Algebra and functions review

Algebra and Functions Review

The SAT doesn’t include:

• Solving quadratic equations that require the use of the quadratic formula

• Complex numbers (a +b i)• Logarithms

Operations on Algebraic Expressions

Apply the basic operations of arithmetic—addition, subtraction, multiplication, and division—to algebraic expressions:

3 5 3

4 3 2 2

24 8

3

x yz z

x y z xy=

4 5 9x x x+ =10 -3 - (-2 ) 2 12 - z y z y z y+ =

2( 3)( - 2) - 6x x x x+ = +

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.

Exponents

4x x x x x= ⋅ ⋅ ⋅

( )a

b a abbx x x= =

1

2x x=

Exponent

Definitions:

0 1a =

33

1y

y− =

• To multiply, add exponents • To divide, subtract exponents

• To raise an exponential term to an exponent, multiply exponents

2 3 5 a b a bx x x x x x +⋅ = ⋅ =

5 23 3

2 5 3

1

mm n

n

x x xx x x

x x x x− −= = = =

3 4 2 6 8(3 ) 9 ( )x y a ax ayx y x y m n m n= =

Evaluating Expressions with Exponents and Roots Example 1 If x = 8, evaluate .

Example 2 If , what is x ?

23 2 338 8 64 4 or use calculator [ 8 ^(2/3)]= = =

2

3x

3

2 64x =

3 33 34 4x = ⋅ →

223 332 (64) x

⎛ ⎞= →⎜ ⎟

⎝ ⎠

3 264x = → ( )( )2

33 4x = →

4 4 16x x= ⋅ → =

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.

"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

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 = 3

z y−

Solving Equations Involving Radical Expressions

Example

3 4 = 16x +

3 12x =3 12

3 3

x=

4 x = → ( )22 4 x = → 16x =

Absolute Value

Absolute value

• distance a number is from zero on the number line

• denoted by • examples

x

5 5 4 4− = =

• Solve an Absolute Value Equation

Example first case second case

thus x=-7 or x=17 (need both answers)

5 12x− =

5 12 5 -12x x− = − =- 7 - -17x x= =

-7 17x x= =

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 • 10 divided into a number b

4 - m⇒

- 60x⇒

10

b⇒

2(3 - 5)x⇒

- 3y⇒

10

b⇒

Inequalities

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

• < (less than or equal to)

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 < -3

-2 6 >

-2 -2

x

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.

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

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)

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

Rational Equations and Inequalities

Rational Expression• quotient of two polynomials•

Example of rational equation

2 3

4

x

x

−+

34

3 2

x

x

+= ⇒

−3 4(3 2)x x+ = −

3 12 8 x x+ = − ⇒ 11 11x = ⇒ 1x =

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 8k 1

4k

1

4k

1(- )(3)

4y 3

4y

Inverse Variation or Inversely Proportional

• for some constant k

Example x and y are inversely proportional when x is 8 and y is -2. If x is 4, what is y?

• Using • Using k = -16,

ky

x

-28

k,

ky

x

-16

4y

-16k

- 4y

Word ProblemsWith 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.

Mathematical Expressions

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.

Example

2( ) ( ) 2 1xf x x g x

3Using ( ) 2 1 , g(3) = 2 + 1 = 8+1=9 xg x

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 2x – 1 > 0 x >

( ) 2 1f x x 1

21 1

domain or ,2 2

x

range 0 or 0,y

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 difference of y'sslope slope=

run difference of x's

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.

Translations and Their Effects on Graphs of Functions

Given f (x), what would be the translation of:

1( )

2f x

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)