Algebra 2 Trig A Final Review 2007. #1 Hyperbola  Center (0, 0)  a = 8, b = 7, c =  Vertices: (+8, 0)  Foci: (, 0)  Slopes of asymptotes: +7/8 Hyperbola.

Slides:

Advertisements

Similar presentations

Conics Review Your last test of the year! Study Hard!

Advertisements

Intermediate Algebra Optional Pre-Final Exam Review

6.7 Notes – Inverse Functions. Notice how the x-y values are reversed for the original function and the reflected functions.

Parabolas $ $300 $300 $ $ $ $ $ $ $ $ $ $ $ $ $ $100.

Math 143 Final Review Spring 2007

7.4 Inverse Functions p Review from chapter 2 Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 ways to.

Jeopardy 100 Composite Functions Combining Functions Combining Functions Inverse Functions and Relations.

Functions and Graphs.

Warm-up Arithmetic Combinations (f+g)(x) = f(x) + g(x) (f-g)(x) = f(x) – g(x) (fg)(x) = f(x) ∙ g(x) (f/g)(x) = f(x) ; g(x) ≠0 g(x) The domain for these.

Combinations of Functions & Inverse Functions Obj: Be able to work with combinations/compositions of functions. Be able to find inverse functions. TS:

What is the standard form of a parabola who has a focus of ( 1,5) and a directrix of y=11.

JEOPARDY Rational Functions and Conics By: Linda Heckman.

Review Day! Hyperbolas, Parabolas, and Conics. What conic is represented by this definition: The set of all points in a plane such that the difference.

Solve. 1. x 5/2 = x 2/ = x 3/4 = 108 x 3/4 = 27 (x 5/2 ) 2/5 = 32 2/5 x 2/3 = 9 x = 2 2 x = (32 1/5 ) 2 x = 4 (x 2/3 ) 3/2 = 9 3/2.

Inverse Functions Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other. Symbols: f -1(x) means “f.

50 Miscellaneous Parabolas Hyperbolas Ellipses Circles

& & & Formulas.

Conics This presentation was written by Rebecca Hoffman Retrieved from McEachern High School.

Q $100 Q $200 Q $300 Q $400 Q $500 Q $100 Q $200 Q $300 Q $400 Q $500 Final Jeopardy CirclesEllipsesHyperbolasParabolasPersonal.

Cheryl Corbett, Ivy Cook, and Erin Koch 5 th Period.

Algebra II Honors Problem of the Day Homework: p , 9, 13, 15, odds and worksheet Paper folding activity is the problem of the day.

Jeopardy CirclesParabolasEllipsesHyperbolasVocabulary Q $100 Q $200 Q $300 Q $400 Q $500 Q $100 Q $200 Q $300 Q $400 Q $500 Final Jeopardy Source:

EXAMPLE 1 Graph the equation of a translated circle Graph (x – 2) 2 + (y + 3) 2 = 9. SOLUTION STEP 1 Compare the given equation to the standard form of.

College Algebra Practice Test 3 This review should prepare you for the third test in College Algebra. Read the question, work out the answer, then check.

Snapshot of the School Year Algebra 2 Honors Class Alex Asemota Mrs. Vittese Period 4 May 3, 2012.

Conic Sections Advanced Geometry Conic Sections Lesson 2.

Algebra Conic Section Review. Review Conic Section 1. Why is this section called conic section? 2. Review equation of each conic section A summary of.

Conic Sections & Rational Functions MATHO Algebra 5/Trig.

Jeopardy CirclesParabolasEllipsesHyperbolas Mixed Conics Q $100 Q $200 Q $300 Q $400 Q $500 Q $100 Q $200 Q $300 Q $400 Q $500 Final Jeopardy.

Conic Sections.

What am I?. x 2 + y 2 – 6x + 4y + 9 = 0 Circle.

Conics This presentation was written by Rebecca Hoffman.

Find the distance between (-4, 2) and (6, -3). Find the midpoint of the segment connecting (3, -2) and (4, 5).

Finding Inverses (thru algebra) & Proving Inverses (thru composition) MM2A5b. Determine inverses of linear, quadratic, and power functions and functions.

Operation of Functions and Inverse Functions Sections Finding the sum, difference, product, and quotient of functions and inverse of functions.

Review Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 ways to show the same relation. y = x 2 x y

7.4 Inverse Functions p. 422 What is an inverse relation? What do you switch to find an inverse relation? What notation is used for an inverse function?

MATH JEOPARDY functionsinversesradicalsWord problems potpourri.

Jeopardy Evaluating Functions Properties of Logarithms Conic Sections zeros Exponential Models Misc. Q $100 Q $200 Q $300 Q $400 Q $500 Q $100 Q $200.

College Algebra Review for final exam. Chapter 2 – Functions and Graphs Increasing/decreasing parts of functions Combining functions with +,-,x,/ Composing/decomposing.

Conics Memory Aid Math SN5 May 25, Circles Locus definition of a circle: The locus of points a given distance from a given point in that plane.

Hyperbolas Objective: graph hyperbolas from standard form.

Conics Name the vertex and the distance from the vertex to the focus of the equation (y+4) 2 = -16(x-1) Question:

Analyzing Conic Sections

EXAMPLE 1 Graph the equation of a translated circle

Algebra 2 Final Review.

College Algebra Practice Test 3

College Algebra Final Review

Vertices {image} , Foci {image} Vertices (0, 0), Foci {image}

Factor Theorems.

This presentation was written by Rebecca Hoffman

Review Circles: 1. Find the center and radius of the circle.

Parabolas Mystery Circles & Ellipses Hyperbolas What am I? $100 $100

Inverse Functions.

Test Dates Thursday, January 4 Chapter 6 Team Test

College Algebra Final Review

6.4 Use Inverse Functions.

3 Inverse Functions.

Chapter 3 Graphs and Functions.

Final Radicals and rational exponents Solving radical equations

Analyzing Conic Sections

7.4 Inverse Functions p. 422.

Warm Up Determine the domain of f(g(x)). f(x) = g(x) =

6.3 Perform Function Operations & Composition

7.4 Inverse Functions.

Chapter 3 Graphs and Functions.

Jeopardy Solving for y Q $100 Q $100 Q $100 Q $100 Q $100 Q $200

Evaluate x = 3; 2x + 6.

Presentation transcript:

Algebra 2 Trig A Final Review 2007

#1 Hyperbola  Center (0, 0)  a = 8, b = 7, c =  Vertices: (+8, 0)  Foci: (, 0)  Slopes of asymptotes: +7/8 Hyperbola  Center (0, 0)  a = 8, b = 7, c =  Vertices: (+8, 0)  Foci: (, 0)  Slopes of asymptotes: +7/8

#2 y 2 = x 2  Circle: x 2 + y 2 = 121  Center: (0, 0)  Radius = 11  Circle: x 2 + y 2 = 121  Center: (0, 0)  Radius = 11

#3 y = 2(x - 2)  Parabola  Center/Vertex: (2, 1)  AOS: x = 2  DOO: up  Focus: (2, 9/8)  Directrix: y = 7/8  Parabola  Center/Vertex: (2, 1)  AOS: x = 2  DOO: up  Focus: (2, 9/8)  Directrix: y = 7/8

#4 6x y 2 = 96  Ellipse:  Center: (0, 0)  a = 4, b =, c =  M vertices: (±4, 0)  Foci: (, 0)  LMA = 8  lma =  Ellipse:  Center: (0, 0)  a = 4, b =, c =  M vertices: (±4, 0)  Foci: (, 0)  LMA = 8  lma =

#5 x 2 - 2x + y - 8 = 0  Parabola: y = -(x - 1)  Center/Vertex: (1, 9)  AOS: x = 1  DOO: down  Focus: (1, 8 3/4)  Directrix: y = 9 1/4  Parabola: y = -(x - 1)  Center/Vertex: (1, 9)  AOS: x = 1  DOO: down  Focus: (1, 8 3/4)  Directrix: y = 9 1/4

#6 x 2 = 2x + y 2 - 4y + 7  Hyperbola  Center: (1, 2)  a = 2, b = 2, c =  Vertices: (3, 2), (-1, 2)  Foci: (1±, 2)  Slopes of Asymptotes: ±1  Hyperbola  Center: (1, 2)  a = 2, b = 2, c =  Vertices: (3, 2), (-1, 2)  Foci: (1±, 2)  Slopes of Asymptotes: ±1

#7 x 2 +4y 2 + 2x - 24y + 33 = 0  Ellipse  Center: (-1, 3)  a = 2, b = 1, c =  Vertices:(-3, 3),(1, 3)  Foci:  LMA = 4  lma = 2  Ellipse  Center: (-1, 3)  a = 2, b = 1, c =  Vertices:(-3, 3),(1, 3)  Foci:  LMA = 4  lma = 2

#8 x 2 + y 2 = x + 2  Circle  Center: (1/2, 0)  Radius= 3/2  Circle  Center: (1/2, 0)  Radius= 3/2

#9 Find f(x) + g(x)  f(x) = x 2 -x+3 g(x) = x+8  f(x)+g(x) = (x 2 -x+3) + (x+8)  f(x)+g(x) = x  f(x) = x 2 -x+3 g(x) = x+8  f(x)+g(x) = (x 2 -x+3) + (x+8)  f(x)+g(x) = x

#10 Find f(x) - h(x)  f(x) = x 2 -x+3 g(x) = x+8  f(x) - h(x) = (x 2 - x + 3) - (3x 2 +1)  f(x) - h(x) = x 2 - x x  f(x) - h(x) = -2x 2 - x + 2  f(x) = x 2 -x+3 g(x) = x+8  f(x) - h(x) = (x 2 - x + 3) - (3x 2 +1)  f(x) - h(x) = x 2 - x x  f(x) - h(x) = -2x 2 - x + 2

#11 Find f(g(x))  f(x) = x 2 -x+3 g(x) = x+8  f(x) = x 2 - x + 3  f(g(x)) =(x+8) 2 - (x+8) + 3  f(g(x)) = x x x  f(g(x)) = x 2 +15x + 59  f(x) = x 2 -x+3 g(x) = x+8  f(x) = x 2 - x + 3  f(g(x)) =(x+8) 2 - (x+8) + 3  f(g(x)) = x x x  f(g(x)) = x 2 +15x + 59

#12 Find f(h(x))  f(x) = x 2 -x+3 h(x) = 3x 2 +1  f(x) = x 2 - x + 3  f(h(x)) = (3x 2 +1) 2 - (3x 2 +1) + 3  f(h(x)) = 9x 4 +6x x  f(h(x)) = 9x 4 +3x 2 +3  f(x) = x 2 -x+3 h(x) = 3x 2 +1  f(x) = x 2 - x + 3  f(h(x)) = (3x 2 +1) 2 - (3x 2 +1) + 3  f(h(x)) = 9x 4 +6x x  f(h(x)) = 9x 4 +3x 2 +3

#13 Find g(f(x))  g(x) = x+8 f(x) = x 2 -x+3  g(x) = x + 8  g(f(x)) = (x 2 - x + 3) + 8  g(f(x)) = x 2 - x + 11  g(x) = x+8 f(x) = x 2 -x+3  g(x) = x + 8  g(f(x)) = (x 2 - x + 3) + 8  g(f(x)) = x 2 - x + 11

#14 Find h(f(x))  h(x) = 3x 2 +1 f(x) = x 2 -x+3  h(x) = 3x  h(f(x))= 3(x 2 - x + 3)  h(f(x))= 3(x 4 -2x 3 +4x 2 -3x+9)+1  h(f(x))= 3x 4 -6x 3 +21x 2 -18x+27+1  h(f(x))= 3x 4 -6x 3 +21x 2 -18x+28  h(x) = 3x 2 +1 f(x) = x 2 -x+3  h(x) = 3x  h(f(x))= 3(x 2 - x + 3)  h(f(x))= 3(x 4 -2x 3 +4x 2 -3x+9)+1  h(f(x))= 3x 4 -6x 3 +21x 2 -18x+27+1  h(f(x))= 3x 4 -6x 3 +21x 2 -18x+28

#15 Find h(g(x))  h(x) = 3x 2 +1 g(x) = x+8  h(x) = 3x  h(g(x)) = 3(x + 8)  h(g(x)) = 3(x x + 64)+1  h(g(x)) = 3x x  h(g(x)) = 3x x  h(x) = 3x 2 +1 g(x) = x+8  h(x) = 3x  h(g(x)) = 3(x + 8)  h(g(x)) = 3(x x + 64)+1  h(g(x)) = 3x x  h(g(x)) = 3x x + 193

#16 Find f(-3)  f(x) = x 2 - x + 3  f(-3) = (-3) 2 - (-3) + 3  f(-3) =  f(-3) = 15  f(x) = x 2 - x + 3  f(-3) = (-3) 2 - (-3) + 3  f(-3) =  f(-3) = 15

#17 Find h(f(4))  h(x) = 3x 2 +1 f(x) = x 2 -x+3  f(4) = (4) 2 - (4) + 3  f(4) = 15  h(x) = 3x  h(15) = 3(15)  h(f(4)) = 676  h(x) = 3x 2 +1 f(x) = x 2 -x+3  f(4) = (4) 2 - (4) + 3  f(4) = 15  h(x) = 3x  h(15) = 3(15)  h(f(4)) = 676

#18 Find g(h(2))  g(x) = x+8 h(x) = 3x 2 +1  h(2) = 3(2)  h(2) = 3(4) + 1  h(2) = 13  g(13) =  g(h(2)) = 21  g(x) = x+8 h(x) = 3x 2 +1  h(2) = 3(2)  h(2) = 3(4) + 1  h(2) = 13  g(13) =  g(h(2)) = 21

#19 Inverse of f(x) = 4x + 5  y = 4x + 5  x = 4y + 5  x - 5 = 4y  x/4 - 5/4 = y  y = 4x + 5  x = 4y + 5  x - 5 = 4y  x/4 - 5/4 = y

#20 Inverse of g(x) = 3x  y = 3x  x = 3y  x + 12 = 3y 2  x/3 + 4 = y 2  y = 3x  x = 3y  x + 12 = 3y 2  x/3 + 4 = y 2

#21 f(x)=1/2x+2 g(x)=2x-4  f(g(x))=1/2(2x - 4) + 2  f(g(x)) = x  f(g(x)) = x  f(g(x))=1/2(2x - 4) + 2  f(g(x)) = x  f(g(x)) = x

#22 f(x) = 3x-9 g(x) = -3x+9  f(x) = 3x-9  y = 3x - 9  x = 3y - 9  x + 9 = 3y  x/3 + 3 = y  Not equal to g(x)  f(x) = 3x-9  y = 3x - 9  x = 3y - 9  x + 9 = 3y  x/3 + 3 = y  Not equal to g(x)

#23 {(1,3),(1,-1),(1,-3),(1,1)}  {(3,1),(-1,1),(-3,1),(1,1)}  Domain: 3, -1, -3, 1  Unique x - coordinates  {(3,1),(-1,1),(-3,1),(1,1)}  Domain: 3, -1, -3, 1  Unique x - coordinates

#24 Simplify  Simplify:

#25Simplify  Simplify

#26Simplify  Simplify:

#27Simplify  Simplify:

#28Absolute value equation  Solve:

#29Absolute Value Inequality  Solve:

#30 Find f(-5)  If f(x) = 4x 3 - x + 1  f(-5) = 4(-5) 3 - (-5) +1  f(-5) =  f(-5) = -494  If f(x) = 4x 3 - x + 1  f(-5) = 4(-5) 3 - (-5) +1  f(-5) =  f(-5) = -494

#31Do the math  (8x 3 + 2x 2 + 3x)÷(2x + 3)

#32Simplify  Simplify:

#33Factor: 27a b 3  Factor:27a b 3  (3a + 5b)(9a ab + 25b 2 )  Factor:27a b 3  (3a + 5b)(9a ab + 25b 2 )

#34 Factor: 9x x + 4  Factor: 9x x + 4  (3x -2) 2  Factor: 9x x + 4  (3x -2) 2

#35Factor: 7y - 12x + 4xy - 21  Factor: 7y - 12x + 4xy - 21  7y xy - 12x  7(y - 3) + 4x(y - 3)  (y - 3)(7 + 4x)  Factor: 7y - 12x + 4xy - 21  7y xy - 12x  7(y - 3) + 4x(y - 3)  (y - 3)(7 + 4x)

#36Factor: 15a 3 b - 5a 2 b ab 3  Factor: 15a 3 b - 5a 2 b ab 3  5ab(3a 2 - ab - 2b 2 )  5ab(3a 2 - 3ab +2ab - 2b 2 )  5ab[3a(a - b) + 2b(a - b)]  5ab(a - b)(3a + 2b)  Factor: 15a 3 b - 5a 2 b ab 3  5ab(3a 2 - ab - 2b 2 )  5ab(3a 2 - 3ab +2ab - 2b 2 )  5ab[3a(a - b) + 2b(a - b)]  5ab(a - b)(3a + 2b)

#37Simplify:  Simplify:

#38 Simplify:  Simplify:

#39Simplify:  Simplify:

#40Solve:  Solve:

#41Solve:x = 0  Solve: x =0  x 2 = -441  x =  x = ±21i  Solve: x =0  x 2 = -441  x =  x = ±21i

#42Simplify: (9 - 3i) - (3 + 5i)  (9 - 3i) - (3 + 5i)  i - 5i  6 - 8i  (9 - 3i) - (3 + 5i)  i - 5i  6 - 8i

#43Simplify: (5 + 4i)(3 - 7i)  Simplify: (5 + 4i)(3 - 7i)  (5 + 4i)(3 - 7i)  i + 12i - 28i 2  i - 28(-1)  i + 28  i  Simplify: (5 + 4i)(3 - 7i)  (5 + 4i)(3 - 7i)  i + 12i - 28i 2  i - 28(-1)  i + 28  i

#44Simplify:  Simplify:

#45Simplify: (7 - 3i)(7 + 3i)  Simplify: (7 - 3i)(7 + 3i)  i - 21i - 9i 2  (-1)   58  Simplify: (7 - 3i)(7 + 3i)  i - 21i - 9i 2  (-1)   58

#46Simplify: i 10 i 21 i 30  Simplify: i 10 i 21 i 30  i =i 61 = i 4(15)+1 = i 1 = i  Simplify: i 10 i 21 i 30  i =i 61 = i 4(15)+1 = i 1 = i

#47Simplify  Simplify:

#48Solve: x 2 + 5x + 13 = 0  x 2 + 5x + 13 = 0

#49Solve: 6x 2 + 7x = 3  Solve:

#50Solve:2x 2 + 3x - 13 = 0  Solve:

#51 Word Problem  h(t) = -16t t + 50  h(1) = -16(1) (1) + 50  h(1) = 44 feet  0 = -16t t + 50  h(t) = -16t t + 50  h(1) = -16(1) (1) + 50  h(1) = 44 feet  0 = -16t t + 50

#52 Simplify, combine like terms  (4b 4 + 6b 2 - 3b + 5) - (2b 3 + 3b - 2)  4b 4 - 2b 3 + 6b 2 - 6b + 7  (4b 4 + 6b 2 - 3b + 5) - (2b 3 + 3b - 2)  4b 4 - 2b 3 + 6b 2 - 6b + 7

#53 Simplify, remove parentheses  (y + 2)(y 2 - 4y + 1)  y 3 - 4y 2 + y + 2y 2 - 8y + 2  y 3 - 2y 2 - 7y + 2  (y + 2)(y 2 - 4y + 1)  y 3 - 4y 2 + y + 2y 2 - 8y + 2  y 3 - 2y 2 - 7y + 2

#54Do the arithmetic  (2x 3 - 3x 2 + 4x - 5) ÷ (x - 2)  Synthetic Division     2x 2 + x + 6x + 7/(x-2)  (2x 3 - 3x 2 + 4x - 5) ÷ (x - 2)  Synthetic Division     2x 2 + x + 6x + 7/(x-2)

#55 Factor  64x 2 y z 2  (8xy - 5z)(8xy + 5z)  64x 2 y z 2  (8xy - 5z)(8xy + 5z)

#56Find the zeros  y = 3x 2 + 5x + 2  0 = 3x 2 + 5x + 2  0 = (3x 2 + 3x) + (2x + 2)  0 = 3x(x + 1) + 2(x + 1)  0 = (x + 1)(3x + 2)  x = -1, -2/3  y = 3x 2 + 5x + 2  0 = 3x 2 + 5x + 2  0 = (3x 2 + 3x) + (2x + 2)  0 = 3x(x + 1) + 2(x + 1)  0 = (x + 1)(3x + 2)  x = -1, -2/3

#57Find the max or min of #56  y = 3x 2 + 5x + 2  DOO: up, therefore a minimum  x = -b/2a x = -5/2(3) = -5/6  y = 3(-5/6) 2 + 5(-5/6) + 2  y = 25/ /6 + 2  y = -1/12  Vertex is (-5/6, -1/12)  y = 3x 2 + 5x + 2  DOO: up, therefore a minimum  x = -b/2a x = -5/2(3) = -5/6  y = 3(-5/6) 2 + 5(-5/6) + 2  y = 25/ /6 + 2  y = -1/12  Vertex is (-5/6, -1/12)

#58 Solve Systems of Equations  x + y = 2  x – 3y =6  -x-y=-2 x+(-1)=2  x-3y=6 x=3  -4y=4  y=-1  x + y = 2  x – 3y =6  -x-y=-2 x+(-1)=2  x-3y=6 x=3  -4y=4  y=-1

#59  (3x-5y=6)*7 21x-35y=42  (2x+7y=12)*5 10x+35y=60  (3x-5y=6)*7 21x-35y=42  (2x+7y=12)*5 10x+35y=60

#60 Solve Systems of Equations  7x+3y=-1  2x-y=9 y=2x-9  7x+3y=-1  2x-y=9 y=2x-9

#61 Solve a system 3x+2y=-24 6x-5y=30 -6x-4y=48 6x-5y=30 -9y=78 y=-78/9 3x+2y=-24 6x-5y=30 -6x-4y=48 6x-5y=30 -9y=78 y=-78/9

#62  Is (1, 2) a solution to the following system of inequalities?  x>1, y 2x-1  1>1, 2 2(1)-1  Yes. Yes. No.  The answer is NO.  Is (1, 2) a solution to the following system of inequalities?  x>1, y 2x-1  1>1, 2 2(1)-1  Yes. Yes. No.  The answer is NO.

END of Review