ALGEBRA 2 REVIEW
Translating Verbal Expressions into Mathematical Expressions
Verbal Expressions / Examples / Math TranslationAddition / added to / 6 added to y / 6+y
more than / 8 more than x / 8+x
the sum of / the sum of x and z / x+z
increased by / t increased by 9 / t+9
the total of / the total of 5 and y / 5+y
Subtraction / minus / x minus 2 / x-2
less than / 7 less than t / t-7
subtracted from / 5 subtracted from 8 / 8 – 5
decreased by / m decreased by 3 / m-3
the difference between / the difference between y and 4 / y-4
Multiplication / times / 10 times 2 / 10 X 2
of / one half of 6 / (1/2) X 6
the product of / the product of 4 and 3 / 4 X 3
multiplied by / y multiplied by 11 / 11y
Division / divided by / x divided by 12 / x/12
the quotient of / the quotient of y and z / y/z
the ratio of / the ratio of t to 9 / t/9
Power / the square of / the square of x / x2
the cube of / the cube of z / z3
squared / y squared / y2
Equivalency / equals / 1+2 equals 3 / 1+2 = 3
is / 2 is half of 4 / 2 = (½)X4
is the same as / ½ is the same as 2/4 /
yields / 3+1 yields 4 / 3+1 = 4
represents / y represents x+1 / y = x + 1
Comparison
greater than / -3 is greater than -5 / -3 > -5
less than / -5 is less then -3 / -5 < -3
greater than or equal to / x is greater than or equal to 5 / x ≥ 5
at least / x is at least 80 / x ≥ 80
no less than / x is no less than 70 / x ≥ 70
less than or equal to / x is less then or equal to -6 / x ≤ -6
at most / y is at most 23 / y ≤ 23
no more than / y is no more than 21 / y ≤ 21
Strategy for Solving Basic Algebraic Equations (Degree 1):
1. Use the distributive property to remove parentheses:
3(x - 3) + 3 = 18 – 5x becomes 3x – 9 + 3 = 18 – 5x
2. Combine like terms on either side of the equation.
-9 and 3 can be added to get -6.
3x – 6 = 18 - 5x
3. Use the addition or subtraction properties of equality to get the variables on one side of the = symbol and the constant terms on the other.
3x and 5x are like terms. Add 5x to each side to get the variable terms on the left.
3x + 5x – 6 = 18 -5x + 5x
8x - 6 = 18
4. Continue to combine like terms whenever possible.
6 and 21 are like terms. Since 6 is subtracted from 8x, add 6 to both sides to move it to the other side.
8x - 6 + 6 = 18 + 6
8x = 24
5. Undo the operations of multiplication and division to isolate the variable (Multiply both sides by the reciprocal of the coefficient of x).
Divide both sides by 8 to get x by itself.
8x/8 = 24/8
x = 3
6. Check the results by substituting your found value for x into the original equation.
3(x -2) + 5x = 18
3(3-2) + 5(3) = 18 ?
3(1) + 5(3) = 3 + 15 = 18 ? Yes.
So x = 3 is the solution to the equation.
Solving Linear Inequalities in one variable:
Complete the same process as above, but if in Step 5 you have to multiply or divide by a negative number, then you must switch the inequalities. The solution could be written as an inequality or graphed on a number line.
GEOMETRY and LINEAR EQUATIONS
x- intercept - point where graph crosses x-axis. y = 0 at this point.
y-intercept – point where graph crosses y-axis. x = 0 at this point.
slope= rise/run of a line, aka change in y/change in x.
If two points of a line are known, (x1,y1) and (x2,y2), the slope can be found with this equation, the slope,
The slope-intercept form of a linear equationis y = mx + b, where m is the slope and ( 0 ,b ) is the y-intercept.
Example:
Slope, m = - ½ and the y-intercept is (0, 1)
The point slope formula for a line with point
(x1, y1) and slope m is
y – y1 = m(x – x1)
Vertical angles (opposite angles formed from intersecting lines) are congruent (the same).
Parallel lines – have the same slope.
Perpendicular lines- form right angles. Their slopes are negative reciprocals of each other. Example: y=2x+1 and y= -½ x -2 are perpendicular.
You can find an equation for a line parallel or perpendicular to another line if you know the equation of the other line and a point on the line that is parallel or perpendicular to it.
Example: Find the equation of a line parallel to y = ½ x + 2 containing the point (2, 4).
Use the point slope formula. Since parallel lines have the same slope, use m = ½ If the line we wanted is perpendicular to to y = ½ x + 2, we would use m = -2/1
Use given point (2,4) in the point-slope formula.
This is the equation of a line PARALLEL to y = ½ x-2 which contains the point (2,4).
To graph a
LINEAR INEQUALITY IN TWO VARIABLES,
First rewrite the inequality to solve for y.
If the resulting inequality is y > ….,
Then make a DASHED line and shade the area ABOVE the line.
If the resulting inequality is y < …..,
Then make a DASHED line and shade the area BELOW the line.
If the resulting inequality is y≥ …..,
Then make a SOLID line and shade the area ABOVE the line.
If the resulting inequality is y ≤ …….,
Then make a SOLID line and shade the area BELOW the line.
If when isolating y, you must divide both sides of inequality by a negative number,
Then the inequality sign must be SWITCHED.
COMPOUND INEQUALITIES:
A UNION is ALL SHADED AREAS and an INTERSECTION is only where the SHADED AREAS OVERLAP.
Triangles
Scalene TriangleIsosceles TriangleEquilateral TriangleRight Triangle
(no sides the same)(2 sides the same) (3 sides the same)(one angle is 90°)
a2 + b2 = c2
Angles of a triangle add up to 180 degrees.
Perimeter – distance around the edges of an object
Perimeter of a square = 4s, where s = length of one side
Perimeter of a rectangle = 2W + 2L, where W = width and L = length
Perimeter of a triangle = side 1 + side 2 + side 3
Area – amount of surface covered by an object.
Area of a square = s2
Area of a rectangle = L*W
Area of a triangle = ½ bh, where b = length of base and h = height
FUNCTIONS
Function Operations: Example:
(f+g)(x) = f(x) + g(x) f(x) = 2x2 g(x) = 3x (f+g)(-1) = 2(-1)2 + 3(-1) = 2 + -3 = -1
(f – g(x) = f(x) – g(x) (f – g)(x) = 2x2 – 3x
(f/g) (x) = f(x)/ g(x) (f/g)(x) = 2x2 / 3x = 2x/3
(f · g)(x) = f(x)*g(x) (f · g)(x) = 2x2 * 3x = 6x3
(f o g)(x) = f( g(x) ) (f o g) (x) = 2(3x)2=2*9x2 = 18x2
WORD PROBLEMS
Mixture Problems
Quantity of a Substance in a solution = % of concentration * Amount of Solution
Example: Forty ounces of a 60% gold alloy means that the quantity of gold in the alloy is .60 * 40 = 24 ounces of gold.
Distance Problems
Distance = rate * time
If distance is in miles, and rate is in mph,
then time must be in hours.
If time is in minutes, then
multiply time by 1 hr/60min
to convert to hours.
Discount Problems
Discount = Discount Rate * Original Price
Discount Rate = Discount/Original Price (in decimal form)
Sale Price = Regular Price – Discount
Original Price = Cost of Item to the Seller
If a jeweler buys a diamond for $500 and discounts it 40%. What is the sale price of the diamond?
Original Price = Cost of Item to the Seller = $500
Discount = Discount Rate * Original Price = .40*500 = $200
Sale Price = Original Price - Discount = $500 - $200 = $300
Setting up word problems:
1)Find out what you are being asked to find. Set a variable to this unknown quantity. Make sure you know the units of this unknown (miles?, hours? ounces?)
2)If there is another unknown quantity, use the given information to put that unknown quantity in terms of the variable you have chosen.
(For example, if total distance traveled is 700 miles, then part of the trip is x miles and the other part of the trip is 700 – x miles.)
3)Set up a table with a row for each unknown and columns made up of the terms of one of equations above (r*t = d, Pr = I, etc..)
4)Use the given information to combine the equations of each row of the table into one equation with one variable to solve for.
5)Once one variable is solved for, you can find the other unknown. (For example, is x = 100 miles, then the other part of the trip is 700 – 100 = 600 miles.)
6)Check your equation by plugging in your value for x and seeing if your equation is true.
GEOMETRY WORD PROBLEMS:
Isosceles Triangle – Triangle with 2 sides the same lenth
Equilateral Triangle – Triangle with all 3 sides the same length
For all triangles, the 3 angles add up to 180 degrees.
Perimeter – distance around the edges of an object
Perimeter of a square = 4s, where s = length of one side
Perimeter of a rectangle = 2W + 2L,
where W = width and L =lengthPerimeter of a triangle = side 1 + side 2 + side 3
Area – amount of surface covered by an object.
Area of a square = s2
Area of a rectangle = L*W
Area of a triangle = ½ bh,
where b = length of base and h = height
Simple Interest
Interest earned ($$) = Principal * Rate * time
(initial investment) * (interest rate )* (1 year for annual interest)
Interest Rate must first be converted to a decimal.
Example: 7 ½ % = 7.5% = .075
Time must in in year units.
Example: 18 months = 18months * 1 yr/12 months = 1.5 years
Example: Interest earned in 1 year on an account earning 3.5% interest with initial investment of $1000 is
I = Prt = (1000)(.035)(1) = $35
EXPONENT RULES
Exponent - A number or symbol, as 3 in (x + y)3, placed to the right of and above another number, variable, or expression (called the base), denoting the power to which the base is to be raised. Also called power.
The exponent (or power) tells how many times the base is to be multiplied by itself.
Example 1:
(x + y)3 = (x + y)(x+y)(x+y)
Example 2:
(-3)4 = (-3)(-3)(-3)(-3) = 81
Scientific Notation
Scientific notation uses powers of 10 to write decimal number. Numbers written in scientific notation contain a number between1 and 10 (called the mantissa) multiplied by a power of 10. For example, the number 3.1 x 102 is in scientific notation.
Example 1 Change 45,000 to scientific notation.
Move the decimal place until the number is between 1 and 10. You have to move it 4 places to the LEFT, so the exponent will be 4.
4.5 x 104
Example 2 Change .00000637 to scientific notation.
Move the decimal place until the number is between 1 and 10. You have to move it 6 places to the RIGHT, so the exponent will be -6.
6.36 x 10-6
Example 3 Rewrite 3.4 x 103 in decimal form.
Since the exponent, 3 is positive, move then decimal point 3 places to the RIGHT.
3,400
Example 4 Multiply (5.4 x 103)(2.2 x 105)
Rearrange in to decimals and powers of 10
(5.4 x 2.2)(103x105) Since both powers have base 10, just add the exponents.
11.88 x 108 Unfortunately this is not in scientific notation because 11.88 is not between 1 and 10.
Since you have to move the decimal point one place to the left to get the mantissa to be between 1 and 10, then you must 1 to the power.
11.88 x 108 = 1.188 x 109
Linear Model Applications
At sea level, the boiling point of water is 100° C. At an altitude of 2 km, the boiling point of water is 93° C. This data can be written in table format, where the input represents the altitude above sea level and the output represents the boiling point of water.
Altitude above sea level (km), x / Boiling point of water (°C), y0 / 100°
2 / 93°
(a)Plot the two points on the grid below and sketch the line containing them. Extend the line so that it intersects the vertical axis.
(b)Determine the slope of the line. What are its units of measurement? What is the practical meaning of the slope in this situation?
(c)Write an equation for the line containing the two points you graphed.
(d)What is the vertical intercept of the line you graphed and whose equation you wrote? What is the practical meaning of the vertical intercept in this situation?
(e)Use your equation to predict the boiling point of water on the top of Mount Everest, which is approximately 8.85 km above the sea level. Round to the nearest degree.
(f)Use your equation to determine the altitude above the sea level if the boiling point of water is 81 C.
POLYNOMIALS
polynomial – is a term or sum of terms in which all variables have whole number exponents. Example: 3x, or x2 + 1, or -3x2 + 3x + 1
monomial – a number, a variable, or a product of numbers and variables.
Example: 3, 2x, -4x2 are all monomials.
binomial – the sum of two monomials that are unlike terms.
trinomial – the sum of three monomials that are unlike terms.
like terms – terms of a variable expression that have the same variable and the same exponent.
Example: 3x and 3x2 are unlike terms, but 3x and 2x are like terms.
factor – (in multiplication) a number being multiplied.
Example: What are the factors of 121? 1, 11, and 121.
121 = 11 X 11, 121 = 1 X 121
to factor a polynomial – to write a polynomial as a product of other polynomials
to factor a trinomial of the form ax2 + bx + c - to express the trinomial as the product of two binomials.
Example: x2 + 5x + 6 = (x+2)(x+3)
to factor by grouping – to group and factor terms in a polynomial in such a way that a common binomial factor is found.
Example: 2x(x+1) – 3(x+1) = (x + 1)(2x – 3)
factor completely – to write a polynomial as a product of factors that are nonfactorable over the integers.
FOIL method – A method of finding the product of two binomials in which the sum of the products of the First terms, of the Outer terms, of the Inner terms, and of the Last terms is found.
Example: (x+2)(x+3) =x2+ 3x + 2x + 2*3 = x2 + 5x + 6
common factor – afactor that is common to two or more numbers.
Example: What are the common factors of 12 and 16x2?
The factors of 12x are 1,2,3,4,6,12, and x
The factors of 16x2are 1,2,4,8,16, x, x
The common factors of 12x and 16x2 are 1x, 2x, and 4x
The Greatest Common Factor of 12x and 16x2 is 4x.
Perfect-Square Trinomials
(a+b) 2 = a 2 + 2ab + b 2
Difference of squares
a 2 - b 2 = (a+b)(a-b)
Rules for Variable Expressions:
Only like terms can be added, and when adding like terms, do not change the exponent of the variable.
5x2 + 3x2 = 8x2
When multiplying variable expressions, add exponents of like variables
(5xy3)(2y2)=10xy3+2 = 10xy5
When taking powers of variable expression that is a monomial (one term), multiply exponents of EVERY term inside the parentheses.
(2x3y4)3 = 23x3*3y4*3 = 8x9y12
When taking powers of a variable expression that is a binomial, trinomial or some other polynomial, use the rules of polynomial multiplication.
For example: (x+2)2 ≠ x2 + 22
(x+2)2 = (x+2)(x+2) = x2+4x + 4 (FOIL METHOD)
Example 2:
(x2+3x+5)2 = (x2+3x+5)(x2+3x+5)
=(x2+3x+5)x2 + (x2+3x+5)3x + (x2+3x+5)5 (DISTRIBUTIVE PROPERTY)
EXPRESSIONS / EQUATIONSExamples: 3+2(1-4)2, -3x+x, (3x+2)2 / Examples: 3x + 2 = 5, 5x + 5y=10, x(x-5) = -6
Can be simplified
Ex1: Don’t forget order of operations!
3+2(1-4)2 can be simplified to
3+2(-3)2 which becomes
3+2(9) which becomes
3+18 which becomes 21.
Ex2:
-3x + x can be simplified to
-2x / Can be solved:
Can be evaluated: (3x+2)2 can be evaluated at x= -1. (3(-1)+2)2 is (-3+2)2 which is (-1)2 which is 1. / Can be evaluated to see if a solution is true.
Is (3,4) a solution of y=-x+2?
4 = -3+2=-1 NO
Can be reduced:
Examples:
A General Strategy for Factoring a Polynomial
- Do all the terms in the polynomial have a common factor? If so, factor out the
Greatest Common Factor.Make sure that you don’t forget it in your final answer.
Example: 24x4 - 6x2 = 6x2(4x2 - 1). Also look to see if the other polynomial factor and be factored more. (4x2-1)=(2x-1)(2x+1), so the final answer is
24x4 - 6x2 =6x2(2x-1)(2x+1),
- After factoring out the GCF (if there is one) count the number of terms in the remaining polynomial.
Two terms: Is it a difference of squares? Factor by using: a2-b2 = (a+b)(a-b)
Example: 36x2 – 49 = (6x)2 – 72 = (6x-7)(6x+7)
If the polynomial can’t be factored, it is PRIME.
Three terms:Is it a perfect square trinomial?
If it is it would be in the form a2x2 + 2abx + b2 , which is factored as (a+b)2
or a2x2 + 2abx + b2 which is factored as (a-b)2
Example: 4x2 + 12x + 9 = (2x)2 + 2(2)(3)x + 32 = (2x + 3)2
Is it of the form x2 + bx + c?
Factor by finding two numbers that multiply to c and add to b.
Example: x2 -3x - 4 = (x+1)(x-4) because 1*-4 = -4 and 1 + -4 = -3
Can’t find the numbers? Maybe the polynomial is PRIME.
Is it of the form ax2 + bx + c?
Try factoring by the Grouping Method (or ac Method)or Trial and Error.
Example: 2x2 + 13x + 15 (the a*c method means multiply 2*15 which is 30.
Find factors of 30 that add up to the middle term’s coefficient, which in this case is 13. 3*10=30 and 3+10 = 13. Split the middle term into two parts:
2x2 + 10x + 3x + 15 and then factor by grouping.
2x(x+5)+3(x+5) = (2x+3)(x+5)
Those methods don’t work? Maybe the polynomial is PRIME.
Four terms:Try Factoring by Grouping. Group the 1st two terms and the last two terms. Factor out the Greatest Common Factor from each grouping. Then factor out the common binomial term.
- Always factor completely. Double check that each of your factors can not be factored more.
- Check your work by multiplying the factors together. Does it result in the original polynomial?
SOLVING POLYNOMIAL EQUATIONS (2 SOLUTIONS):
Put Equation in standard form, ax2 + bx + c = 0.
FACTORING METHOD:If product ac has two factors that add up to b, then it is factorable. Factor it and use the Zero Product Product to find the solutions. This says that if A*B=0, then A=0 or B=0. Example: x2 -2x = -3
Standard form: x2 – 2x + 3 = 0.
Factored: (x+1)(x-3) = 0
So x+1 = 0, which gives x = -1, or another possible solution is x-3 = 0, which gives x = 3.
RATIONAL EXPRESSIONS
A rational expression is a fraction in which the numerator or denominator is a variable expression (such as a polynomial). A rational expression is undefined if the denominator has a value of 0.
A rational expression is in SIMPLEST form when the numerator and denominator have no common factors other than 1.
Reducing to simplest form – factor the numerator and denominator, then cancel out any common factors in the numerator and denominator (not common factors that are both in the numerator or both in the denominator, e.g. side by side).
Multiplying Rational Expressions – factor the numerators and denominators then cancel out common factors as above, then multiply the numerators and multiply the denominators.
Dividing Rational Expressions – change to a multiplication problem by changing the DIVISOR into it’s RECIPROCAL.
Adding and Subtracting Rational Expressions –
Step 1: Factor the denominators, then find the LCM. The LCM of two polynomials is the simplest polynomial that contains the factors of each polynomial. To find the LCM of two or more polynomials, first factor each polynomial completely. The LCM is the product of each factor the greater number of times it occurs in any one factorization.
Step 2: Change each rational expression so that the new denominator will be the LCM. You will multiply the numerator and denominator of each expression by whatever it takes to get the LCM as the new denominator.
Step 3: Add the two new fractions by adding the numerators and keeping the denominator (the LCM) the same.
Step 4: Now factor the resulting expression and cancel out any common factors in the numerator and denominator.
Simplify Complex Fractions – Complex fractions are just rational expressions with fractions within fractions. To simplify, find the LCM of all the denominators of every fraction in the expression, then multiply the main numerator and denominator by that LCM. Then simplify as usual.
LCD= x3
Solving Equations with Fractions – multiply BOTH SIDES of the equation by the LCM of all denominators in the equation. Then solve as usual.
If the equation is one fraction set equal to another, this is called a PROPORTION. Solve by CROSS-MULTIPLYING, then isolating the variable.
Similar Triangles
Triangles are similar if at least 2
corresponding angles are the same
in each triangle.
WORK
Rate of Work * Time Worked = Part of Tasked Completed
If someone can do a job in 60min, their rate of work is 1/60min.
If someone else can do the same job in 40minutes, their rate of work is 1/40min.
The TIME to get the same job done TOGETHER can be found by
Adding their parts together to make 1 whole job.
Find LCD of 60 and 40. LCD = 120.
Multiply both sides of the equation by 120 to remove fractions.