Topic: Expressions & Operations AII.1

Transcription

1 Topic: Epressions & Operations AII.1 AII.1 The student will identify field properties, aioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, comple numbers and matrices. Notes and/or Formulas 1. Which of the following is an eample of the commutative property of addition? Properties Identity Property a b a b a ( b c) ( a b) c (+) a0 a a ( b c) a ( c b) () a(1) a ab c a bc Inverse Property (+) a ( a) 0. Which property justifies the statement ( a c) a c? () 1 1 a a Associative Property (+) a ( b c) ( a b) c () a( bc) ( ab) c Commutative Property (+) a b b a () ab ba Distributive Property a( b c) ab ac Aioms of Equality Refleive a a Symmetric If a b, then b a Transitive If a b, and b c then a c Order of Operations Parenthesis Eponent Multiply* Divide* Add** Subtract** * Multiply/Divide in order from left to right ** Add/Subtract in order from left to right F. Associative Property of Multiplication G. Commutative Property of Multiplication H. Associative Property of Addition J. Distributive Property. For which of the following operations is the commutative property not valid? Multiplication of integers Multiplication of comple numbers Multiplication of matrices Multiplication of negative real numbers 4. Use the distributive property to simplify: 7( y) F. y G. 14 y H. 14 y J. 14 y. Use the distributive property to simplify: 9(6 4 y) 4 6y 4 4y 6 6y 6 4y 6. Simplify: ( ) 9 F. G. 6 H. 700 J Simplify: 10 (8 )

2 Notes and/or Formulas 8. Solve the equation by using the properties of equalities: ( ) 8 F. 14 G. 14 H. 10 J Solve the equation by using the properties of equalities: 7 ( ) 7 No Solution Identity 10. If a 4, which of the following statement is true? F. a 7 G. 4a 16 H. 4a 14 J. a 10

3 Topic: Epressions & Operations AII. AII. The student will identify and factor completely polynomials representing the difference of squares, perfect trinomials, the sum and difference of cubes and general trinomials. Notes and/or Formulas 1. Which is the factored form of 64 1? Ways to Factor: (4 1)(4 4 1) 1) Greatest Common Factor y w ( y w) (41)(16 1) (4 1)(16 4 1) ) Difference of Squares a b ( a b)( a b) (4 1)(16 4 1) ) Sum of Cubes* a b ( a b)( a ab b ) 4) Difference of Cubes* a b ( a b)( a ab b ) *Square-Multiply-Square- Opposite Plus ) Trinomials 6) Factor by Grouping (leading coefficient) 7) Completing the square. Factor: 100 F. ( 0)( 0) G. ( 10)( 10) H. ( 10)( 10) J. ( )( 4). Factor: ( 7 ) ( 7 ) 6 ( 7 ) 6 (9 1 ) 4. Factor: 7 4 F. (1)( 4) G. (4)( 1) H. (1)( 4) J. (1)( 4). Factor: ( 1)( ) ( 1)( ) ( 1)( ) ( )( ) 6. Factor: 18u u 1 F. (6u1)( u 1) G. (6u1)( u 1) H. (6u1)( u 1) J. (6u1)( u 1)

4 Notes and/or Formulas represents the area of a rectangle. Which of the following could represent the length of one side of the rectangle? 4 1 represents the area of a rectangle. Which of the following could represent the length of one side of the rectangle? F. G. 1 H. J. 9. Factor: b 64 ( b 4) ( b4)( b4)( b 4) ( b 4)( b 4b 16) ( b 4)( b 4b 16) 10. Find the term that must be added to both sides of the equation so that the equation can be solved by the method of completing the square 8 1 F. 16 G. 64 H. 1 J. 11. Find the term that must be added to both sides of the equation so that the equation can be solved by the method of completing the square

5 Topic: Epressions & Operations AII. AII. The student will add, subtract, multiply, divide, and simplify rational epressions, including comple fractions. Notes and/or Formulas a 1a Rules for fractions: 1. Simplify: a a 1) Always factor any squared terms a completely watch a for greatest a common factors a ) Addition & a 1 Subtraction of fractions require a a common a denominator a ) When dividing fractions flip the second fraction and 1 y multiply. Simplify: 1 1 4) Comple Fractions simplify y numeration, 1 F. simplify the y denominator, then 1 Divide G. y H. y J.. Simplify: m m m 1 m 1 m 1 m 1 m m1 m 1

6 Notes and/or Formulas 4. Multiply: F. G. H. J. 4 y 4 y 4 y 4 y. Multiply: 4y 4y 4y 7 4y y 4 y y 4y Divide: F. 4 G. H. J. 7. Divide:

7 Notes and/or Formulas Simplify: F. G. H. 7 J Simplify:

8 Topic: Epressions & Operations AII.17 AII.17 The students will perform operations on comple numbers and epress the results in simplest form. Simplifying the results will involve using patterns of the powers of i. Notes and/or Formulas 4i 1. Simplify: i Calculator (TI-8 or TI-84) i Use i button on the calculator, however 11i remember to use the parentheses to separate operations i i EX. i i. Simplify: ( i) ( i) 7 4i 1 47 F. 6 6 i 1 47 G. 6 6 i 1 47 H. 6 6 i 1 47 J. 6 6 i 7 i. Simplify: 8 i 7 i 6 7 i 6 7 i 6 7 i 6 Don t Forget: i 1 4. Simplify: ( i)(8 i) F. 1 46i G. 1 4i H. 1 4i J. 1 46i. Simplify: (1 7 i)( 9 4 i) 7 9i 19 9i 19 67i 7 67i 8

9 Notes and/or Formulas 6. Simplify: (8 i) (7 6 i) F. 10 i G. 10 i H. 69 8i J. 4 14i 7. Simplify: (6 6 i) (1 i) 7 8i 7 8i 6 18i 4i 8. Simplify: ( 4 i) (i 6) F. 4 1i G. 9 14i H i J. i To simplify powers of i Change to ( i ) Power then change ( i ) to (- 1) Eample 11 i ( i ) i ( 1) 11 ( 1) i i i 9. Simplify: 1 1 i i 10. Simplify: F. 1 G. i H. i J i 7 i Don t Forget: 1 i 11. Write the given epression in terms of i 8 8i i i 8 i 1. Write the given epression in terms of i 64 F. 8 G. 8i H. 8i J. 8 9

10 Topic: Epressions & Operations AII. AII. The student will add, subtract, multiply, divide, and simplify radical epressions containing radical epressions containing positive rational numbers and variables and epressions containing rational eponents; and write radical epressions as epressions containing rational eponents, and vise versa. Notes and/or Formulas 9 1. Simplify: 7y y y 6 y. Simplify: 18 y F. G. H. J. y y 4 y y y y 4 y y Multiplying Radicals n n n a b ab n n a b n a b. Simplify: 4 y 6 y 6 6 y 4y y y 6 6 4y 6 y 4 4. Simplify: F. G. H. J. 6. Rationalize the denominator:

11 Notes and/or Formulas Only radical epressions with like radicands (stuff under the radical) can be added or subtracted. 6. Rationalize the denominator: F. G. H. J Simplify: Simplify: F. 4 7 G. 6 7 H J Multiply: Divide: F. 18 G H. 1 J

12 Notes and/or formulas 11. Rewrite 6 using rational eponents. Don t Forget: a a b a b b Rewrite 8 7 using rational eponents F. G. H. J Properties of Rational Eponent 1. a a a. ( a m ) n a mn m n m n. ( ab) m a m b m m 1 4. a m a m a mn. a n a 6. a b m a b m m 1. Simplify and write in simplest radical form: Simplify and write in simplest radical form: 6 F. G. H. J Simplify: y 8 y y y 16. Simplify: F. G. H. J. 4 1 y y y y 4 ( 4 y )( y ) 4 ( y )( 6 y ) 1 1 1

13 Notes and/or Formulas 17. Simplify: y y y y y 4 6 9y Simplify: F. G. H. J. 4 y 6 4 y 6 6 4y 4 9 y 8 y 7 8y 7 1

14 Topic: Equations & Inequalities AII.4 AII.4 The student will solve absolute value equations and inequalities graphically and algebraically. Graphing calculators will be used both as a primary method of solution and to verify algebraic solutions. Notes and/or Formulas To Solve Absolute Value Equations Set = to positive value Set = to negative value 1. Which graph represents the solution of 6 9? To Solve Absolute Value Inequalities 1. Write equation as is. Write equation, switch inequality symbol, change to negative value To Graph Absolute Value Inequalities GreatOR Than Less ThAND OR. (Open Left & Right) (Closed Left & Right) (Open Between) (Closed Between) Solve: 4 F. 0 and 4 G. 4 H. 0 J. No Solution. Solve: 4 1,1,,, 1 4. Solve and Graph: 9 F G H J

15 Topic: Equations & Inequalities AII.6 AII.6 The student will select, justify and apply a technique to solve a quadratic equation over the set of comple numbers. Graphing calculators will be used for solving and for confirming the algebraic solutions. Notes and/or Formulas 1. Solve: Methods for Solving Quadratics 1. Factor. Complete the Square. Square Root Method 4. Quadratic Formula b b 4ac a negative number use i Calculator Hints: To Find Roots, Solutions, Zeros 1. Graph quadratic in y =. Press nd Trace. Press # for zeros 4. Left Enter. Right Enter 6. (Guess) Enter,, 4, 1,. Solve: 14 8 F., G. i. i H., J. i, i. Solve: 0 1, 1,, Solve: ( 1) 8 F., 1 G., H. 1,1 J. No Solution. Solve: , 4, 4 4, 4 4 4, 1

16 Notes and/or Formulas 6. Solve: 10 0 F.,1 G. 1, H.,1 J., 1 7. Solve: 6 1 i 1 8. Which statement is true for the quadratic equation F. The product of the roots is 4. G. The product of the roots is 4. H. The sum of the roots is 4. J. The sum of the roots is Describe nature of roots 1. You can graph the quadratic in the calculator look for the number of times the graph touches the -ais. Touches once One real solution Touches twice Two real solutions Does not touch Two Imaginary solutions. Use the discriminant b 4ac Discriminant > 0 Two real solutions Discriminant < 0 Two imaginary solutions 9. Find the quadratic equation with roots and Find the quadratic equation with roots 1 and F. G. H. J Describe the nature of the roots of the equation 0 One real root Two imaginary roots One real root and one imaginary root Two real roots Discriminant = 0 One real solution 16

17 Notes and/or Formulas 1. Describe the nature of the roots of the equation F. Two imaginary roots G. Two real roots H. One real root J. One real root and one imaginary root

18 Topic: Equations & Inequalities AII.7 AII.7 The student will solve equations containing rational epressions and equations containing radical epressions algebraically and graphically. Graphing calculators will be used for solving and confirming algebraic solutions. Notes and/or Formulas Solve a radical equation Isolate the radical Square or cube both sides 1. Solve: y y 4 y 0 y y 0 and y No Solution. Solve: 9 F. 7 G. 7 H. 7 and 7 J. 0 and 7. Solve: and 1 4. Use the quadratic formula to solve: F. 4i G. 8i H. 8i J. 4i

19 Topic: Relations & Functions AII.8 AII.8 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step and eponential functions) and convert between a graph, table and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators. Notes and/or Formulas To find the -intercept, plug in zero for y To find the y-intercept, plug in zero for. 1. Graph 7y by determining its - and y-intercepts. Graph y1 by determining its - and y-intercepts. F. G. H. J. 19

20 Notes and/or Formulas Slope = rise run. A line goes through the point (1, ) and has a slope 1. Graph this line. For Positive Slope Count, Count OR Count, Count For Negative Slope Count, Count OR Count, Count If slope is a whole number put a 1 underneath to make it a fraction. 4. A line goes through the point (, ) and has slope 1. Graph this line. F. G. H. J. 0

21 Notes and/or Formulas. Identify the function f Greatest Integer Absolute Value Direct Variation Constant ( ) Which is an identity function? F. f( ) 1 G. f ( ) H. f ( ) J. f ( ) 7. Identify the type of function for the graph below. Greatest Integer Constant Identity Absolute Value 8. Graph: y 1 F. G. H. J. 1

22 Notes and /or Formulas 9. Graph: y 10. Write the following equation in the form y 4 y a( h) k and graph. F. G. H. J. Degree X X X X 4 X Name Linear Quadratic Cubic Quartic Quintic The degree of a polynomial is the highest power of. 11. Identify the polynomial function Quartic Cubic Quadratic Quintic 4 f ( ) 6 9.

23 Notes and/or Formula 1. Identify the polynomial function, give the degree and the maimum number of 4 real zeros : g( ) F. Quintic, 4, G. Quartic, 4, 4 H. Quintic,, J. Not a polynomial function 1. Determine whether the degree of the function below is odd or even. How many real zeroes does the function have? Real zeroes occur where the graph crosses the - ais. Odd; 4 Zeroes Even; Zeroes Even; 4 Zeroes Odd; Zeroes 14. Which the following represents the graph of F. G. f ( ) ? H. J.

24 Topic: Relations & Functions AII.9 AII.9 The student will find the domain, range, zeros, and inverse of a function, the value of a function for a given element in its domain, and the composition of multiple functions. Functions will include those that have domains and ranges that are limited and/or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions, including eponential and logarithmic. Notes and/or Formulas. Find the range of the relation {( 4, 1),(,1),(,4)}. Domain: -values Range: y-values {, 4, } {, 1,1} { 4,,4} { 1,1,4} 4. Find the domain of the relation {(,),(, 6),(0, )} F. { 6,0,} G. {0,,} H. {,,} J. { 6,,}. What is the range of the function f ( ) if the domain is { 4, 1,}? {,1,} {4,19,8} { 8, 4,1} { 8, 19, 4} To be a function: No s may be the same OR Passes the vertical line test 6. Which of the following is NOT a function? F. {(, 1),( 4,4),(, 1)} G. 4y H. y4 J. y 7. Which of the following is NOT a function? {(, ),(,0),(4, )} y 1 y y To find the value of a function, substitute the value into the function for every 8. Find f () given F. 9 G. H. 9 J. 16 f ( ) 19 4

25 Notes and/or Formula 9. If Q( ), find Q( 4) Composite Function f(g()) Substitute the entire epression for g() into all of the s in the epression for f() 10. Find f () given F. G. H. 41 J. 7 f ( ) Find g( f ( )) where f ( ) 1and g ( ). 1. If f ( ) and g( ) 6, find g( f ( )). F. ( ) 6 G. ( 6) H. J. 1 ( 6) 1. Find g( f ( )) where f ( ) 7 and g ( ) 14. Solve for by factoring. F., 1 G. -, 1 H. -, -1 J. -1, 0

26 Notes and/or Formulas 1. Find all the real zeros., -, -, - -, Find all real zeros of the function. F. 0, 6, 8 G. 0, 6 H. -7, 0, 6 J. None of these answers y Rational Zero Theorem The factors of the leading coefficient will be Q. The factors of the constant will be P. To find all the possible rational roots for a polynomial, every P over every Q 17. Find all real zeros of the function. -, 0 -, 0, 1 -, -, 0 None of these answers y List all the possible rational zeros of the polynomial 4 f ( ) 6 14 according to the rational zero theorem. F. 1,, 7, 14 G ,,,,,, H ,, 7, 14,,,,,, ,,,,,, J ,,,,, Given that one zero is 4, which of the following is NOT a zero of P(). P( ) Given that one zero is i, which of the following is NOT a zero of P(). P( ) F. i G. i H. -1 J. 1 6

27 Notes and/or Formulas To find an inverse 1. Switch and y. Solve for y 1. Which of the following is a zero of the function 4 f ( ) ?. Find the inverse given the function f ( ) 9. F. 1 9 G. 9 H. ( 9) 1 J Given f ( ) 4 1, find ( 1) f 1 ( ). 7

28 Topic: Relations & Functions AII.1 AII.1 The student will recognize the general shape of polynomial, eponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions. Notes and/or Formulas 1. Identify the polynomial function f ( ) 6 9 Cubic Quadratic Quartic Quintic. Identify the polynomial function, give the degree and the maimum number of 4 real zeros. g( ) F. Not a polynomial function G. Quintic, 4, H. Quartic, 4, 4 J. Quintic,,. Use synthetic division to perform the following ( 6 7) ( 1) 8

29 Topic: Relations & Functions AII.16 AII.16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term and evaluating summation formulas. Notation will include and a. Notes and/or Formulas Term Formulas A: a a1 ( n 1) d G: n a a r n 1 n 1 n 1. Insert two arithmetic means between - and 1. 1, 10, 8, 9 4, 7 Sum Formulas n( a1 a ) A: S n 1 G: (1 n a r S ) 1 r a1 IG: S 1 r If you forget the formulas, just list series out and count terms or add terms on calculator.. Evaluate: F. 984 G H. 10 J. 00 n1 (n ). Which is an arithmetic sequence?,, 9, , 0, 1., , 10, 17, , -4, -, If an () nwhich of the following represents a F. 1 G. 18 H. 4 J. 7. Which of the following represents n (n ) 6. Find the net term in the sequence 8,,, -1. F. - G. - H. -4 J. - 9

30 Topic: Relations & Functions AII.0 AII.0 The student will identify, create, and solve practical problems involving inverse variation and a combination of direct and inverse variation. Notes and/or Formulas 1. The area (A) of a circle varies directly as the square of the radius (r). If k is the constant of proportionality, which is the formula for this relationship? Inverse Variation k k A Function y r A kr k y A kr Direct Variation r ka Function y k. The frequency of a radio signal varies inversely as the wave length. A signal of y frequency 100 kilohertz (khz), which might be the frequency of an AM radio k station, has a wave length 0 m. What frequency has a signal wave length of 400m? F. 8 khz G. 70 khz H. 10 khz J. 190 khz 0

31 Topic: Analytical Geometry AII.10 AII.10 The student will investigate and describe through the useof graphs the relationships between the solution of an equation, zero of a function, -intercept of a graph, and factors of a polynomial epression. Notes and/or Formulas 1. Which of the following functions has -intercepts at 1 and -? y Zeros are where the graph y crosses the -ais. y 1 Factored form ( zero) E: If zero at, then factor y 1 is ( ). Find all real zeros If zero at -, then factor is F. -, - ( + ) G., H. -, J. -,. Use the graph to determine the roots of the equation. 1 and 4 1 and -4 - None of these answers 4. Find all the real zeros of the function. F. 0,, 6 G. 0, H. 0,, 7 J. None of these answers. Find all the real zeros of the function., 1,1, 6, 4 y

32 Notes and/or Formulas 6. Find all the real zeros of the function. F., 41 0 G., H., J. 10, Which of the following could not be a factor of the function? ( ) ( ) ( ) ( ) 8. What are the factors of the given graph? F. ( 1)( )( 4) G. ( 1)( 4)( ) H. ( 1)( 4)( ) J. ( 1)( 4)( ) 9. What type of polynomial function is illustrated in the graph? Linear Quadratic Cubic Quartic

33 Topic: Systems of Equations & Inequalities AII.1 AII.1 The student will solve practical problems using systems of linear inequalities and linear programming, and describe the results both orally and in writing. A graphing calculator may be used to facilitate solutions. Notes and/or Formulas 1. Choose the system of linear inequalities shown by the graph. y m b y dotted line, shade above y y m b dotted line, shade below y y m b y solid line, shade above y m b y solid line, shade below y y y y 6. Graph the system of inequalities: y0 F. G. H. J.

34 Notes and/or Formulas y. Graph the system of inequalities: y0 4. Find the maimum and minimum values of the function subject to the given constraints. y 66y1 8y416 f (, y) 7y F. The maimum value of f is 76 at (9, 7). The minimum value of f is 9 at (, 0). G. The maimum value of f is 6 at (7, ). The minimum value of f is at (1, 0). H. The maimum value of f is 7 at (10, 6). The minimum value of f is 0 at (0, 0). J. The maimum value of f is 66 at (8, 6). The minimum value of f is 6 at (, 0).. Eleanor raises only free-range chickens and turkeys. She wants to raise no more than 60 animals with no more than 0 turkeys. She spends $1 to raise a chicken and $4 to raise a turkey. She has at most $10 to spend on the animals. Find the maimum profit Eleanor can make if she makes a profit of $ per chicken and $8 per turkey. How many chickens should she raise? 4 1 4

35 Topic: Systems of Equations & Inequalities AII.14 AII.14 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. Notes and/or Formulas Number of solutions Number of points of intersection on graph 1. How many solutions are there for this system? 0 1. y 41 Solve: y 7 F. (-4, -4) G. (4, ) H. (, 8) J. (-4, -19). Which is the solution to the system below? {(0, 7),(0,7)} {( 7,0),(7,0)} {(0, ),(0,)} 4. Solve the system graphically: F. {(0, 8),(0,8)} G. {(0, 9),(0,9)} H. J. {( 8,0),(8,0)} y 49 y

36 Topic: Statistics AII.19 AII.19 The student will collect and analyze data to make prediction and solve practical problems. Graphing calculators will be used to investigate scatter plots and to determine the equation for the curve of best fit. Notes and/or Formulas Negative slope & correlation Positive slope & correlation 1. Determine the correlation for the scatter plot. Strong positive correlation. Strong negative correlation. No correlation Not enough information given. Line of best fit: Look for slope & y-int. Calculator: 1. Put data into lists Stat-edit-L 1 -L. Stat-calc-4:Lin Reg-enter. Equation for line of best is y = a + b and substitute the values given for a and b. Look at the given scatter plot. This data best fits what type of equation? F. Linear G. Eponential H. Logarithmic J. Quadratic. Which of the following equations represents the line of best fit for the following data? X Y y4.8 y y y.4 4. The table shows the number of students enrolled in the Honors Algebra-Trig program at Menchville High School the first years since its initiation. What is your prediction for the number of students in the eighth year? F. 10 G. 10 H. 140 J. 10 Year (X) Number of Students (Y)