Review Guide for Algebra 2 Honors Final
Review Guide for Algebra 2 Honors Final
Name_____________________________
The final exam in this class is worth 20% of your final grade, so you should study for it as it can greatly affect your overall grade in class!
Format of the exam:
27 Multiple Choice Problems
5 Short Answer Problems:
One dealing with each of the following topics
1. Equations of lines
2. Functions
3. Quadratic Functions
4. Exponential and Logarithmic Functions
5. Polynomials
Topics you should be comfortable with for the exam:
Exponents
Foiling
Factoring
Slope Intercept Form
Point Slope Form
Parallel and perpendicular lines
Function
Domain and range
Solving square roots
Completing the square
Quadratic Formula
Growth and Decay
Logarithmic Functions
Exponential Functions
Synthetic Division
Dividing Rationals
Multiplying Rationals
Rationalizing a Denominator
Axis of Symmetry
Vertex
Graphing Polynomials
End Behavior
Lets Start at the Beginning with…
Slope Intercept Form and Point Slope Form
Slope Intercept Form: y=mx+b
Point Slope Form: (y-y1)=m(x-x1)
Parallel Lines: same slope
Perpendicular Lines: inverse reciprocal slope ex: 3/2 and -2/3
Slope: m=(y2-y1)/(x2-x1)
X intercept: plug in zero for y and solve for x
Y intercept: plug in zero for x and solve for y
Domain: x values: on a graph you read domain left to right
Range: y values: on a graph you read range bottom to top
Determining if something is a function:
X: 2 3 4 5
Y: -1 -1 -1 -1 is a function because no x values repeat
X: 2 3 2 5
Y: -1 0 -4 6 is not a function because the x value 2 repeated twice
Quadratic Functions:
Methods of Factoring:
Greatest Common Factor: Divide each term by the GCF
When to use it:
Any time you notice common factors among all terms
Example: 6x2-18x+24 is 6(x2-3x+4)
Difference of Squares:
When to use it:
When both terms are perfect squares separated by a “-“ sign
When you cannot use it: When both terms are perfect squares separated by a “+” sign
How to use it:
Break into two parentheses. One parentheses gets a + sign and the other gets a – sign. Take the square root of each term and put in each parentheses.
Example:
9x2-25y2 = (3x+5y)(3x-5y)
Factoring a Trinomial when the leading coefficient is 1:
When and how to use it:
Find the factor of the last term that adds or subtracts to give the middle term
Example:
X2-x-12= (x-3)(x+4)
Factoring a Trinomial when the leading coefficient is not 1:
When and how to use it:
Both method: Factors of the a term go in the top row of boxes and factors of the c term go in the bottom row. Nothing in the top row is ever to be negative. You cross multiply to get the numbers outside the box. These numbers need to add or subtract to give you the middle term. You write the parentheses for the problem going down each column of the box. The top number always gets the variable.
Example:
8x2+30x+27= (2x+3)(4x+9)
The Quadratic Formula:
[pic]
When to use it: any time you need to factor anything!
Example:
3x2+9x-4 becomes [pic]
Completing the Square:
When and how to use it:
Get all x terms on one side of the equal sign and all other on the other side. Add (b/2)2 to both sides. Write as a perfect square binomial keeping in mind you will either have (x- b/2)2 or (x+b/2)2
Example:
Solving Square Roots
Example: 5x2=50
X2=10
[pic]
Move all variables to one side and all constants to the other. Take the square root of each side, keeping in mind there will be a positive and negative answer.
Polynomials:
Degree of a polynomial: # of the highest exponent
Even or Odd Polynomial: highest exponent is even/odd number
Positive or Negative Polynomial:
Is the leading coefficient positive or negative?
End Behavior:
Even + means both sides are up
Even – means both sides fall
Odd+ left side falls, right side rises
Odd – right side falls, left side rises
Y Intercept of a polynomial: last term of the polynomial that does not have a variable. If there is no term without a variable , then the y intercept is zero!
Synthetic Division:
Example:
Factoring a Polynomial if given one factor
Example: If given x=3 is a factor of a specific polynomial, but the 3 in the box of the synthetic division problem and solve as taught in synthetic division. When you reach the end with your new polynomial, factor the quadratic to get the other zeros
Graphing a Polynomial:
Look at the highest exponent- even or odd?
Look at the leading coefficient- even or odd?
Describe the end behavior of the graph as noted above
Locate zeros using the synthetic division, calc tool on the graph, or the table (in table you are looking for x values when the y value is 0)
Identify the y intercept as outlined above
Rationals:
Simplifying Rationals: Factor and cancel factors when possible
Example:
Multiplying Rationals:
Factor first and then cross cancel factors
Example:
Dividing Rationals:
Multiply by reciprocal and repeat using steps from multiplying rationals steps above
Example:
Logarithms and Exponents:
Exponential Form: bx=y
Logarithmic Form:
Logby=x
Growth and Decay Model:
A=a(1+-r)t where A is the final amount, a is the initial amount
R is the rate in decimal form
You use a + for growth and a – for decay
T is the number of years
Miscellaneous:
Exponents:
To the zero power: x0=1
Negative exponents: move to denominator and it becomes positive
Multiplying Polynomials: (x4)(x6)=x10
Add exponents
Power to power: (x4)6=x24
Multiply exponents
Dividing Exponents: (x6)/(x4)=x2
Subtract exponents
Rationalizing a Denominator:
When do we do it?
When there is a square root in the denominator
How do we do it?
Multiply by the opposite to get rid of the square root in the denominator
Example:
[pic][pic]=[pic]
Practice Problems:
1. Which is the equation of the line that is perpendicular to the line x=4.5 and passes through the point (3,2)?
X=4.5 is a vertical line. A horizontal line will be perpendicular at y=2
2. Find the intercepts and graph 5x+7y=-6
X intercept: 5x+7(0)=-6
5x=-6
X=-6/5
Y intercept 5(0)+7y=-6
7y=-6
Y=-6/7
3. Write the equation of the line in slope intercept form: A line passes through (2,4) and (6,1)
y-4=m(x-2) need to find m m=(4-1)/(2-6)=-3/4
y-4=-3/4(x-2)
y-4=-3/4x+6/4
y=-3/4x+6/4+16/4
y=-3/4x+22/4
4. Write the equation of line in slope intercept form: A line passing through (1,0) parallel to the line that contains the points in the table
|X |0 |0 |0 |
|Y |-1 |0 |1 |
The table is a vertical line at x=0, so you must have another vertical line to have a parallel slope. If it passes through 1,0 it must be a vertical line at x=1
5. Solve : |x-8|=20
x-8=20 x=28
x-8=-20 x=-12
6. Solve : |[pic]|=12
Absolute value her is actually with the x-6. |x-6|=60
x-6=60 x=66
x-6=-60 x=-54
7. Solve 4|3x-8|+16=2
4|3x+8|=-14
|3x+8|=-14/4
|3x+8|=-7/2
3x+8=-7/2 3x=-7/2-16/2 3x=-23/2 x=-23/6
3x+8=7/2 3x=7/2-16/2 3x=-9/2 x=-3/2
8. Solve |9x-2|=15
9x-2=15 9x=17 x=17/9
9x-2=-15 9x=-13 x=-13/9
9. Tickets to the math club dance (woohoo!!) cost $5 if bought in advance and $6 at the door. The math club needs to make a total of at least $600 from ticket sales for the dance.
a. Let x be the number of tickets sold in advance and y be the number of tickets sold at the door. Write and graph an inequality for the total amount in ticket sales that the math club needs.
[pic]
b. If the math club sells 30 tickets in advance, how many tickets must be sold at the door for the math club to reach its goal?
5x+6y≥600
5(30)+6y≥600
y≥75 tickets
10. A local theater charges $7.50 for adult tickets and $5.00 for discount tickets. The theater needs to make at least $240 to cover the rent of the building. How many of each type of ticket must be sold to make a profit? If 20 discount tickets are sold, how many adult tickets must be sold?
a. Create an equation describing the inequality
7.5x+5y≥240
x≥19 tickets
b. Graph the line, by putting in slope intercept form or finding the x and y intercepts of the line. Be careful to use the correct type of line, and remember to shade the correct region.
[pic]
Solve for the roots:
11. 14x+x2=24 quad formula: x=[pic]
12. x2=3x+4 factor in standard form x2-3x-4 (x+1)(x-4) x=-1, x=4
13. 2a2-a+1=0 quad form [pic]
14. 5n2+3n+1=0 quad form [pic]
15. 3x2+4x+8=0 quad form [pic]
16. -149=x2-24 quad form [pic]
17. 2x2+16=0 quad form [pic]
18. Divide x4+3x3+1 by x+1 synthetic division x3+2x2-2x+2 R -1
19. Sketch a complete graph of the function x3-9x
a. Is the function even or odd? odd
b. What will the end behavior look like? Pos leading coefficient left side falls, right side rises
c. What is the y intercept?
0
d. Can you locate any zeros?
-3, 3, 0 zero finder or table in calc
21. Kyle estimates that his business is growing at a rate of 5% per year. His profits in 2005 were $67,000. Estimate his profits for 2010 to the nearest hundred dollars
A=67000(1+.05)5 A=855100
22. A parcel of land Jason bought in 2000 for $100,000 is appreciating in value at a rate of about 4% each year.
a. Write a function to model the appreciation of the value of the land 100,000 _ 1.04 _ x
b. Graph the function.
c. In what year will the land double its value? [pic]
A=100000(1.04)t
17.47 years= 18 years
23 . A certain car depreciates about 15% each year.
a. Write a function to model the depreciation in value for a car valued at $20,000.0 _ 0.85 _ x
b. Graph the function. [pic]
c. Suppose the car was worth $20,000 in 2005. What is the first year that the value of this car will be worth less than half of that value?
20000(1-.15)t
5 years
24. Simplify. (2x2)(4x3y2)
8x6y2
25. (5x2)(4x)
20x3
26. (3x6y)2
9x12y2
27. [pic]x6/y6
28 . [pic] 2s3
29. (3y6)(x5y2z)
3x5y8z
30. [pic] [pic]
31. [pic] [pic]
32. [pic] [pic]
33. [pic] [pic]
34. [pic] [pic]
Simplify
35. [pic] [pic]
36. [pic] [pic]
37. [pic] [pic]
38. [pic] n/2
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