PowerPoint Project: Exam Review

By: Jack Dunn and Max Scheurell

Properties Addition Property (of equality):

Example: If x = y then 5 + x = y + 5 Hint: if you add a 5 to both sides the answer does not change

Multiplication Property (of equality) Example: If a = b then ca = cb Hint: If you multiply both sides by “c” the answer does not change

Reflexive Property (of equality) Example: If a is a real number then a = a Hint: a number equals itself

Symmetric Property (of equality) Example: If a and b are real numbers then a = b and b = a Hint: a real number = a real number

Transitive Property (of equality) Example: If a = b and b = c then a = c Hint: can be assumed If a = b and b = c then a = c

Associative Property of Addition Example: (a + b) + c = a + (b +c) Hint: if the signs are the same they can be moved and the answer will not change

Associative Property of Multiplication Example: (ab)c = a(bc) Hint: if the signs are the same they can be moved and the answer will not change

Commutative Property of Addition Example: a + b = b + a Hint: you commute to school one way and back another way

Commutative Property of Multiplication Example: ab = ba Hint: you commute to school one way and back another way

Distributive Property Example: a(b + c) = ab + ac Hint: you distribute the a to the b and the c and the answer doesn’t change

Property of Opposites or Inverse Property of Addition Example: a + (-a) = 0 Hint: if you add the opposite the answer will be zero

Property of Reciprocals or Inverse Property of Multiplication Example: b × 1/b = 1 Hint: If you multiplied by a reciprocal to get 1

Identity Property of Addition Example: a + 0 = a Hint: a keeps its identity the answer does not change because zero is added

Identity Property of Multiplication Example: a × 1 = a Hint: if you multiply by one “a” keeps its identity (does not change)

Multiplicative Property of Zero Example: a × 0 = 0 Hint: any number multiplied by zero = 0

Closure Property of Addition Example: if “a” is a real number and “b” is a real number then a + b is real number Hint: a real number added to a real number is a real number

Closure Property of Multiplication Example: if “a” is a real number and “b” is a real number then ab is real number Hint: a real number times a real number is a real number

Product of Powers Property Example: 72 × 76 = 7(2 + 6) = 78

Hint: when finding the product of powers you add them

Power of a Product Property Example: (ab)m = am × bm

Hint: the exponent outside the parenthesis applies to all that is in the parenthesis Power of a Power Property

Example: (am)n = amn

Hint: in Power of a product property you multiply the exponents Quotient of Powers Property

Example: Hint: property states that to divide powers having the same base, subtract the

exponents. Power of Quotient Property

Example: Hint: power of a quotient can be obtained by finding the powers of numerator and

denominator and dividing them Zero Power Property

Example: (3)0 = 1 Hint: a number to the zero exponent equals one

Negative Power Property Example: Hint: move the number to the denominator if in the numerator

Zero Product Property Example: (1 + a) (6 + a) then 1 + a = 0 or 6 + a = 0 Hint: A product of factors is zero if and only if one or more of the factors is zero.

Product of Roots Property Example:√a ∙ √b = √ab Hint: the variables are multiplied

Quotient of Roots Property Example: Hint: take the square of both the numerator and the denominator

Root of a Power Property Example: = Hint: none

Power of a Root Property Example: Hint: none

Solving First Power Inequalities in one Variable (Practic Number lines

Solving inequalities with only one inequality sign (try problems before looking at the answer

Problem: x < 4 (put on number line)

Hint: the problem reads “x” is less than four. Therefore the circle is open and the arrow goes from the four toward the negative and carries on. If the problem had this sign ≤ (less than or equal to) the circle would be colored in rather than open

Continued

Conjunction Problem: -2 < x and x≤ 4

Hint: The reason this is a conjunction is because it contains the word “and” which tells you that both inequalities need to be contained in you number line. If the problem read -2 < x ≤ 4 it is also implied that both inequalities need to be contained in you number line. The problem reads “x” is greater than negative two “and” “x” is less than or equal to four. You have to find an answer that works for both. A special case is also a conjunction and in this special case the answer is null set (Ø) because it is impossible to put both on the same number line

Example: -2 < x and x > 4 These would go in different directions on the number line and have no common

points

Continued

Disjunction Example: x < -3 or x > -2

Hint: in a disjunction both inequalities can be labeled. There is a special case with disjunctions in which all real number or the whole number line is contained. An example would be:

x > -3 or x < -2 Contained would be all real numbers because it is a disjunction and the lines

would go toward one another making them go on forever in both directions

Linear Equations in two variables

Slopes A slope is told by rise over run and in slope-intercept form (y = mx + b) is

“mx” and the y intercept is “b” Rising Line: Has a positive rise over run or slope and all rising lines will

enter quadrant I and quadrant III eventually. A possible equation is y = 2x + 3.

Is not the answer to the equation but is an example of a rising line

Falling Line: Has a negative slope or rise over run and all falling lines will enter quadrant II and quadrant IIII eventually. A possible equation is y = - 2x -9

Is not the answer to the equation but is an example of a falling line

Vertical line: Has an undefined slope and a rise over run of 1/0. The x-coordinate stays the same in a vertical line. A possible equation is x = 5 it is as simple as that because there is no slope or y-intercept.

Horizontal line: Has a slope of 0 and a rise over run of 0/1. The y-coordinate stays the same in a horizontal line. A possible equation is y = 7 because there is no slope worth writing in the equation and the y-intercept is seven

Standard Form: for a linear equation in two variables, x and y, is usually given as:

Ax + By = C A, B, and C are integers, and A is non-negative, and, A, B, and C have no common

factors other than 1 Also called general form You can convert it to slope-intercept form by isolating the “y” variable

Point-slope formula: is used to take a graph and find the equation of that particular line.

Just plug x and y values into your point-slope formula above and you can find the equation of a line simple as that

Plug in 5 for “m” and plug in points (5,5) and when you find your answer click Answer: y = 5x -5 If you have an x and y coordinate and the slope you can find the slope-intercept

form

How to Graph a line When you have your equation in slope-intercept form you can easily graph

the line A refresher:

Slope-intercept form is y= mx + b “b” is the y-intercept “m” is the slope In the problem y = -5x + 6 six is the y-intercept and can be put on the y axis at

positive six. -5x is the slope and x really has no meaning here -5 is the same as -5 over 1 so therefore you go down 5 pts. And over one making this line a falling line

7

6

5

4

3

2

1

-2 2

f x = -5x+6

This is the equation above graphed

Another possible equation which would give you a rising line is Y = 6x -5 The y intercept is -5 and the slope is 6 and over one F(x) = 6x – 5 is the same problem f(x) is the same as y and is a function

A Function is only a relation when it has a distinct input and a distinct output. Not all relations are functions though all functions are relations

See if you can graph this problem yourself

The problem will appear when you click

This is the equation above graphed

2

1

-1

-2

-3

-4

-5

-2 2 4

A

Another possible equation which would give you a Horizontal line is: Y or f(x) = 2 This problem results in a horizontal line because the y coordinate is always equal

to 2 and will go across the graph See if you can graph this

The problem will appear when you click

This is the equation above graphed

It is hard to see but the horizontal line across the graph is the line

4

3

2

1

f x = 2

Another possible equation which would give you a Vertical line is: X = 3 The x coordinate will always equal 3 and will go straight up the graph See if you can graph this equation

The problem will appear when you click

This is the equation above graphed

6

4

2

-2

-4

-5 5

f y = 3

Linear Systems Substitution Method

Replace a variable with an equal expression To be able to do substitution method you need to be able to isolate a variable and

leave it with a coefficient of one Take the equality of the variable and substitute it in for the variable in the second

equation and then solve Example (see if you can complete the problem and find the right answer)

y = x + 7 and 2x + 3y = 11 Steps: 2x + 3 (x + 7) = 11 2x + 3x + 21 = 11 5x + 21 = 11 5x = -10 x = -2 Then you must plug the x variable into the beginning problem y = -2 + 7 Final answer is: (-2,5)

Elimination Method Combine the two equations to cancel out a variable Find the LCM of either the x or y variable and its coefficient then multiply them

by the factor then add the variables together you will end up with either the x or the y variable eliminated then solve the remaining for the variable and back substitute

Example (try and solve by yourself first) 3x + 2y = 5 and 5x -3y = 15 9x + 6y = 15 and 10x – 3y = -34 19x = -19 x = -1 Then back substitute -3 + 2y = 5 2y = 8 y = 4 Final answer is: (-1,4)

Consistent system Parallel lines don’t intersect and have the same slope Parallel lines are a consistent system

4

2

-2

-4

Inconsistent System Perpendicular lines which intercept and have opposite reciprocal Perpendicular lines are an inconsistent system

4

2

-2

-5 5

g x = 1

3 x

f x = -3x+1

Dependent System Two line with the same domain and range All points on the line

4

2

-2

-4

Factoring and Rational Expressions

GCF For any numbers of terms Examples of GCF try to do them before looking at the answer

10x^3 - 15x^2 Solution:

5x^2(2x - 3) Both 10x^3 and 15x^2 are multiples of five and each contain 5x^2

Difference of Squares Difference of squares factors into two conjugates Examples of Difference of Squares

x^2 – 16 Solution:

(x+4)(x-4) Unlike the sum of squares this can be factored and is factored into two conjugates

Sum and Difference of Cubes End product will be a binomial and a trinomial Example:

X^3 - 27 Solution:

You have to first cub the roots with the original sign in the middle. Then you square those roots. The middle product is the opposite product of the cubed roots:

(x-3)(x^2+3x+9)

PST A trinomial that is the exact square of a binomial Example:

x^2 + 4x + 4 Solution:

(x+2)(x+2) When factored the solution is two of the same binomials If the 1st and third are squares and the middle number is twice the product the

answer is a PST

Reverse FOIL Trial and Error Example:

q^2+5q+6x Solution:

(q+3)(q+2) You foil by trial and error but if the signs are both addition signs then all of your

signs in the binomial are addition signs Grouping 2 X 2

Find the GCF and then the leftover factors which will leave you w/two binomials Example:

5x+10y-ax-2ay Solution:

First regroup and find the GCF: 5(x+2y) - a(x+2y) x+2y is the common factor and can be taken out the final answer is: (x+2y)(5-a)

Grouping 3 X 1 If an expression consists of four terms and part of it is a perfect square, then the

expression can be factorized by grouping 'Three and One‘ Example:

a^2-6a+9-16y^2 Solution:

(a^2-6a+9) - 16y^2 (a-3)2 - (4y)2 Final answer: (a-3+4y)(a-3-4y)

Quadratic Equations In One Variable

By Factoring Set the equation equal to zero so that all variables and numbers are on one side of

the equation Then factor the problem Example y2 + 15y + 56 = 0 Now factor for your answer (y + 8)(y + 7)

Square root of each side Example: 9x^2 = 64 Solution: 9x^2 – 64 = 0 (3x + 8)(3x – 8) = 0 3x = -8 or 8 X = -8/3 or 8/3

Solve by completing the square x^2 + bx __?__ Find half the coefficient of x Square the result of Step 1 Add the result of Step 2 to x^2 + bx + (b/2)^2 Example: X^2 + 14x + ___?___ X^2 + 14x + 49 = (x + 7)^2 (14/2)^2 = 49

Quadratic Formula Y = Y = ax^2 + bx + c

A goes in for “a” in the formula B goes in for “b” in the formula etc…

Discriminant Example: X^2 +4x + 1 = 0

2 4

2

b b ac

a

24 4 4(1)(1)

2(1)

4

2

-2

-5

4 12 4 2 3

2 2

2 3

2

Functions

Domain The domain of a function is the set of all possible x values which will make the

function "work" and will output real y-values The range of a function is the complete set of all possible resulting values of the

dependent variable of a function, after we have substituted the values in the domain.

Example: The domain of the line below is all real numbers. (X= All real numbers)

If the line is vertical the domain would be what X is equal to. If X were 14 the domain could only be 14 no other numbers because no point on that line would have a coordinate of 14.

Range The range is the set of all possible output values (usually y), which result from

using the function formula. The Range of the Line below on the left is all real numbers. (Y = all Real numbers) The Range of the line below on the right is only 14 because all the points on that

line will have a Y coordinate of 14

How to find a linear function from two ordered pairs of data Use the Formula: (y2 – y1)/(x2 – x1) to find the slope of the linear function To find the Y intercept you set x= 0 in slope intercept form after you find the slope Example if you have (-2, 5) and (4,8) as your ordered pairs Step 1: find the slope (m = slope

m = (8 – 5)/(4 – (-2)) m = (8 – 5)/(4 + 2) m = 1/2

Step 2: substitute the slope or 1/2 for the variable m in y = mx + b y = 1/2x + b

Choose one of the points, say (4,8). Now substitute 4 for x and 8 for y.

y = 1/2x + b 8 = 1/2(4) + b Now solve for b 8 = 2 + b 6 = b

Then substitute 6 in for the variable b in the point slope formula (y = mx + b) like you did with the slope for the variable m

Your final equation should be y = 1/2x + 6

Quadratic Functions

How to graph a quadratic function Example 1: f(x) = x^2 – 2x - 2 Solution find the coordinates of selected points as shown in the table below. Plot the

points and connect them with a smooth curve

x X^2 – 2x - 2

-2 (-2)^2 – 2(-2) – 2 = 6

-1 (-1)^2 – 2(-1) – 2 = 1

0 (0)^2 – 2(-1) – 2 =1

1 1^2 – 2(1) – 2 = -3

2 2^2 – 2(2) -2 = -2

3 3^2 – 2(3) -2 = 1

4 4^2 – 2(4) -2 = 6

6

4

2

-2

-4

5

g y = 1

f x = x2-2x-2

This curve shown on the graph below (same graph as slide before) is called a parabola. This parabola opens upward and has a minimum point at (1, -3). The y-coordinate of this point s the least value of the function.

You can decide whether the parabola will open upward or open downward (close) is if a in the quadratic function f(x) or y = (a)x^2 – bx – c is positive then the parabola will open upward and if it is negative then the parabola will be closed or open downward

The dotted vertical line x= 1, containing the minimum point, is called the axis of symmetry of the parabola. Meaning that the two halves are equally the same.

The equation for the axis of symmetry’s line is x = -b/2a The maximum point is the highest point or ordered pair of a parabola and the y –

coordinate this point is the greatest value

6

4

2

-2

-4

5

g y = 1

f x = x2-2x-2

The maximum point is the highest point or ordered pair of a parabola and the y – coordinate this point is the greatest value

The minimum or maximum point of a parabola is called the vertex. The average of the x- coordinates of any such pair of points is the x-coordinate of the

vertex The average of these x-coordinates is -b/2a The x-coordinate of the vertex of the parabola y = ax^2 + bx + c is -b/2a

Note: a cannot equal 0 Example 1: find the vertex of the graph of H: x 2x^2 + 4x -3

Use the vertex and four other points to graph H Identify and graph the axis of symmetry

Solution Step 1: x – coordinate of vertex = -b/2a

= - 4/2(2) = -4/4 = -1

Step 2: To find the y-coordinate of the vertex, substitute -1 for x y = 2x^2 + 4x – 3 y = 2(-1)^2 + 4(-1) - 3 y = 2 – 4 – 3 y = -5

The vertex is (-1, -5) because our solution for x = -1 and y = -5

Step 3: For values of x, select two numbers greater than -1 and two numbers less that -1 to obtain paired points with the same y coordinate.

Step 4:Plot the points and connect hem with a smooth curve. Step 5: The axis of symmetry is the line x = -1 (dashed line) Red = vertex x and y coordinates

X 2x^2 + 4x – 3 = y

-3 2(-3)^2 + 4(-3) – 3 = 3

-2 2(-2)^2 + 4(-3) - 3 = 3

-1 2(-1) + 4 (-1) - 3 = 3

0 2(0)^2 +4(0) - 3 = 3

1 2(1)^2 + 4(1) - 3 = 3

6

4

2

-2

-4

-5

g y = -1

f x = 2x2+4x -3

Vertex

Simplifying Expressions With Exponents simplify with exponents, don't feel like you have to work only from the for exponents. It is

often simpler to work directly from the definition and meaning of exponents. For instance: Simplify x6 × x5 The rules tell me to add the exponents. But I when I started algebra, I had

trouble keeping the rules straight, so I just thought about what exponents mean. The " x6 " means "six copies of x multiplied together", and the " x5 " means "five copies of x multiplied together". So if I multiply those two expressions together, I will get eleven copies of x multiplied together. That is:

x6 × x5 = (x6)(x5)              = (xxxxxx)(xxxxx)    (6 times, and then 5 times)             = xxxxxxxxxxx         (11 times)              = x11  

Thus: x6 × x5 =   x11 Simplify the following expression: The exponent rules tell me to subtract the exponents. But let's suppose that I've forgotten the

rules again. The " 68 " means I have eight copies of 6 on top; the " 65 " means I have five copies of 6 underneath.

How many extra 6's do I have, and where are they? I have three extra 6's, and they're on top.

Simplify (–46x2y3z)0 This is simple enough: anything to the zero power is just 1. (–46x2y3z)0 = 1 Simplify –(46x2y3z)0 The parentheses still simplifies to 1, but this time the

"minus" is out front, out from under the power, so the exponent doesn't touch it. So the answer is:

–(46x2y3z)0 = –1

Simplify the following expression: I can cancel off the common factor of 5 in the number part of the

fraction: Now I need to look at each of the variables. How many extra of each do I

have, and where are they? I have two extra a's on top. I have one extra b underneath. And I have the same number of c's top and bottom, so they cancel off entirely. This gives me:

In the context of simplifying with exponents, negative exponents can create extra steps in the simplification process. For instance:

Simplify the following: The negative exponents tell me to move the bases, so:

   Then I cancel as usual, and get:  

Simplifying Expressions with Radicals

Square root problems are not difficult for a person when they are just one number or a variable squared etc… but when we come across a problem that cannot be squared we have a little more difficulty

These problems (an example is: ) we have more difficulty but in truth the problem is not that hard

can simply be reduced to and that can be reduced to since square can be taken of four you can take a two out as the square of four and your final answer will be:

The answer does look complicated but with the simple steps you can find the answer now you try one (click and examples will appear on the next page

56

56 7 8 4 2 7

2 7 2

Examples (try these before you look at the answer):

32

16 2

4 2

84

12 7

4 3 7

2 3 7

50

2 25

5 2

Word Problems

Mimi is four years older than Ronald. Five years ago she was twice as old as he was. Find their ages now. (try to solve)

Step 1 the problem asks for Mimi’s age and Roald’s age now. Step 2 Let m = Mimi’s age now, and let r = Ronald’s age now Step 3 Use the facts of the problem to write two equations

M = 4 + r (now) M – 5 = 2(r – 5) (five years ago)

Step 4 Simplify the equations and solve M = 4 + r M – 5 = 2(r-5) That goes to m – r = 4 and m – 2r = -5 Then it goes to m-r = 4 m - 9 = 4 M = 13 is your final answer

Bicyclists Brent and Jane started at noon from points 60 km apart and rode toward each other, meeting at 1:30 pm Brent’s speed was 4km/h greater than Jane’s speed. Find their speeds.

Solution Step 1 the problem asks for Brent’s speed and Jane’s speed. (Draw a sketch to help

yourself out) Step 2 Let r = Jane’s speed. Then r +4 = Brent’s speed Step 3 The sketch helps you write the equation

1.5(r + 4) + 1.5r = 60 Step 4

1.5r + 6 + 1.5r = 60 3r + 6 = 60 3r = 54 R = 18 (Jane’s speed) R + 4 = 22 (Brent’s speed)

Raoul says he has equal numbers of dimes and quarters and three times as many nickels as dimes. The value of his nickels and dimes is .50$ more than the value of his quarters. How many of each kind of coin does he have?

Solution Step 1: the problem asks for the numbers of nickels, dimes, and quarters Step 2: :et x = the number of dimes and x = the number of quarters Then 3x = the

number of nickels Step 3: value of nickels + value of dimes = value of quarters + 50

15x + 10x = 25x + 50 Step 4: 25x = 25x + 50 Then 0 = 50 giving you a false statement and letting you know that the problem

doesn’t have a solution

Hector Herrera made a rectangular fish pnd surrounded by a brick walk 2m wide. He had enough bricks for the are of the walk to be 76 sq. m. Find the dimensions of the pond if it is twice as long as it is wide.

Solution: Step 1: The problem asks for the dimensions of the pond. Make a sketch. Step 2: Let x = the width of the pond. Then 2x = the length of the pond Step 3: Area of walk = Area of pond and walk – area of pond

OR 76 = (2x + 4)(x + 4) – (2x)(x) Step 4: 76 = 2x^2 + 12x + 16 – 2x^2 76 = 12x + 16 60 = 12x 5 = x and 2x = 10

Line of Best Fit or Regression Line

When do you use this? When you are sure you cannot graph the line using simple arithmetic

How does your calculator help? Your calculator can more quickly find and solve the function that you plug in to

it, but becomes not useful when you use your calculator for simple easy problems that you can do on your own

Here is a problem for you to plug into your calculator and find out if you can find the regression equation

Please get your calculator out Do you remember how to find the regression equation?

Plug in: (4,6), (8,6), (4,4) and (6,8) (when you have found the answer click) Answer: y = 0.37x + 3.9