x-values when y=0/where line
cuts x-axis/x-intercepts
𝑦2−𝑦1
m = gradient =
𝑥2−𝑥1
𝑦 = 2𝑥 means when x increases by 1, y-increases by 2


7
𝑦 = −3𝑥 means when x increases by 1, y decreases by 3 because gradient is negative
𝑦 = 6.56𝑥 +
36646.9565
5535
means when x increases by 1, y increases by 6.56
If 𝑦 = 2.5𝑥 and they ask how much does x change when y increase by 1, just put
equation in terms of x(make x the subject)
so 𝑥
=
meaning

𝑦
2.5
𝑥=𝑦×
midpoint =
𝑥1+𝑥2
2
,
1
2.5
so when y increases by 1, x increases by
𝑦1+𝑦2
2
1
2.5

To save yourself from khuwaari of finding c, use this formula: 𝑦
− 𝑦1 = 𝑚(𝑥 −
𝑥1)

e.g: equation of line passing through (9,1) and (7,6)
first find gradient so
𝑚=
6−1
7−9
=−
5
2
choose any one set of coordinates and put in values of y1 and x1, leave y and x as the same
5
𝑦 − 1 = − (𝑥 − 9)
2
5
45
𝑦−1=− 𝑥+
2
2
5
47
𝑦=− 𝑥+
2
2


For perpendicular lines(lines meeting at θ=90˚), if first lines gradient is m1 then
second line’s gradient is negative reciprocal of m2. In other words:
𝑚1 × 𝑚2 = −1
For scatterplots, if for example a city’s population starts from 10,000 in 1990 and
increases every year after then your y-intercept will be 10,000.
Remember one thing about scatterplots, if for example they gave you a
scatterplot of units produced on y-axis and time on x-axis and ask you value of
units produced per time or rate of units produced then that is gradient as gradient
is y over x so units produced over time.
FUNCTIONS






y and f(x) are the SAME, if they ask you for value of f(x) that means value of y
Domain of function means for what x values does the graph exist
Range of function means for what y values does graph exist
A function is valid(a function exists) if one x-value gives one y-value or one yvalue gives one x-value
A function is valid(a function exists) one y-value gives more than 1 x-value
A function is NOT valid(a function DOES NOT exist) if one x-value gives more
than 1 y-value
Figure 1 VALID FUNCTION

To check if function is valid or not make a vertical line on graph anywhere, if it cuts the
graph in two places then it is an INVALID function
Figure 2 Examples of invalid functions

To find inverse of function:
1.change f(x) to y
2.make x subject
3.replace y with x and x with f-1(x)
e.g:
𝑓(𝑥 ) = 2𝑥 + 7
𝑦 = 2𝑥 + 7
𝑦−7
2
𝑥−7
𝑓 −1 (𝑥 ) =
2
𝑥=
f-1(x) is a reflection of f(x) along the line of 𝑦 = 𝑥

To see if inverse of a function exists, make a horizontal line anywhere on graph. If it cuts
the graph at two places then inverse DOES NOT exist
Figure 3 Examples of functions of which inverse
function DOES NOT exist

-f(x) is reflection of f(x) along x-axis, just multiply entire equation by -1 to get -f(x)

f(-x) is reflection of f(x) along y-axis, just put -x instead of x in original equation to get
f(-x)


If u want to translate graph upward by for example 4 units then just add 4 to the
equation
If u want to translate graph downward by for example 5 units then just subtract 5 from
the equation
𝑓(𝑥) = 2𝑥 + 7

When translated 4 units up:
𝑓(𝑥) = 2𝑥 + 11


If u want to translate graph for example 6 units to RIGHT then replace 𝑥 with 𝑥 − 6
If u want to translate graph for example 8.9 units to left then replace 𝑥 with 𝑥 + 8.9
𝑓(𝑥 ) = 2𝑥 + 7

When translated 5.7 units to RIGHT:
𝑓(𝑥 − 5.7) = 2(𝑥 − 5.7) + 7
𝑓(𝑥 − 5.7) = 2𝑥 − 11.4 + 7
𝑓 (𝑥 − 5.7) = 2𝑥 − 4.4
𝑓(𝑥 ) = 𝑥 2 + 5

When translated 3 units to left:
𝑓 (𝑥 + 3) = (𝑥 + 3)2 + 5

There are two ways of finding turning point of function:
1.find midpoint of x-intercepts and put the x-coordinate of midpoint in equation of f(x) to
find y-coordinate of midpoint. The x and y of midpoint are the turning point.
2. if x-intercepts don’t exist then use completing square method
𝑦 = (𝑥 − ℎ)2 + 𝑘
Where turning point/vertex/maximum point/minimum point is (ℎ, 𝑘)
Circles

Equation of circle is: (𝑥
− ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
Where (ℎ, 𝑘) is center of circle and 𝑟 𝑖𝑠 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒

If given coordinates of diameter then midpoint of coordinates is center of circle and
length of coordinates divided by 2 is radius.
Figure 4: RED + BLUE = 180 degrees
INEQUALITIES
If −5𝑥 + 5 ≤ −6
−5𝑥 ≤ −11

SIGN WILL FLIP BECAUSE YOU ARE CHANGING SIGN OF X/BECAUSE YOU ARE
MULTIPLYING BOTH SIDES WITH -1/BECAUSE YOU ARE DIVIDING BOTH SIDES BY -1
SO 𝑥 ≥
11
5
If −5𝑥 + 5 ≤ 6
−5𝑥 ≤ 1

SIGN WILL FLIP BECAUSE YOU ARE CHANGING SIGN OF X/BECAUSE YOU ARE
MULTIPLYING EVEN IF ONLY ONE SIDE WITH -1/BECAUSE YOU ARE DIVIDING EVEN IF
ONLY ONE SIDE BY -1
SO 𝑥 ≥ −

5
For graphing inequalities, make graph of normal equation and if it’s a lesser than or
greater than sign then dotted line, otherwise if lesser than equal to or greater than
equal to then solid line. If its lesser than or lesser than equal to than shade below line(if
its vertical line then shade left of line),if its greater or greater than equal to then shade
above line(if its vertical line then shade right of line).
Figure 6:
Figure 5:
1
y <= 2x + 7
y > 0.5x + 9
PERCENTAGES AND PROPORTIONS


If its directly proportional then cross multiply
If its inversely proportional then straight multiply



If x increases by 10 percent then 𝑥 = 𝑥0 (1 + 10%)
If x decreases by 67 percent then 𝑥 = 𝑥0 (1 − 67%)
Where 𝑥0 is original value of 𝑥.
STATISTICS
Range = Highest Value – Lowest Value
Mode = value with highest frequency
Median = middle term of values

For grouped data, median(50% of data) =
𝑓+1
2
(THIS IS MEDIAN TERM NOT
MEDIAN, YOU WILL USE THIS TO FIND MEDIAN)

Lower Quartile (25% of data)
=
1+(𝑚ⅇ ⅆⅈ𝑎𝑛𝑡ⅇ𝑟𝑚−1)
2
(THIS IS Lower Quartile
TERM NOT LOWER QUARTILE, YOU WILL USE THIS TO FIND Lower
Quartile)

Upper Quartile (75% of data)
=
(𝑚ⅇ ⅆⅈ𝑎𝑛𝑡ⅇ𝑟𝑚+1)+ℎⅈ𝑔ℎⅇ 𝑠𝑡 𝑓 𝑟 ⅇ𝑞𝑢ⅇ𝑛𝑐𝑦
2
(THIS IS
Upper Quartile TERM NOT UPPER QUARTILE, YOU WILL USE THIS
TO FIND Upper Quartile)
 BOX AND WHISKER PLOT
∑𝑥

Mean for ungrouped data =

Mean of grouped data =

Standard deviation for ungrouped data √
𝑛
𝛴𝑓𝑥
𝛴𝑓
where x are the data and n is number of data items
(add up all the f’s multiplied by x’s)
∑𝑥 2
𝑛
− (𝑥̅ )2 where n is number of data
items and 𝑥̅ is mean of data


Standard deviation for grouped data √
∑𝑓𝑥 2
− (𝑥̅ )2
𝛴𝑓
x2 ’s) where f is frequency and 𝑥̅ is mean of data
variance = square of standard deviation
(add up all the f’s multiplied by
EXPONENTS AND POWERS
QUADRATIC EQUATIONS AND THEIR
GRAPHS
𝑎𝑥 2 + 𝑏𝑥 + 𝑐









If 𝑏 2 − 4𝑎𝑐 < 0 then no roots(no solutions)
If 𝑏 2 − 4𝑎𝑐 = 0 then only one root
If 𝑏 2 − 4𝑎𝑐 > 0 then more than one roots
If a is positive then happy face
If a is negative then sad face
To make the graph find turning point and y-intercept and x-intercepts(solutions)
If no solution then graph wont touch x-axis
c is the y-intercept
find turning point by completing square method
Shapes and angles and similarity and
congruency
(𝑛 − 2) × 180

Sum of interior angles =

Each angle of REGULAR shape =
(𝑛−2)×180

𝑛
where n is the number of sides of shape

When triangles are congruent angles and lengths are same
If 𝐴𝐷𝐵 ≡ 𝑇𝑅𝑄 then length of AD=TR, DB=RQ, AB=TQ
Angle A = Angle T and so on


When triangles are similar, angles are same but lengths are not same. Lengths are
PROPORTIONAL. If triangle ABC is similar to triangle PQR and AB=5 and PQ=15
Then PQ is 3 times more than AB in length, which means all lengths of PQR are 3 times
more than the CORRESPONDING lengths of triangle ABC BUT ANGLES ARE SAME
ANGLE A = ANGLE P AND SO ON
TRIGONOMETRY

To convert from degrees to radians 𝑥 0
×

𝑟
×
To convert from radians to degrees 𝑥
𝜋
180
180
𝜋
Figure 5: ASTC (ADD SUGAR TO COFFEE)
7

If

Draw triangle

Fill in the missing length using Pythagoras theorem and find cos(x).
sin(𝑥 ) =
8
then what is cos(𝑥 )?
sin(–x) = –sin(x)
cos(–x) = cos(x)
sin 𝑥
tan 𝑥 = cos 𝑥
tan(–x) = –tan(x)
0
30
SIN(X)
0
1
2
COS(X)
TAN(X)
1
0
Just do sin(x) over cos(x)
undefined
45
√2
2
60
√3
2
√3
2
√2
2
1
2
90
1
0
1
Just do sin(x) over cos(x)
SHAPES FORMULAS
Graphs
Figure 7: y=x square
Figure 8: y = a to the power x. even if it is 2 to power x
or 50 to power x graph basic shape will be like this
Figure 6: y= -x square
Figure 10: y=sin(x)
Figure 9: y = cos(x)
Figure 12: y= 1/x, note that if it becomes 1/x-sqaure then graph
becomes steeper but shape will be same
Figure 11: y=x cube