American Certificate
(ACT/EST)
Final Revision
4 Edition
th
Name: ………………………………
Phone: ………………………………
Centre: ………………………………
Eng\ Kariom Alaa Eldin
Mob:010 3397 2664
Date: /
Contents
Introduction ...................................................................................................... 3
The Language of Algebra................................................................................... 3
Chapter 1 : Linear Equations ............................................................................. 4
Chapter 2 :Statistics and Data Analysis ........................................................... 12
Chapter 3 Law of Exponents and Polynomials ................................................ 17
Chapter 4 : Quadratic Functions ..................................................................... 19
The imaginary number .................................................................................... 24
Chapter 5 :Composition of Functions ............................................................. 25
Chapter 6 Direct and Inverse Variations ......................................................... 29
Chapter 7 :Trigonometric Functions ............................................................... 30
Chapter 8 : Geometry ..................................................................................... 31
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For Example
There is no simplifying between two different variables
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An equation is a mathematical sentence with an equal sign.
To translate a word sentence into an equation, choose a
variable to represent one of the unspecified numbers or
measures in the sentence. This is called defining a variable.
Then use the variable to write equations for the unspecified
numbers.
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Some equations have variables on both sides. To solve such
equations, first use the Addition or Subtraction property of
Equality to write an equivalent equation that has all of the
variables on one side. Then use the multiplication or Division
Property of Equality to simplify the equation if necessary.
When solving equations that contain grouping symbols, use
the Distributive Property to remove the grouping symbols
Use (shift solve) if you have only one missing from first
degree
And (mode 5eqn 3) if you have one unknown from 2nd
degree
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Y=c
x=c
The slope-intercept form of the equation of a line is
𝐴𝑥 + 𝐵𝑦 = 𝐶, in which
−𝐴
𝐵
𝐶
is the slope and is the y𝐵
intercept.
𝑦 = 𝑚𝑥 + 𝑏 , in which m is the slope and b is the yintercept.
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Parallel and Perpendicular Lines
Lines in the same plane that do not intersect are called parallel lines.
Parallel lines have the same slope.
If two nonvertical lines have the same slope, then they are parallel
For example:
Write the equation in point-slope form of the lines through
point(1,2) that are
(a) parallel to (b) perpendicular to, 3x-y=-2
Sol.
a=3 b=-1
−𝑎
𝑏
=
− 3
−1
finding the coefficients
=3
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calculating the slope
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A)
B)
Y=mx+c
finding the
equation
Y=3x+c 2=3*1+c
c=-1
Y=3x-1
Y= mx+c
−1
𝑦 = 𝑥+𝑐
3
𝑦=
−1
𝑥
3
−1
×
3
2=
1+𝑐
7
+3
Use: (mode 5 eqn 1) to get the solution
Or
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NO solution
Zero intersection
infinite solutions
many intersections
Ax+by=c
Mx+ny=k
Ax+by=c
Mx+ny=k
Use
𝑎
𝑏
=𝑛
𝑚
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Use
𝑎
𝑏
𝑐
=𝑛=𝑘
𝑚
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Solving Word Problems Using Linear Models
In SAT verbal problems, the construction of mathematical models that
represent real-world scenarios is a
critical skill. Linear equations can be used to model many types of real
life situation word problems, such
as cost, profit, speed, distance and time problems. To solve the verbal
problems, you need to interpret the
situation described in the problem into an equation, then solve the
problem by solving the equation.
Plan for Solving a Word Problem
1. Find out what numbers are asked for from the given information.
2. Choose a variable to represent the number(s) described in the
problem. Sketch or a chart may be helpful.
3. Write an equation that represents relationships among the numbers
in the problem.
4. Solve the equation and find the required numbers.
5. Answer the original question. Check that your answer is reasonable.
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The word percent means hundredth or out of every hundred.
To write a decimal or a fraction as a percent, multiply the decimal or the
fraction by 100 and add the % sign.
Convert the fraction to decimal.
To write a percent as a decimal or a fraction, multiply the percent by , and
1
drop the % sign.
100
Simplify the fraction.
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𝑛
𝑚
√𝑥 𝑚 = 𝑥 𝑛
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5
2
√𝑥 2 = 𝑥 5
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The standard form of a quadratic function
y=f(x) can be written in the form f(x)=ax2+bx+c,
in which 0≠a. The graph of a quadratic function
is called a parabola.
 If 0>a, the graph opens upward and the
vertex is the minimum point.
 If 0<a, the graph opens downward and the
vertex is the maximum point.
 The maximum or minimum point of a
parabola is called the vertex.
 The equation of the axis of symmetry is 𝑥 =
−𝑏
2𝑎
 The coordinates of the vertex are (
−𝑏
2𝑎
,f(
−𝑏
2𝑎
))
 The vertex form of a quadratic function can
be written in the form f(x)=a(x-h)2+k , in
which (h,k)
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 The factored form of a quadratic function can
be written in the form f(x)= a(x-L)(x-M)
𝑙+𝑚
 roots. vertex =(
2
𝑙+𝑚
,𝑓(
2
))
 To factorize the quadratic
y=x2 - sum x +product
 The solutions of a quadratic function are
the values of x for which f(x) =0. Solutions of
functions are
also called roots or zeros of the function. On
a graph, the solution of the function is the xintercept(s).
 To find the y-intercept of a parabola, let x
equal to zero in the equation of the parabola
and solve for y.
 If the parabola has two x-intercepts, then the
x-intercepts are equidistant from the axis of
symmetry.
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 The sum of roots equals
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−𝑏
𝑎
 The product of roots equals
𝑐
𝑎
 Determining the type of roots for the
second degree equation. There are 3 types.
Use the Discriminant to know: 𝑏 2 − 4𝑎𝑐
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''i" is defined as the number whose √−1 = 𝑖
i.e. 𝑖 2 = −1
i2 = - 1
i1 = √−1
i3 = - i
i4 = 1
If we have an 'i' with a power greater than 4 what we'll do
in divide n by 4
𝑛
4
= 𝑎. 𝑏 take the decimal value b
and multiply it by 4
for Example:
i23
23
4
= 5.75 then 0.75×4=3
so, i23 =i3 = -i
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Complex numbers consist of two parts (real + img)
a + bi
real
img no
no
Two Complex numbers are equal if and only if( ⟺) the two
real parts are equal and the two imaginary parts are equal.
i.e. if : (a + bi) and (c + di) are two Complex numbers and if :
a = c , b = d , then :
a +bi = c + di and vice versa : if : a + bi = c + di ,
then : a = c , b = d
The two numbers : (a + bi) and (a – bi) are called Conjugate
numbers.
Note : Take care that the Complex number and its
Conjugate differ only in the sign of their imaginary parts.
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Exponential Growth and Decay
Compound Interest Formulas
If initial amount P is invested at annual interest rate r,
the investment will grow to final amount A
in t years. 𝐴 = 𝑃(1 + 𝑟)𝑡
We can use the same formula for the population or
value of goods that is increasing or decreasing.
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Finding the nth Term of an
Arithmetic Sequence
Tn = a1 + (n - 1)d Where:
Tn is the nth term in the sequence
a1 is the first term n is the number of the term
d is the common difference
Infinite Arithmetic series
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A ratio of the lengths of sides of a right triangle is called a
trigonometric ratio.
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𝛱=180
For not right angled triangle
1
Area= x S1 x S2 sin(θ)
2
In any triangle the sum of two
smaller sides is greater the
biggest one
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Regular Polygons
A regular polygon is a convex polygon with all sides
congruent and all angles congruent.
A polygon is inscribed in a circle and the circle is
circumscribed about the polygon where each vertex
of the polygon lies on the circle. The radius of a regular
polygon is the distance from the center to a vertex
of the polygon. A central angle of a regular polygon is an
angle formed by two radii drawn to consecutive
vertices. The apothem of a regular polygon is the distance
from the center to a side.
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Circles
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X2+y2+ax+by+c=0
Center = (
−𝑎 −𝑏
2
,
2
)
−𝑎
−𝑏
2
2
Radius = √( )2 + ( )2 − 𝑐
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