This document provides an overview of key algebra and functions concepts covered on the SAT, including operations on algebraic expressions, factoring, exponents, evaluating expressions, solving equations, inequalities, systems of equations, quadratic equations, rational expressions, direct and inverse variation, word problems, functions, and translations of functions. Key points are that the SAT will not include solving quadratic equations using the quadratic formula or complex numbers. Factoring may be required to solve or evaluate expressions. Functions are represented using function notation and defined by their domain and range.

1. Algebra and
Functions Review

2. The SAT doesn’t include:
• Solving quadratic equations that require
the use of the quadratic formula
• Complex numbers (a +b i)
• Logarithms

3. Operations on Algebraic
Expressions
Apply the basic operations of arithmetic—addition,
subtraction, multiplication, and division—to
algebraic expressions:
4x + 5x = 9x
10z -3y - (-2z) + 2 y = 12z - y
(x + 3)(x - 2) = x2 + x - 6
x yz z
x y z xy
3 5 3
4 3 2 2
24 =
8
3

4. Factoring
Types
of
Factoring
• You are not likely to find a question
instructing you to “factor the following
expression.”
• However, you may see questions that
ask you to evaluate or compare
expressions that require factoring.

5. Exponents
x4 = x × x × x × x
y - 3
=
1
3
y
a
xb = b xa = ( b x )
a
1
x2 = x
Exponent
Definitions:
a0 = 1

6. • To multiply, add exponents
x2 × x3 = x5 xa × xb = xa+b
• To divide, subtract exponents
x 5 x 2
x x x x
x x x x
= 3 = - 3
= = -
1
2 5 3
m
m n
n
• To raise an exponential term to an exponent,
multiply exponents
(3x3 y4 )2 = 9x6 y8 (mxny )a = maxnay

7. Evaluating Expressions
with Exponents and Roots
Example 1
2
If x = 8, evaluate x3
.
2
83 = 3 82 = 3 64 = 4 or use calculator [ 8 ^(2/3)]
Example 2
If , what is x ?
3
x2 = 64
2
æ 3 ö
3 2
ç x2 ÷ = (64)3
®
è ø
x = 3 642 ® (( ) )2
x = 3 43 × 3 43 ®
3
x = 3 4 →
x = 4× 4® x = 16

8. Solving Equations
• Most of the equations to solve will be
linear equations.
• Equations that are not linear can usually
be solved by factoring or by inspection.

9. "Unsolvable" Equations
• It may look unsolvable but it will be workable.
Example
If a + b = 5, what is the value of 2a + 2b?
• It doesn’t ask for the value of a or b.
• Factor 2a + 2b = 2 (a + b)
• Substitute 2(a + b) = 2(5)
• Answer for 2a + 2b is 10

10. Solving for One Variable in Terms of Another
Example
If 3x + y =z, what is x in terms of y and z?
• 3x = z – y
• x =
z - y
3

11. Solving Equations Involving Radical
Expressions
Example
3 x + 4 = 16
3 x = 12
3 12
3 3
x =
x = 4 ®( )2
x = 42 → x = 16

12. Absolute Value
Absolute value
• distance a number is from zero on the number
line
• denoted by
• examples
x
-5 = 5 4 = 4

13. • Solve an Absolute Value Equation
Example
5 - x = 12
first case second case
5 - x = 12 5 - x = -12
-x = 7 -x = -17
x = -7 x = 17
thus x=-7 or x=17 (need both answers)

14. Direct Translation into
Mathematical Expressions
• 2 times the quantity 3x – 5
• a number x decreased by 60
• 3 less than a number y
• m less than 4
• 10 divided by b
Þ 4 - m
• 10 divided into a number b
Þ x - 60
10
b
Þ
Þ 2(3x - 5)
Þ y - 3
Þ b
10

15. Inequalities
Inequality statement contains
• > (greater than)
• < (less than)
• > (greater than or equal to)
• < (less than or equal to)

16. Solve inequalities the same as equations except
when you multiply or divide both sides by a
negative number, you must reverse the
inequality sign.
Example
5 – 2x > 11
-2x > 6
x
-2 > 6
-2 -2
x < -3

17. Systems of Linear
Equations and
Inequalities
• Two or more linear equations or
inequalities forms a system.
• If you are given values for all variables in
the multiple choice answers, then you can
substitute possible solutions into the
system to find the correct solutions.
• If the problem is a student produced
response question or if all variable
answers are not in the multiple choice
answers, then you must solve the system.

18. Solve the system using
• Elimination
Example 2x – 3y = 12
4x + y = -4
Multiply first equation by -2 so we can eliminate the x
-2 (2x - 3y = 12)
4x + y = -4
-4x + 6y = -24
4x + y = -4

19. Example 2x – 3y = 12
4x + y = -4 continued
Add the equations (one variable should be eliminated)
7y = -28
y = -4
Substitute this value into an original equation
2x – 3 (-4) = 12
2x + 12 = 12
2x = 0
x = 0
Solution is (0, -4)

20. Solving Quadratic
Equations by Factoring
Quadratic equations should be factorable
on the SAT – no need for quadratic
formula.
Example
x2 - 2x -10 = 5
x2 - 2x -15 = 0 subtract 5
(x – 5) (x + 3) = 0 factor
x = 5, x = -3

21. Rational Equations and
Inequalities
Rational Expression
• quotient of two polynomials
•
2 x
3
x
4
Example of rational equation
-
+
3 4
x
x
+ = Þ
-
3 2
x + 3 = 4(3x - 2)
x + 3 = 12x - 8 Þ 11x = 11Þ x = 1

22. Direct and Inverse
Variation
Direct Variation or Directly Proportional
• y =kx for some constant k
Example
x and y are directly proportional when x is 8 and y
is -2. If x is 3, what is y?
Using y=kx,
Use ,
2 - = k ´8
1
4
k = -
k = - (- 1)(3)
1
4
y = 3
4
4
y = -

23. Inverse Variation or Inversely Proportional
• y k
=
for some constant k
x
Example
x and y are inversely proportional when x is 8
and y is -2. If x is 4, what is y?
• Using
y =
k
, -2
= k x
8
• Using k = -16,
-16
4
y =
k = -16
y = - 4

24. Word Problems
With word problems:
• Read and interpret what is being asked.
• Determine what information you are given.
• Determine what information you need to know.
• Decide what mathematical skills or formulas you
need to apply to find the answer.
• Work out the answer.
• Double-check to make sure the answer makes
sense. Check word problems by checking your
answer with the original words.

25. Mathematical Expressions

26. Functions
Function
• Function is a relation where each element of the
domain set is related to exactly one element of
the range set.
• Function notation allows you to write the rule or
formula that tells you how to associate the domain
elements with the range elements.
f (x) = x2 g(x) = 2x +1
Example
Using g(x) = 2x +1 , g(3) = 23 + 1 = 8+1=9

27. Domain and Range
• Domain of a function is the set of all the values,
for which the function is defined.
• Range of a function is the set of all values, that
are the output, or result, of applying the function.
Example
Find the domain and range of
f (x) = 2x -1
2x – 1 > 0 x >
1
2
domain 1 or 1 ,
= ìí x ³ üý êé ¥ö¸ î 2 þ ë 2
ø
range = { y ³ 0} or [ 0,¥)

28. Linear Functions: Their Equations and Graphs
• y =mx + b, where m and b are constants
• the graph of y =mx + b in the xy -plane is a line
with slope m and y -intercept b
•
rise slope slope= difference of y's
run difference of x's
=

29. Quadratic Functions: Their Equations and
Graphs
• Maximum or minimum of a quadratic
equation will normally be at the vertex. Can
use your calculator by graphing, then
calculate.
• Zeros of a quadratic will be the solutions to
the equation or where the graph intersects
the x axis. Again, use your calculator by
graphing, then calculate.

30. Translations and Their Effects on Graphs of
Functions
Given f (x), what would be the translation of:
1 f ( x
)
2
shifts 2 to the left
shifts 1 to the right
shifts 3 up
stretched vertically
shrinks horizontally
f (x +2)
f (x -1)
f (x) + 3
2f (x)