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Exam Review

Exam Review. By: John Misiewicz Jeremiah Johnson. 1 st Power Equations. -Has no exponents -Make the equation equal the variable. 1 st Power Examples. Ex1: x-2=3. Answer: x= 5. Ex2: x+12=x+12. Answer=x so the answer is All real numbers. 1 st power Examples cont. Ex3:.

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Exam Review

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  1. Exam Review By: John Misiewicz Jeremiah Johnson

  2. 1st Power Equations -Has no exponents -Make the equation equal the variable

  3. 1st Power Examples Ex1: x-2=3 Answer: x= 5 Ex2: x+12=x+12 Answer=x so the answer is All real numbers

  4. 1st power Examples cont. Ex3: Answer: x= 20

  5. Properties • Addition/ Subtraction property of Equality- If the same number is added/ subtracted to equal numbers the sums/ differences are equal • a+c=b+c; a-c=b-c • Multiplication/ Division property of Equality- If equal number are multiplied/ divided by the same [(non zero) in the case of division] number the products/ quotients are equal • ca=cb;

  6. Properties • Reflexive property- all numbers equal themselves. a=a • Symmetric Property- If one number equals another than the other number equals the first number. a=b, b=a • Transitive Property- If a=b and b=c than, a=c

  7. Properties • Associative Property of Addition- (a+b)+c=a+(b+c) • Associative Property of Multiplication- (ab)c=a(bc) • Commutative property of Addition- a+b=b+a • Commutative Property of Multiplication- ab=ba

  8. Properties • Distributive Property-a(b+c)=ab+ac • Property of Opposites- a+(-a)=0; -a+a=0 • Property of Reciprocals- • Identity Property of Addition- a+0=a; 0+a=a • Identity Property of Multiplication- • Multiplicative property of zero- • Closure property of addition-a+b • Closure property of multiplication- ab

  9. Properties • Power of a product property-(2t)2 = 22 x t2 • Product of a power property- 84 x 46 • Power of a power property- (84)5 • Quotient of powers property- • Power of a quotient property- • Zero power property-(x+5)(x-8)=0

  10. Properties Negative power property- -3-3=1/27 Zero product property-(x+10)(x-15)=0 x=-10 or x=+15 Product of roots property-√9 x√5 = √45 Quotient of roots property-√225=√9x25 =3x5 = 15

  11. Inequalities in one variable • x-8<15 solve and graph • 3x-5>1 solve and graph • 13-7n<8 solve and graph • 4y+3>7 solve and graph • 2x+7>3 solve and graph

  12. Conjunctions 2m<4 and 12+2m<0 solve and graph • 2d-5>-8 and -2d-5<d-3 solve and graph • -3+2<x-2+2<4+2 solve and graph • 5-2p>-8 and -2d-5<d-3 solve and graph

  13. Disjunction • 2w-1<3 or 3w>w+10 solve and graph • 5-2p>11 or 5-2p<-1 solve and graph • -6>18 or 12+3r>0 solve and graph • H+5<-2 or h+5>2 solve and graph

  14. IMPORTANT!!!!!!!!!!!!!!!!!!!!!! When solving these equations conjunctions join the sentences together using the word and (just like in a sentence in english class). In a conjunctions you will find that both of the sentences are true. In a disjunction the word or is used. This means that at least ONE sentence is true and it cancels out the other sentence.

  15. Linear Equations in Two Variables 4x+3y=10 here are a list of ordered pairs (4,-2) (-2,6) (2,3) (8,5) plug these in to see which one fits the equation.

  16. All Equations For All Lines Standard/General- ax+by=c Point slope form- y-y1=m(x-x1) Slope intercept form- y=mx+b equation of a horizontal line-y=b Equation of a vertical line- x=a

  17. Slopes for all lines The slope for a rising line is positive. The slope of a falling line is negative. A line that is parallel with the bottom of the graph has a slope of zero because it never rises. A verticle line has a slop that is undefined because it it has no run.

  18. How to Graph Using the equation y=mx+b y= the equation of the line mx= the rise over the run or the slope of the line And b is the where the line crosses the y-axis

  19. Linear System • Substitution Method- 2x+y=25 4x+3y=32 Solution- y=25-2x (plug this equation into your second one) 4x+3(25-2x)=32 X=21 y=67

  20. Linear Systems cont. • Addition/Subtraction Method- 5-y=12 3x+y=4 solution is- 8x=16 Final solution is: x=2 y=-2

  21. Linear Solution Commonalities Solution- Intercept point Consistent System Solution- A.R.N. Dependent System Solution- Null set Inconsistent System

  22. Factoring Polynomials • 5a3+b3 Solution is : • 5(a3+b3) • ax+bx-bx-by solution is: • a(x+y)-b(x-y) • 5a3-5a2+10a-10 solution is: • (a-1) (5a2+10)

  23. Quadratic Equations • Factoring- 20x2-49x+30 • (5x-6) (4x-5) • Quadratic Equation-ax2+bx+c=0

  24. Functions • Quadratic Functions-Graphing the Parabola y = ax2 + bx + c • 1.    Determine whether the parabola opens upward or downward. • a.    If a > 0, it opens upward. • b.    If a < 0, it opens downward. • 2.    Determine the vertex. • a.    The x-coordinate is . • b.    The y-coordinate is found by substituting the x-coordinate, from          Step 2a, in the equation y = ax2 + bx + c. • 3.    Determine the y-intercept by setting x = 0. • 4.    Determine the x-intercepts (if any) by setting y = 0, i.e., solving the equation        ax2 + bx + c = 0. • 5.    Determine two or three other points if there are no x-intercepts. • Remember point slope form and domains and ranges

  25. Simplifying expressions with exponents • (2r3s2)(-5r5s) • Solutions is – • 10r8s3 • Important to note that the powers add together not multiply only the numbers multiply together

  26. Simplifying Expression with Radicals Rationalizing Denominators-

  27. Now Onto the Word Problems

  28. One rectangle’s length is five more (ft.) than twice as long as the width. If the area of the rectangle is 35ft what is the width 2w+5 w W=the square root of 15 So your equation is And your solution is 2w2+5=35

  29. The sum of two numbers is 100. Five times the smaller is 8 more than the larger. Find the number(s) The numbers are 18 and 82

  30. A $200 coat is on sale for $166. What is the percent of discount. . 17%

  31. You put $1000 into an investment yielding 6% annual interest; you left the money in for two years. How much interest do you get at the end of those two years? I=Prt I=Prt I=Prt $120

  32. The sum of two numbers is 100. Five times the smaller is 8 more than the larger. Find the number(s) The numbers are 18 and 82

  33. Line of Best Fit • Use this when- you have a linear regression and you need to find a general estimate of the best line. • Your Calculator helps by having the technology to find the exact estimate and pick any one of the x or y axis’s.

  34. For extra help try these websites- http://www.mathtv.com/ http://www.coolmath.com/

  35. The End

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