Algebra and functions review

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)