Summary for a n = b b number of real roots when n is even number of real roots when n is odd

Transcription

1 Day Roots and Radical Expressions Warm Up Write each number as a square of a number. For example: 25 = Write each expression as a square of an expression. For example: 4. x x 4 y x 6 y 12 I. Roots and Radical Expressions Since 5 2 =, is a square root of Since 5 3 =, is a cube root of Since 5 4 =, is a fourth root of Since 5 5 =, is a fifth root of Definition: nth Root For any real numbers a and b, and any positive integer n: If a n = b, then a is an nth root of b. Since 2 4 = and (-2) 4 =, both and are fourth roots of Since there is no real number x such that x 4 = -16, then has no real fourth root. Since (-5) 3 =, then is a cube root of. Notice that -5 is the only real cube root of Some roots, such as the square roots of 10, are irrational numbers. Nevertheless, there is a square root and a square root of 10: Since (- 10) 2 = and ( 10) 2 =, both and are square roots of Summary for a n = b b number of real roots when n is even number of real roots when n is odd positive 0 negative

2 1 Example - Finding All Real Roots Find all the real square roots of each number. a. 36 b. c d. -25 Find all the real cube roots of each number. a. 8 b. c d Find all the real fourth roots of each number. a. 256 b. c d Example - Finding Roots Find each real number root. Property: nth Root of a n (a<0) For any negative real number a, 3 Example - Simplifying Radical Expressions Simplify each radical expression.

3 Day Multiplying and Dividing Radical Expressions Warm Up 1. Find all the real square roots of each number. a. 121 b. -49 c. 64 d. 2. Find all the real cube square roots of each number. a b. 3. Find each real-number root. 4. Simplify each radical expression. Property: Multiplying Radical Expressions 1 Example - Multiplying Radicals Multiply. Simplify if possible. The property does not apply because -2 is not a real number.

4 2 Example - Simplifying Radical Expressions Simplify each expression. Assume that all variables are positive. Then absolute value symbols are never needed in the simplified expression. 3 Example- Multiplying Radical Expressions Multiply and simplify.

5 Day 16½ 7.2 Dividing Radical Expressions (continued) Warm Up: Simplify. Assume all variables are positive. a. b. c. d. e. f. Property: Dividing Radical Expressions 4 Example - Dividing Radicals Divide and simplify. Assume that all variables are positive. a. b. c. d.

6 e. 5 Example - Rationalizing the Denominator Rationalize the denominator of each expression. Assume that all variables are positive. a. b. c. d. e. f.

7 Day Binomial Radical Expressions Warm Up: Simplify. Rationalize all denominators. a. b. c. d. e. I. Adding and Subtracting Radical Expressions Like Radicals - have the same index (nth root) and the same radicand (what s under the radical) 1 Example - Adding and Subtracting Radical Expressions Add or subtract if possible. a. b. c. d. e. 2 Example - Word Problem In the stained-glass window design, the side of each square is 5 inches. Find the perimeter of the window to the nearest tenth of an inch.

8 3 Example - Simplifying before Adding or Subtracting Simplify. a. b. II. Multiplying and Dividing Binomial Radical Expressions 4 Example - Multiplying Binomial Radical Expressions (Use FOIL) a. b.

9 Day 17½ 7.3 Binomial Radical Expressions (continued) Warm Up: Recall: Conjugates differ only in the sign of the second terms: and are conjugates. 5 Example - Multiplying Conjugates Multiply. a. b. 6 Example - Rationalizing Binomial Radical Denominators Rationalize the denominator. a. b.

10 Day Rational Exponents Warm Up: Simplify I. Simplifying Expressions with Rational Exponents Rational exponent - another way to write a radical expression: 1 Example - Simplifying Expressions with Rational Exponents a. b. c. Definition: Rational Exponents

11 2 Example - Converting to and from Radical Form Write in radical form. a. b. c. Write in exponential form. a. b. c. Summary: Properties of Rational Exponents Let m and n represesnt rational numbers. Assume that no denominator equals 0.

12 You can simplify a number with a rational exponent by using the properties of exponents or by converting the expression to a radical expression... 4 Example - Simplifying Numbers with Rational Exponents Method 1: Method 2: a. b. c.

13 Day 18½ 7.4 Rational Exponents Warm Up: Simplify Example - Write in simplest form. a. b. c. c. d.

14 Day 19 Review Warm Up: Simplify Day 20 Quiz Day Solving Radical Equations Warm Up: Solve by factoring. 1. x 2 = -x x 2 + x = x 2 = -8x + 5 I. Solving Radical Equations Radical Equation - has a variable (i.e. x) under the radical sign. 3 + x = 10 is a square root equation x + 3 = 10 is a NOT square root equation To solve a radical equation, isolate the radical on 1 side of the equation. Then raise both sides of the equation to the same power. 1 Example - Solve the Square Root Equation a. b. To solve, raise both sides of the equation to, which is the reciprocal of. Then 2 Example - Solve the Radical Equation with Rational Exponents a. b.

15 Day 21½ Solving Radical Equations (continued) Warm Up: Example - Solve. Then check for extraneous solutions. a. b. If an equation contains 2 radicals (or 2 rational exponents), then put them on opposite sides of the equal sign. 5 Example - Solving Equations with 2 Rational Exponents Solve. Then check for extraneous solutions. a. b.

16 Day Function Operations Warm Up: Find the domain and range of each function. 1. {(0, -5), (2, -3), (4, -1)} 2. f(x) = 2x g(x) = x 2 Evaluate each function for the given value of x. 4. f(x) = 3x + 4, f(2) 5. g(x) = 2x 2-3x + 1, g(-3) I. Operations with Functions Definition: Function Operations Addition Multiplication Subtraction Division For a sum, difference, product, or quotient funtion, the domain is the x-values that are in the domains of... However, the domain of a quotient function excludes any x-value such that g(x) =... 1 Example - Adding and Subtracting Functions Given f(x) and g(x). Find f + g and f - g. State their domains. f(x) + g(x) f(x) - g(x) a. f(x) = 3x + 8 g(x) = 2x - 12 b. f(x) = 5x 2-4x g(x) = 5x + 1

17 2 Example - Multiplying and Dividing Functions Let f(x) and g(x). Find f g and f g. State their domains. a. f(x) = x 2-1 g(x) = x + 1 b. f(x) = 6x 2 + 7x - 5 g(x) = 2x - 1 II. Composition of Functions Composite Function - combining 2 functions by applying 1 after the other. The output from the 1st function becomes the input for the 2nd function. diagram: Definition: Composition of Functions The composition of function g with function f can be written as 1. g f 2. (g f)(x) 3. g(f(x)) The domain of g f is the values a in the domain of f for which f(a) is in the domain of g. (g f)(x) = g(f(x)) Work inside out : Evaluate f(x) first. Then use your result and plug it into g(x).

18 3 Example - Composition of Functions a. Let f(x) = x - 2 and g(x) = x 2. Find (g f)(-5). Method 1: Plug f(x) into g(x). Then evaluate x = -5. Method 2: Evalute f(-5). Then plug result into g(x). b. Now find (f g)(-5). c. What do you notice from parts a and b? Make a conclusion about compositions of functions. Day 22½ 7.6 Function Operations Warm Up 1. Let f(x) = -2x + 6 and g(x) = 5x 2 + x - 7. Find g(x) - f(x) and -2f(x) + g(x). 2. Let f(x) = x and g(x) = x 4-1. Find f(x) g(x) and f(x) g(x).

19 3. Let f(x) = x 3 and g(x) = x Find g(f(2)). 4 Example - Word Problem Suppose you are shopping in the store. You have a coupon worth $5 off any item. a. Use functions to model discounting an item by 20% and to model applying the coupon. b. Use a composition of functions to model how much you would pay for an item if the clerk applies the discount first and then the coupon. c. Use a composition of functions to model how much youwould pay for an item if the clerk applies the coupon first and then the discount. d. How much more is any item if the clerk applies the coupon first?

20 Day Inverse Relations and Functions Warm Up Recall: How do you determine whether a graph is a function? I. The Inverse of a Function Inverse Relation - if (x, y) is an ordered pair of a relation, then (y, x) is an ordered pair of its inverse. diagram: relation r inverse of r domain range domain range

21 1 Example - Finding the Inverse of a Relation a. Find the inverse of the relation. relation x y inverse x y b. Now graph the relation and its inverse. c. How is the line y = x related to the graphs of the relation and its inverse? d. Is the relation a function? Is its inverse a function? Remember to use the Conclusion for Relation and its Inverse: 1. Graphically: The graph of a relation and its inverse is a reflection in the line y = x 2. Algebraically: To get from a relation (x, y) to its inverse (y, x), just switch the x and y. 2 Example - Interchanging x and y a. Find the inverse of y = x Step 1: Switch x and y. Step 2: Solve for y. Step 3: Square root each side. b. Is y = x a function? c. Is its inverse a function? Explain.

22 3 Example - Graphing a Relation and its Inverse Quick Check: 1a. Given the function y = 3x - 10, find its inverse. b. Is the inverse a function? c. Graph the function and its inverse. 2. (HW #35) Find the inverse of f(x) = 1.5x 2-4. Is the inverse a function?

23 Day Inverse Relations and Functions (continued) Warm Up Given f(x) = -x 2-2. Find its inverse. Then graph them both. The notation for a function is, meaning f(x) must pass the The notation for the inverse of a function is. We say the inverse of f or f inverse. 4 Example - Finding an Inverse Function Consider the function f(x) = (x+1). a. Find the domain and range of f. b. Find f -1. c. Find the domain and range of f -1. d. Is f -1 a function? Explain. 5 Example - Word Problem The function is a model for the distance d in feet that a car with locked brakes skids in coming to a complete stop from a speed of r mi/h. Find the inverse of the function. What is the best estimate of the speed of a car that made skid marks 114 feet long?

24 Inverse Functions - If f and f -1 are both functions, and f maps x to y, then f -1 must map y to x. f and f -1 are called inverse functions. Property: Composition of Inverse Functions If f and f -1 are inverse functions, then 6 Example Since f is a linear function, f-1 is also a linear function. Quick Check: a. Find the domain and range of f. b. Find f -1. Is f -1 a function? c. Find the domain and range of f -1. a. Find the domain and range of f. b. Find f -1. Is f -1 a function? c. Find the domain and range of f -1.

25 Day Graphing Radical Functions Warm Up: Graph the functions on the same axes. State the transformation compared to the parent function. 1. y = x 2 (parent) 4. y = -(x + 4) 2 2. y = (x - 3) 2 5. y = -x y = (x + 2) 2 6. y = -(x + 1) I. Radical Functions function? function? You can restrict the domain of f so that its inverse is a function. The inverse of g(x)=x 3 is a function function? function?

26 Summary: Families of Radical Functions Square Root Function Radical Function Parent Reflection in x-axis Vertical Stretch a>1 Vertical Shrink 0<a<1 Reflection in x-axis Translation: Horizontal by h Vertical by k Combined 1 Example - Translating Square Root Functions Vertically 2 Example - Translating Square Root Functions Horizontally

27 3 Example - Graphing Square Root Functions Quick Check: State the domain and range of both. State the domain and range of both.

28 Day Graphing Radical Functions (continued) Warm Up 1. Graph the cube function. y = x 3 2. Graph the cube root function. y = x 4 Example - Graphing Cube Root Functions 5 Example - Physics (Calculator) You can model time t (seconds) an object takes to reach the group falling from height H (meters). g=9.81 m/s 2. From what height does an object fall if it takes 7 seconds to reach the ground?

29 6 Example - Rewriting Radical Functions Quick Check: Day 27 Review Chapter 7 Day 28 Test Chapter 7