Algebra and functions review
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Transcript of Algebra and functions review
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Algebra and Functions Review
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The SAT doesn’t include:
• Solving quadratic equations that require the use of the quadratic formula
• Complex numbers (a +b i)• Logarithms
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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+ = +
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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.
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Exponents
4x x x x x= ⋅ ⋅ ⋅
( )a
b a abbx x x= =
1
2x x=
Exponent
Definitions:
0 1a =
33
1y
y− =
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• 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= =
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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= ⋅ → =
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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.
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"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
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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−
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Solving Equations Involving Radical Expressions
Example
3 4 = 16x +
3 12x =3 12
3 3
x=
4 x = → ( )22 4 x = → 16x =
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Absolute Value
Absolute value
• distance a number is from zero on the number line
• denoted by • examples
x
5 5 4 4− = =
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• 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= =
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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⇒
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Inequalities
Inequality statement contains • > (greater than)• < (less than)• > (greater than or equal to)
• < (less than or equal to)
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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
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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.
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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
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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)
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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
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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 =
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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
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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
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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.
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Mathematical Expressions
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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
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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
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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
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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.
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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)