Unit 3: Ra.onal and Radical Expressions. 3.1 Product Rule M1 5.8, M , M , 6.5,8. Objec.ve. Vocabulary o Base. o Scien.fic Nota.

Transcription

1 Unit 3: Ra.onal and Radical Expressions M1 5.8, M , M , 6.5,8 Objec.ve 3.1 Product Rule I will be able to mul.ply powers when they have the same base, including simplifying algebraic expressions and scien.fic nota.on. I will be able to evaluate expressions with ra.onal exponents. Vocabulary o Base o Scien.fic Nota.on 1

2 3.1 Product Rule Mul.plying Powers When mul.plying powers with the same base, add the exponents. Base number raised to an exponent Exponent a number that tells how many.mes the base should be mul.plied by itself. The rule: a m *a n =a m+n where a 0 and m & n are ra.onal a m *b n =a m b n (cannot combine exponents 3.1 Product Rule in Algebraic Expressions When mul.plying algebraic terms together, mul.ply the coefficients and add the exponents. THE BASE STAYS THE SAME Don t forget to add the assumed 1 if no exponent is listed (Ex: d 2 *d=d 3 ) 2

3 3.1 Product Rule with Scien.fic Nota.on Scien.fic Nota.on any number wriden as a 10 b, where 1 a < 10. a is the coefficient, 10 is the base. If you move the decimal to the right, subtract from the exponent. If you move the decimal to the lee, add to the exponent. Ex: is wriden as Ex: is wriden as Product Rule Can be mul.plied just like algebraic expressions with the same base. In your calculator, appears as 4.567E14. It is okay to leave nega.ve exponents in scien.fic nota.on. 3

4 3.1 Product Rule Simplifying Expressions with Ra.onal Exponents Frac.onal exponents are called ra.onal exponents. If a 1/m =b, then a=b m You can use this (or your calculator) to simplify expressions with ra.onal exponents. If you use your calculator, you MUST put the ra.onal exponent in parentheses!! 3.1 Product Rule Addi.onal Rules to Remember Zero as an Exponent: a 0 =1 where a 0 Nega.ve Exponents Property: a n =1/a n where a 0 and n is ra.onal 4

5 3.1 Product Rule Prac.ce Objec.ve 3.2 Power Rule I will be able to raise powers to powers, raise products to powers, and apply these rules to scien.fic nota.on. Vocabulary None 5

6 3.2 Power Rule An item raised to a power may be raised to a second power by mul.plying the powers. The base stays the same (x m ) n =x mn If a monomial in parentheses is raised to a power, that power must be applied to each factor in the parentheses. (xy) n =x n y n 3.2 Power Rule If a polynomial is parentheses is raised to a power, rewrite as separate factors and use polynomial mul.plica.on NOTE: Many theorems and procedures exist to speed this process and are covered in units on polynomials. Chief among them is Binomial Theorem and FOIL. For numbers in Scien.fic Nota.on: Use the distribu.ve property of exponents Put back in scien.fic nota.on 6

7 3.2 Power Rule Remember your order of opera.ons: Parentheses Exponents Mul.plica.on and Division Addi.on and Subtrac.on Though not a required step in the order of opera.ons, grouping terms can some.mes make it easier to see what goes together. 3.2 Power Rule Prac.ce 7

8 3.3 Quo.ent Rule Objec.ve I will be able to divide powers with like bases and take the power of a quo.ent. I will be able to apply these rules to scien.fic nota.on. Vocabulary None 3.3 Quo.ent Rule To divide two numbers with the same base, subtract the exponents Rule: NOTE: For exponents, division is subtrac.on 8

9 3.3 Quo.ent Rule Remember Nega.ve Exponents mean division Rule: Note: Simplify a single term with exponents means to make it so that each variable appears only once and that all exponents are posi.ve. Remember terms do not have addihon or subtrachon. 3.3 Quo.ent Rule The power of a quo.ent is the quo.ent of the powers Rule: Note: be sure to evaluate powers with numerical bases Bringing it together Break down the problem, then handle each set of numbers and each variable in turn. Group coefficients (numbers), and group the like bases. 9

10 3.3 Quo.ent Rule Division and Scien.fic Nota.on Group the numbers and the 10s just like before. Put it back in scien.fic nota.on. 3.3 Quo.ent Rule Prac.ce 10

11 Objec.ve 3.4 Radicals and Ra.onal Exponents I will be able to iden.fy parts of a radical expression and convert between ra.onal exponents and radical expressions. Vocabulary o Index o Radicand 3.4 Radicals and Ra.onal Exponents Anatomy of a radical: When no index is shown, it is 2: 11

12 3.4 Radicals and Ra.onal Exponents For a real number, a, and posi.ve integers m and n, 3.4 Radicals and Ra.onal Exponents Conver.ng to exponents allows us to simplify radicals that before we could not. Consider : 12

13 3.4 Radicals and Ra.onal Exponents Prac.ce Objec.ve 3.6 Inverse Varia.on I will be able to iden.fy direct and inverse varia.ons. I will be able to calculate the constant of varia.on and set up a model for each varia.on. I will be able to iden.fy, model, and manipulate combined varia.ons. Vocabulary o Direct Varia.on o Asymptote o Joint Varia.on o Inverse Varia.on o Constant of Varia.on o Combined Varia.on 13

14 3.6 Inverse Varia.on Direct Varia.on y=kx (a line through the origin) Constant of Varia.on (k): given a set of data, for each point y/x is constant. Therefore, k=y/ x. Inverse Varia.on y=k/x 3.6 Inverse Varia.on Constant of Varia.on (k): given a set of data, for each point xy is constant. Therefore, k=xy. Inverse varia.on graphs have asymptotes (lines the graph approaches but never touches). 14

15 3.6 Inverse Varia.on Combined Varia.on A rela.on in which one variable varies with respect to each of two or more variables. Ex: y=kz/x Joint Varia.on A direct varia.on of the product of two or more variables. Ex: y=kxz Type of combined varia.on Constant of Varia.on (k): given a set of data, for each point y/xz is constant. Therefore, k=y/ xz. 3.6 Inverse Varia.on Prac.ce 15

16 3.7 Transforma.ons of Inverse Varia.ons Objec.ve I will be able to recognize reciprocal func.ons, analyze how reciprocal func.ons have been transformed, and sketch reciprocal func.ons. Vocabulary o Branch o Reciprocal Func.on 3.7 Transforma.ons of Inverse Varia.ons Reciprocal Func.on Family Parent func.on: y=1/x Always goes through points (1,1) and ( 1, 1) Asymptotes: approaches but never reaches x = 0 and y = 0 Domain: x 0 Range: y 0 16

17 3.7 Transforma.ons of Inverse Varia.ons General Equa.on for a Reciprocal Func.on: Always goes through points (h+1, k+a) and (h 1, k a) Asymptotes: approaches but never reaches x = h and y = k Domain: x h; Range: y k where a 0 Reflec.ons: a < 0 means the graph is flipped over the x-axis Transla.ons: Graph moved right h units, and up k units (lee and down if h and k are nega.ve). 3.7 Transforma.ons of Inverse Varia.ons Prac.ce 17

18 Objec.ve 3.8 Simplifying Radicals I will be able to simplify square roots and radicals of other indexes. I will be able to simplify products of radicals with different indexes. I will be able to ra.onalize the denominator. Vocabulary o Radical Expression o Ra.onalize the Denominator 3.8 Simplifying Radicals Simplifying Square Roots by Perfect Squares Factor out the largest square you can find. Evaluate that square root, carry the rest. Check for any squares you missed. 18

19 3.8 Simplifying Radicals The Perfect Squares 3.8 Simplifying Radicals Simplifying Radicals by Prime Factoriza.on Factor into prime factors Group the factors Divide powers of each factor by the index (use remainders) Simplify Recombine terms Ra.onal Exponents Ra.onal exponents can be used instead of radicals. This is oeen useful when dealing with different indexes. 19

20 3.8 Simplifying Radicals Ra.onalizing the denominator Radicals cannot be lee in the denominator. To ra.onalize the denominator, mul.ply the original frac.on by the radical in the denominator over itself and simplify. 3.8 Simplifying Radicals Prac.ce 20

21 Objec.ve 3.9 Radical Equa.ons I will be able to recognize radical equa.ons and solve them. I will be able to iden.fy and eliminate extraneous solu.ons. Vocabulary o Radical Equa.on o Square Root Equa.on o Extraneous Solu.ons o Square Root Func.on o Radical Func.on 3.9 Radical Equa.ons Radical Equa.ons Equa.ons with the variable in the radicand. Ex: Radical Equa.ons NOT Radical Equa.ons 21

22 3.9 Radical Equa.ons Solving Radical Equa.ons Isolate the radical to one side of the equa.on Raise both sides of the equa.on to the same power Simplify Check your Extraneous Solu.ons Solu.ons that do not work when you check them. Not part of your answer. If all solu.ons are extraneous, there is no solu.on. NOTE: Squares and square roots are inverses (cube and 3.9 Radical Equa.ons Prac.ce 22

23 Objec.ve 3.10 Transforma.ons I will be able to iden.fy transforma.ons to radical func.ons from their equa.ons. I will be able to plot radical func.ons to find zeros. I will be able to simplify radical func.ons to find the transforma.ons. Vocabulary None 3.10 Transforma.ons Square Root Func.ons Parent Func.on: General Form: a < 0 reflects graph over x-axis a > 1 stretches graph; 0 < a < 1 shrinks graph Transla.on: Horizontal: h (posi.ve à right, nega.ve à lee) Ver.cal: k (posi.ve à up, nega.ve à down) 23

24 Radical Func.ons Parent Func.on: General Form: 3.10 Transforma.ons a < 0 reflects graph over x-axis a > 1 stretches graph; 0 < a < 1 shrinks graph Transla.on: Horizontal: h (posi.ve à right, nega.ve à lee) Ver.cal: k (posi.ve à up, nega.ve à down) 3.10 Transforma.ons Solving Radical Equa.ons by Graphing Plot the graph Use your calculator to find the zeros Y1 = (plot func.on) Press 2 nd then TRACE Select 2: zero Move cursor near x-intercept, select lee boundary and right boundary then press enter again. 24

25 3.10 Transforma.ons Rewri.ng a Radical Func.on If you simplify the radical so that x has a coefficient of 1, you can graph the func.on using its transforma.ons. Solve so b = 1 Then c = h 3.10 Transforma.ons Prac.ce 25