FUNCTIONS. Algebra 2 & Trigonometry Mrs. Martin Room 227. Name:

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1 FUNCTIONS Algebra & Trigonometry Mrs. Martin Room 7 Name:

2 Functions Topic Pages Day 1 Relations & Functions / Domain & Range Graphically 3 9 Day One-to-One and Onto Functions / Applications of Functions Day 3 Domain and Range Algebraically 17 1 Day 4 Function Notation and Evaluating Functions / Algebra of functions 6 Day 5 Composition of Functions (1) 7 31 Day 6 Composition of Functions () 3 35 Day 7 Finding an Inverse of a Function Day 8 Transformations of Graphs Day 9 Circles (1) Day 10 Circles () 5 56 Day 11 - Direct and Inverse Variation 57 6 Day 1 Review p

3 Step 1: Is it a relation? Day 1 Relations and Functions / Domain and Range Graphically Definitions: A relation is a set of ordered pairs. A relation can consist of numbers or other items, and may be represented as a list, table, or graph. The domain of a relation is the set of possible x-values. (Independent Variable) The range of a relation is the set of possible y-values. (Dependent Variable) Directions: For problems 1-4, determine whether or not the set is a relation. Explain your answer. If it is a relation, find the domain and the range. 1.) {(-3, 5), (4, 7), (4, 5)}.) {, 4, 6, } 3.) {x : x + = 5} 4.) Set-builder Notation: {(x, y) : y = x + } Step : Is the relation a function? Definitions: A function is a relation in which no two ordered pairs have the same first element (domain). **Note: No x-value repeats** Ex) y = x + 3 is the set of ordered pairs (x, x + 3) and can be written as the function {(x, y): y = x + 3} Vertical Line Test (VLT): tells us that if a vertical line drawn any where over the graph intersects the graph more than once, the relation does NOT represent a function. (Fails the VLT) 5.) a.) Students Birth Month b.) Birth Month Students p3

4 Directions: For problems 6-10, find the domain and the range, and determine whether or not the relation is a function. Explain your answer. 6.) {(-3, 5), (4, 7), (0, 5)} 7.) {(-4, 7), (-4, 8), (3, 4), (11, 8)} Domain: Range: Function: YES or NO Domain: Range: Function: YES or NO 8.) Domain: Range: Function: YES or NO 9.) Domain: Range: Function: YES or NO 10.) Domain: Range: Function: YES or NO 11.) Domain: Range: Function: YES or NO p4

5 1.) f (x) x 4 Domain: Table of Values Range: Function: YES or NO 13.) y x 3 Domain: Table of Values Range: Function: YES or NO p5

6 Practice Problems: Directions: a.) State the domain and range of each relation. b.) Explain whether the relation is or is not a function. 14.) a.) Domain: Range: b.) Function: YES or NO 15.) a.) Domain: Range: b.) Function: YES or NO p6

7 Day 1 Relations and Functions / Domain and Range Graphically HOMEWORK **Complete any Practice Problems from class work that have not been completed** Directions: a.) State the domain and range of each relation. b.) State whether the relation is or is not a function, EXPLAIN your reasoning. 1.) {(Albany, New York), (Bismark, North Dakota), (Juneau, Alaska)} a.) Domain: Range: b.) Function?.) {(1, January), (4, July), (11, November), (5, December)} a.) Domain: Range: b.) Function? 3.) {(3,9), (-,4), (4,16), (-1,1), (-3,9), (5,5)} a.) Domain: 4.) {(,1), (4,5), (5,7), (5,11), (9,15)} a.) Domain: Range: Range: b.) Function? b.) Function? 5.) Domain: Range: Function: YES or NO p7

8 6.) Domain: Range: Function: YES or NO 7.) Domain: Range: Function: YES or NO 8.) Domain: Range: Function: YES or NO Review: 9.) Simplify: b 1 10 b 1 5 p8

9 10.) Solve for x in simplest radical form: x 4x 1 0 Answers: 1.) a) Domain: {Albany, Bismark, Juneau}, Range: {NY, ND, AK}; b) Function: Yes, no x-values repeat.) a) Domain: {1, 4, 11, 5}, Range: {Jan, July, Nov, Dec}; b) Function: Yes, no x-values repeat 3.) a) Domain: {3, -, 4, -1, -3, 5}, Range: {9, 4, 16, 1, 5}; b) Function: Yes, no x-values repeat 4.) a) Domain: {, 4, 5, 9}, Range: {1, 5, 7, 11, 15}; b) Function: No, x-value repeats (5) 5.) a) Domain: 4 x 4, IN: [-4, 4], Range: y 6, IN: [-, 6]; b) Function: No, fails VLT 6.) a) Domain: All Reals or x, IN,, Range: y 0 or 0 y, IN 0,; b) Function: Yes, passes VLT 7.) a) Domain: 0 x 8, IN 0,8, Range: 4 y 5, IN 4,5; b.) Function: No, fails VLT 8.) a) Domain: x or x, IN,, Range: All Reals or y, IN, ; b) Function: No, fails VLT 5 9.) b 5 10.) 5 p9

10 Day One to One and Onto Functions / Applications of Functions Do Now: (Question 1 & ) 1.) Which graph does not represent a function? (1) () (3) (4).) Which diagram represents a relation in which each member of the domain corresponds to only one member of the range? (1) () (3) (4) Step 3: Is the function one to one (1-1) or onto? (or both) Steps to Determine if a Function is 1 to 1: (1-1) A function is said to be one to one if each element of the domain has only one element in the range, and vice versa. (It MUST pass both the HORIZONTAL and VERTICAL line test!) Steps to Determine if a Function is Onto: A function is said to be onto if every element in the range is mapped to at least one element in the domain. (No y-value is left out) Directions: For problems 3 8, draw a sketch of the graph, if necessary, find the domain and the range, and determine whether the function is 1-1, onto or both. Explain your reasoning. 3.) Domain: Range: One-to-One: YES or NO Onto: YES or NO p10

11 4.) Domain: Range: One-to-One: YES or NO Onto: YES or NO 5.) Domain: Range: One-to-One: YES or NO Onto: YES or NO 6.) Domain: Range: One-to-One: YES or NO Onto: YES or NO 7.) {(x,y):y x 3} Domain: Range: Table of Values One-to-One: YES or NO Onto: YES or NO p11

12 8.) y x 1 Domain: Range: Table of Values One-to-One: YES or NO Onto: YES or NO Applications of Functions 9.) A New York subway train slows down as it approaches the 4 nd Street Station, stops at the station for two minutes, and then continues on its route. Which graph below shows the speed of the train compared to the time elapsed? (1) (3) () (4) p1

13 Day One to One and Onto Functions / Applications of Functions HOMEWORK Directions: For problems 1 8, draw a sketch of the graph, if necessary, find the domain and the range, and determine whether the function is 1-1, onto or both. Explain your reasoning. 1.) Domain: Range: One-to-One: YES or NO Onto: YES or NO.) Domain: Range: One-to-One: YES or NO Onto: YES or NO p13

14 3.) Domain: Range: One-to-One: YES or NO Onto: YES or NO 4.) Domain: Range: One-to-One: YES or NO Onto: YES or NO 5.) y x 4x 3 Domain: Range: Table of Values One-to-One: YES or NO Onto: YES or NO p14

15 6.) y 3 Domain: Range: Table of Values One-to-One: YES or NO Onto: YES or NO 7.) Kathryn left for school and walked casually until she realized she d forgotten her calculator. She turned and hurried home, got the calculator, and then ran to school so she wouldn t be late for class. Which graph depicts the situation in which distance from home is a function of time elapsed? (1) () (3) (4) Review: 8.) Express as a single fraction r 3 r p15

16 9.) Solve for x: 5x 4 8 Answers: 1.) Domain: x 4, IN,4, Range: 5 y 7, IN 5,7 ; 1-1: No, fails HLT; Onto: No, different domain and range.) Domain: x 1, IN 1,, Range: y 0, IN 0, ; 1-1: Yes, passes VLT & HLT; Onto: No, different domain and range 3.) Domain: {1, 4, 64, 5}, Range: {3, 15, 6}; 1-1: No, 3 Repeats; Onto: Yes, all y s are used 4.) Domain: {14,, 16}, Range: {0, 1, }; 1-1: Yes, no y s repeat; Onto: Yes, all y s are used 5.) Domain: All Reals or x, IN,, Range: y 1 or 1 y, IN 1,; 1-1: No, fails HLT; Onto: No, different domain and range 6.) Domain: All Reals or x, IN,, Range: y 3, IN 3 ; 1-1: No, fails HLT; Onto: No, different domain and range 7.) Choice (4) r r 3 8.) r 68 9.) or p16

17 Day 3 Domain and Range Algebraically Domain and Range Type of Function Graph Domain Range Example 1: Linear y = x 1 Example : Quadratic y = x 4 Example 3: Fractions Determining Restricted Domain 3 y x 5 Example 4: Radicals y 10 x Example 5: Fractions with Radicals 1 y x 8 Directions: Find the domain of the following functions: 3x 1 1.) y.) x 3x 4 y 7x 5 x p17

18 3.) y x 4 4.) y x 8x 15 5.) The function y x 5x 14 with a domain of 1 x 7. a. Find the smallest element in the range. b. The range as determined by the given domain. Practice Problems: Directions: Find the domain and range for each of the following functions. 6.) y x 10 7.) y 3x ) The function y 4x 7 has a domain of x 7. a. Find the range. b. Find the least element in the range. p18

19 Day 3 Domain and Range Algebraically HOMEWORK **Complete any Practice Problems from class work that have not been completed** Directions: Choose the correct choice for the domain of each function. 3x 1.) What is the domain of the function y? x 49 (1) { x x real numbers, x 7} () { x x real numbers, x 7} (3) { x x real numbers} (4) { x x real numbers, x 0}.) What is the domain of the function (1) all real numbers except 0 () all real numbers except 3 (3) all real numbers except 3 and 3 (4) all real numbers y x x 9? 3.) In the set of real numbers, what is the domain of (1) x > 0 (3) x 4 () x < 4 (4) x > 4 y 4x x 4? 4.) If y 1, what is the domain of y? x 4 (1) x = (3) x () x < (4) x > 4 5.) What is the domain of the function y x (1) x x (3) x x 1 1 () x x (4) x x over the set of real numbers? p19

20 6.) What is the domain of the function (1) x x 5 () x x 5 (3) x x 5 (4) x x 5 y 4 x 5 over the set of real numbers? 7.) What is the domain of (1) x x1 or x 5 () x x5 or x 1 (3) x 1 x 5 (4) x 5 x 1 y x x 4 5? 8.) In the set of real numbers, what is the domain of y x 5? (1) x 5 (3) x > 5 () x 5 (4) x ) Which negative real number is not in the domain of y x 4? 10.) For what values of x will the function y x 4 be real? (1) {x x < 0} (3) {x x 4} () {x x > 0} (4) {x x 4} p0

21 11.) In which function is the range equal to the domain? (1) y x (3) y log x () y x (4) y x 1.) What is the domain of the function y 4 x 1 (1) {x x = 1} (3) {x x < 1} () {x x 1} (4) {x x > 1} over the set of real numbers? 13.) What is the domain of the function y x? (1) {x x 0} (3) {x x } () {x x } (4) {x x } Answers: 1.) ().) (3) 3.) (4) 4.) (4) 5.) (4) 6.) (1) 7.) () 8.) (1) 9.) 10.) (4) 11.) (4) 1.) (4) 13.) () p1

22 Day 4 Function Notation and Evaluating Functions / Algebra of Functions Do Now: (Questions 1 & ) 1.) What is the domain of the function 4x.) Given the function y, find the domain. y x 3? x 4 (1), (3), (), (4) 3, Function Notation f ( x) x 3 Read as: f of x equals two times x plus three **NOTE: f ( x) y Steps for Evaluating Functions: 1. Plug the value in for x.. Evaluate. Directions: Evaluate each function. 3.) f ( x) x x 4.) f ( x) x x Find f (3) Find f ( 1) p

23 5.) If the function h(x) is represented by the mapping below: 6.) Use the graph of the function f(x) shown below: Find: a.) h(-3) = b.) h(1) = c.) h(3) = d.) the value of x if h(x) = 5. Find: a.) f() = b.) f(-1) = c.) f(4) = d.) f(3.5) = e.) f(0) = f.) For how many values of x does f(x) = -? Operations with Functions Directions: Given the functions f ( t) t3 and gt ( ) t 6 evaluate the four operations below. 7.) Find the sum : f ( t) g( t) or ( f g)( t) 8.) Find the difference: f ( t) g( t) or ( f g)( t) 9.) Find the product: f ( t) g( t) or ( fg )( t) 10.) Find the quotient: gt () f () t g or (t) f p3

24 Practice Problems: 11.) If a function f(x) is defined as f ( x) 3 5x x, evaluate a.) f(3)= b.) f(-)= c.) 1 f = 1.) If f(x) is defined as f ( x) x and g ( x) x 4. Find: a.) f ( x) g( x) b.) f ( x) g( x) c.) f ( x) g( x) d.) g( x) f ( x) p4

25 Day 4 Function Notation and Evaluating Functions / Algebra of Functions HOMEWORK **Complete any Practice Problems from class work that have not been completed** Directions: Evaluate each function. 1.) f ( x) 7 x, find f(5)..) h( x) 1 x, find h(10). 3.) f ( x) x, find f(-8). 4.) g( x) 1 x, find g(-4). 5.) The graph of the function f is shown below: Find: a.) f() = b.) f(-5) = c.) f(-) = d.) f(3.5) = Directions: For problems 6 9, f ( x) x x 6 and g ( x) x 3. Find the four given operations below. 6.) f ( x) g( x) 7.) f ( x) g( x) p5

26 8.) f ( x) g( x) 9.) f ( x) g( x) Review 10.) Write the quotient i in simplest 3 i a bi form. 11.) Find the sum of 18 and Answers: 1.).) 3i 3.) 6 4.) 17 5.) a), b) -3, c) -, d) 6.) x 9 7.) x x ) x 4x 3x 18 9.) x + 10.) i 11.) i p6

27 Do Now: (Question 1 & ) 1.) Which function is not one-to-one? (1) {(0, 1), (1, ), (, 3), (3, 4)} () {(0, 0), (1, 1), (, ), (3, 3)} (3) {(0, 1), (1, 0), (, 3), (3, )} (4) {(0, 1), (1, 0), (, 0), (3, )} Day 5 Composition of Functions (1).) If f ( x) 5x 6, evaluate f (3). Notation: f gx or f gx Steps to Evaluate Compositions of Functions 1. Evaluate the innermost function.. Use the answer from step 1 and evaluate the outer (second) function. 3.) If f(x) is defined as f ( x) x and g(x) is defined as g ( x) x 4. Find: a.) f g 3 b.) g f 5 c.) f f d.) g g1 p7

28 4.) Find: a.) g f b.) f g 3 c.) f f 9 Practice Problems: Directions: Evaluate each composition using the following four functions: f ( x) 5x g ( x) x 16 h ( x) x 16 j ( x) x 3x 4 5.) h j4 6.) g f 7.) f g 8.) f j 3 p8

29 f ( x) 5x ( ) x g x 16 h ( x) x 16 j ( x) x 3x 4 9.) j f 3 10.) g h 7 11.) h j 7 1.) g g p9

30 Day 5 Composition of Functions (1) HOMEWORK **Complete any Practice Problems from class work that have not been completed** Directions: For questions 1 8, evaluate each composition using the following four functions: f ( x) 3x g ( x) x h( x) x j ( x) x 1 h 4 g f 1.) j.) 3.) f g 4.) f j 3 5.) j f 3 6.) g h 7 7.) h j 7 8.) f f p30

31 9.) Find: g a.) f 1 b.) f g c.) g g1 Review: 10.) Solve for x: x x x x 11.) For what value(s) of a is the fraction a 5 undefined? 3a 4a 1 Answers: 1.) 17.) 8 3.) 0 4.) 6 5.) 8 6.) 47 7.) 36 8.) 18 9.) a) -1, b), c) ) x = 6 11.) 1,1 3 p31

32 Do Now: (Question 1 & ) 1 1.) If f ( x) x 3and g ( x) x 5, what is the value of g f 4? (1) -13 (3) 3 () 3.5 (4) 6 Day 6 Composition of Functions ().) If f ( x) 3x and g ( x) x 5, find the f g 3. value of Steps to Finding the Rule of Composite of Functions The rule of a composite function is defined as the single function that will perform the same operation as the composition, but in one step. 1. Place the innermost functions expression in place of each x in the outer (second) function.. Simplify 3.) If f(x) is defined as f ( x) x and g(x) is defined as g ( x) x 4. Find: a.) g f x b.) f gx p3

33 4.) If h(x) is defined as h( x) 3x and f(x) is defined as f ( x) x. 3 Find: a.) h x h f x f b.) Practice Problems: Directions: Find the rule of each composition using the following three functions: f ( x) 5x ( ) x g x 16 j ( x) x 3x 4 5.) g f x 6.) f jx 7.) f f x p33

34 Day 6 Composition of Functions () HOMEWORK **Complete any Practice Problems from class work that have not been completed** Directions: Find the rule of each composition using the following four functions: f ( x) 3x g ( x) x h( x) x j ( x) x 1 h x g f x 1.) j.) 3.) f gx 4.) f jx 5.) j f x 6.) g hx p34

35 f ( x) 3x g ( x) x h( x) x j ( x) x 1 7.) h jx 8.) f f x Review: 9.) Simplify x 00x 3 10.) Express 3 5ab in simplest form. Answers: 1.) x 1.) 3x 3.) 3x 6 4.) 3 x 1 5.) 3x 1 6.) x 7.) x 1 8.) 9x p35 9.) 15 x x 10.) 9 6b 5ab 5ab 3

36 Do Now: (Question 1 & ) 1.) If g( x) x and h( x) x, find the rule for h g x. Day 7 Finding an Inverse of a Function.) If f ( x) x and g ( x) x 5, find the rule for g f x. Steps to Finding an Inverse Algebraically: Notation: f 1 ( x) An inverse is a reflection in the line y x. Therefore, inverse means switch the x and y values. 1. Switch x and y.. Solve for y. Directions: Find the inverse of the given functions. 3.) {(8,5), (6,8), (4,11), (,14)} 4.) y 4x 7 5.) y x 6 6.) Find 1 ( x f ), if f ( x) x 4, then evaluate f 1 (1). 3 p36

37 Steps to Finding an Inverse Graphically: 1. Find points.. Switch x and y. 3. Plot the new points. 7.) Find and graph f 1 ( x ). 8.) Given: f ( x) 6x 3 a.) Find g(x), the inverse of f(x). b.) Use the composition of functions to justify that f(x) and g(x) are inverses of one another. p37

38 Day 7 Finding an Inverse of a Function HOMEWORK Directions: Find the inverse of the given functions. 1.) {(-5, 8), (5, 4), (-4, 10), (-, 14)}.) {(-, 3), (-5, 5), (-8, 7), (-11, 9)} 3.) y 3x 1 4.) f ( x) x ) f ( x) x 4 6.) g ( x) 3x 3 6 p38

39 8.) f ( x) 3x 9 7.) y 4x 3 9.) If g ( x) 3x 1, then evaluate g 1 (4)? 1 10.) If h( x) x 3and j ( x) x 1, what is the j h 1 3? solution to p39

40 Review: 11.) What is the domain of the function f ( x) 3x 1? Answers: 1.) {(8, -5), (4, 5), (10, -4), (14, -)}.) {(3, -), (5, -5), (7, -8), (9, -11)} x 1 3.) ) x 15 5.) x 3 4 x 6 6.) 3 3 x 1 7.) 3 x or x 9 8.) 3 9.) 1 10.) 3 11.) x 4 p40

41 Do Now: (Question 1 & ) 1.) What is the inverse of g ( x) x 4? (1) ( ) x 1 g x (3) g ( x) () x ( ) x 1 g x (4) g 1 ( x) 3x 6.) Find the inverse of 1 f (4). Day 8 Transformation of Graphs x 9 f ( x) and evaluate x Reflections / Vertical and Horizontal Shifts Reflections: Reflection in the x-axis, negates the y. Reflection in the y-axis, negates the x. Reflection in the origin, negates both x and y. Vertical Shifts: (up and down) - (Do exactly what it tells you to do!) Moves the graph either up or down along the y-axis. (*Note-not attached to the x.) Horizontal Shifts:(left or right) (Move the OPPOSITE of what it tells you to do!) Moves the graph either right or left along the x-axis. (*Note-attached to the x.) Original Reflection in the x-axis Vertical Shift Horizontal Shift p41

42 Directions: Describe each transformation that occurs compared to the graph y x. 3.) y x 6 4.) y x 4 5.) y x 5 3 Directions: Describe each transformation that occurs compared to the graph y x 6.) y x4 7.) y x1 3 8.) y x1 1 p4

43 Directions: Describe each transformation that occurs compared to the graph y x 9.) y x1 10.) y 3 x 11.) y x 1 p43

44 1.) Describe the shift that occurs to f ( x) x when given f ( x) x 10. Day 8 Transformation of Graphs HOMEWORK.) Describe the shift that occurs to f ( x) x when given f ( x) x 5. 3.) Describe the shift that occurs to given ( x) x 8 f. f ( x) x when 4.) Describe the shift that occurs to given ( x) x 1 f. f ( x) x when 5.) Describe the shift that occurs to f ( x) x when given f ( x) x 3. 6.) Describe the shift that occurs to f ( x) x when given f ( x) 6 x ) Describe the shift that occurs to f ( x) x when given f ( x) 5 x ) Describe the shift that occurs to f ( x) x when 3 given ( x) x 1 f. p44

45 Review: 9.) What is the vertex of the parabola y x 3? 10.) Find f 1 ( x ), given f ( x) x 8, then find f 1 ( 1). Answers: 1.) Right 10.) Left, up 5 3.) Right, down 8 4.) Reflection over x, left 1, down 5.) Left 3, down 6.) Left 1, up 6 7.) Reflection over x, up 5 8.) Right 1, up 9.) (0, 3) 10.) 9 or 4.5 p45

46 Do Now: (Question 1 & ) 1.) The graph below shows the function f(x). Day 9 Circles (1) Which graph represents the function f(x + )? (1) () (3) (4).) The function ( x) x 1 f is a shift of (1) units up and 1 unit to the left. () units down and 1 unit to the right (3) units to the right and 1 unit up (4) units to the left and 1 unit down f ( x) x Standard Form of a Circle (Centered at the Origin) x y r r = radius Center-Radius Form of a Circle (Not Centered at the Origin) ( x h) ( yk) r ( h,k ) = Center - OPPOSITE r = radius Directions: Determine the center and radius of each circle. 3.) x y 5 x 4 y ) x y 64 4.) Directions: Write the equation of each given circle. 6.) Center (0,0) Radius 7.) Center (-1,5) Radius 6 8.) Center (0,-3) Radius 9 p46

47 Determining if a point lies on the circle Substitute the point into the equation to see if it satisfies the equation. Directions: Determine if the point lies on the circle. 9.) x y 5, Point (3,4) 10.) ( x ) ( y3) 18, Point (-1,5) 11.) ( x 1) y 50, Point (,7) Directions: Write the equation of the circle given the picture below. 1.) p47

48 13.) Write an equation for the circle where the diameters endpoints are (, 6) and (6, 10). p48

49 Day 9 Circles (1) HOMEWORK Directions: Select the numeral preceding the choice that best completes the sentence or answers the question: 1.) The center of the circle ( x4) ( y) 9 is.) Which circle has a center of (1, 0) and a radius of (1) (4, -3) (3) (-4, ) length 4? () (4, ) (4) (4, -) (1) x y 4 () ( x1) y 4 (3) ( x1) y 4 (4) x y 4 3.) Which point is not on the circle ( x 4) ( y 3) 5? (1) (8, 6) (3) (0, 0) () (-1, 3) (4) (-4, -3) 4.) The graph of the equation ( x ) ( y 3) 16 is a circle with (1) center (, -3) and radius 4 () center (, -3) and radius 16 (3) center (-, 3) and radius 4 (4) center (-, 3) and radius 16 5.) Which is the equation of the circle shown below? 6.) Which is a graph of the circle whose equation is ( x 3) ( x ) 4? (1) () (1) ( x ) ( y 4) 5 () ( x ) ( y 4) 5 (3) ( x ) ( y 4) 5 (4) ( x ) ( y 4) 5 (3) (4) p49

50 7.) Which circle lies entirely in the fourth quadrant? (1) ( x 4) ( y 6) 5 () ( x 4) ( y 6) (3) ( x 4) ( y 6) 5 (4) ( x 4) ( y 6) ) Jen is playing with a Frisbee whose diameter is 1 inches. If she tosses it onto a coordinate plane, and its center falls on the point (-, 1), what is the equation of the Frisbee? (1) x y 36 () ( x ) ( y 1) 36 (3) ( x ) ( y 1) 36 (4) ( x ) ( y 1) 36 9.) Harmony and Melodie were blowing bubbles when one of them landed on Derek s math homework and burst on the graph paper. The bubble formed a perfect circle on the coordinate grid with a center at (6, -5) and a radius of 4.5. Which equation represents the bubble s circle? (1) ( x 6) ( y 5) 4. 5 () x y 4.5 (3) ( x 6) ( y 5) 0. 5 (4) ( x 6) ( y 5) ) A hot coffee mug stained Mrs. Hilton s coffee table. If the equation of the circle left by the coffee mug is ( x 1) ( y 4) 7. 84, what is the diameter of the mug? (1) 7.84 () 5.6 (3).8 (4) 1.4 Directions (Questions 11-15): Find the center and the length of the radius for each circle: 11.) ( x 7) y ) x + (y 3) = ) ( x ) ( y ) ) (x 5) + (y + ) = ) (x + 1.5) + (y 3.6) = 10 Directions (Questions 17-0): Write an equation for the circle with the given conditions: 16.) Center at (-1, -5), radius of 8 17.) Center at (0, 3) and diameter of 18.) Center at (-, 0) and diameter of 3 p50

51 19.) Diameter with endpoints (, 6) and (6, 10) 0.) Diameter with endpoints (-7, -3) and (3, 5) Review 1.) Given the functions f ( x) x 4 and g ( x) x 3. a. Find g (g(4)) b. Find ( f g)( x) Answers: 1.) (4).) (3) 3.) (4) 4.) (1) 5.) (4) 6.) (4) 7.) (1) 8.) (3) 9.) (4) 10.) () 11.) center (-7, 0), r = ) center (0, 3), r = ) center (-, -1/), r = 4 14.) center (5, -), r = ) center (-1.5, 3.6), r 10 x1 y ) 17.) x y3 1 p51 18.) 9 x y 4 19.) x y 0.) x y ) a.) 5, b.) 4x 1x 13

52 Do Now: (Question 1 & ) 1.) Which equation has a diameter with endpoints (3, -) and (-5, 8)? (1) x 1 y 3 41 () x 1 y 3 41 (3) x 3 y 164 (4) x 3 y 41 Day 10 Circles ().) Write an equation of the circle shown in the graph below. Standard Form of the Equation of a Circle x y DxEyF 0 Steps to write a circle given in center-radius form into standard form: Expand the square of the binomials, and simplify. Directions: Find the standard form of the circle. 3.) x 3 y 34 4.) center: (4, -3), radius = 5

53 Steps to write the center-radius form of a circle given in standard form. 1.) Move the integer to the other side of the equation..) Rearrange the equation into x + Cx + y + Dy = E. 3.) Use completing the square on each set of variables separately. 4.) Add both integer values that were used to complete the squares to the right side of the equation. 5.) Simplify. Directions: Find the center-radius form of the equation of each given circle. 5.) x y x4y1 0 6.) x y x y p53

54 Day 10 Circles () HOMEWORK Directions: (Questions 1 4): Write each equation in center-radius form to find the coordinates of the center and length of the radius. 1.) x y xy7 0.) x y x y ) x y x y ) x y x y p54

55 5.) The endpoints of a diameter of a circle are P(6, 1) and Q(-4, -5). Write the equation of the circle: a.) In center-radius form. b.) In standard form. Review: 6.) Write in simplest form: 5x 49y 10x 14y p55

56 7.) Simplify: x 3 3 x x 3 x Answers: 1.) x 1 y 1 9 Center: (1, 1) Radius: 3.) 3 y 1 16 x Center: (3, -1) Radius: 4 3.) 4 y 4 x Center: (4, -) Radius: 4.) 5 y 3 4 x Center: (-5, 3) Radius: 5.) a.) 1 y 34 6.) 7.) x b.) x y x 4y 9 0 5x 7y x 3 3 p56

57 Day 11 - Direct and Inverse Variation Do Now: (Question 1 & ) 1.) A circle has the equation x y 8x 14y Express this equation in center-radius form..) The radius of the circle with equation x y 1x 10y 57 0 is (1) -1 (3) 57 () (4) 4 Direct Variation x x x or k, where k is the constant of proportionality y y y 3.) If y varies directly as x and y = 1 when x = 3: a.) Find the constant of proportionally for y in terms of x, and use this value to express y as a function of x. b.) Graph the linear function. p57

58 4.) Driving along I-95 at a constant rate, Dave drove 165 miles in 3 hours. If he continued to travel at the same rate, how long would it take him to travel another 143 miles? Indirect Variation (Varies Inversely) xy xy or xy k, where k is the constant of proportionality Graph of Varies Inversely INVERSE HYPERBOLA 5.) If y varies inversely as x, and y = 4 when x = 6, what is the value of y when x = 1? p58

59 6.) The rate at which Valerie travels from home to college varies inversely as the time it takes to make the trip. If Valerie can make the trip in four hours at 45 miles per hour, how many miles per hour must she travel to make the trip in three hours? 7.) The amount of a tip each waiter receives after a wedding is inversely proportional to the number of waiters serving the event. If the total amount of tips at Mr. O Leary s wedding was $100, and n represents the number of waiters, and t represents the tip each waiter received, which represents the relationship between n and t? t t (1) n (3) 100 1, 00 n n 100 () 100 (4) n t t 8.) If a varies inversely as b, what is the missing value in the table? a b 6 9? p59

60 Day 11 - Direct and Inverse Variation HOMEWORK Directions: Answer each question using direct or inverse variation, as appropriate. 1.) If r varies directly as s, and r = 5 when s = 7, what is r when s = 1?.) If x varies inversely as y, and x measures 14 when y is 6, find x when y is 4. 3.) Which expresses a direct variation where k is the constant of proportionality? (1) xy = k () x + y = k (3) k x y x (4) k y 4.) Given the area of a rectangle to be 360 square inches, the length of the rectangle varies inversely as the width. If the length of the rectangle is 0 inches, what is the width? p60

61 5.) If 10 potato chips contain approximately 105 calories, approximately how many calories are there in 5 potato chips? 6.) When David travels to college, his travel time varies inversely as his speed. If he drives at 56 miles per hour, he arrives in 3 hours. How many minutes would he save if he traveled at 60 miles per hour? Directions: For problems 7 8, determine whether the table shows direct or inverse variation, and find the constant of variation. 7.) A ) C B D Review: 9.) Find the center and radius of the following equation of a circle: x y 6x8y5 0 p61

62 10.) Solve for x: x 4 x 5 Answers: 1.) 15.) 1 3.) (4) 4.) 18 5.) ) 1 minutes 7.) Direct, k = 5/6 8.) Inverse, k = 60 9.) x 3 y ) { } p6

63 Day 1 Review 1.) Given the equation a) Graph the equation. Table of Values y x x 4 5 b) What is the vertex? c) What are the domain and the range? d) Is this a function? e) Is this onto, one-to-one, both, or neither? Directions: For problems -5, find the domain for each equation..) f( x) x x 6 3.) gx ( ) x 4 p63

64 1 4.) hx ( ) x 64 5.) f( x) 4 x 3x 1 6.) Given the functions f( x) x 9 and gx ( ) x 6. Find: a) f () b) g (-8) c) f ( g (4)) d) ( g f)(4) e) g (f (x)) f) ( f g)( x) p64

65 7.) Given the functions f( x) 3x and gx ( ) x 5. Find: a) f (x) + g (x) b) gx ( ) f( x) c) f ( x) g( x) d) ( g f)( x) Directions: For problems 8-9, find the inverse of each function. 8.) f( x) 3x 1 9.) gx ( ) x 4 Directions: Describe the movement of the vertex for each equation. 10.) f( x) ( x) 4 11.) gx ( ) ( x5) 4 p65

66 Directions: Given the center and radius, write the equation in standard form. 1.) Center: (, -3), Radius = 4 13.) Center: (1, 0), Radius = 5 Directions: Find the center and the radius. 14.) x y 4x8y ) x y x y 10 0 p66

67 Directions: Solve each question. 16.) Mr. O Leary was driving to Albany to write the Regents. If it took him 3 ½ hours to drive there at a speed of 55 miles per hour, how long did it take him to drive home if he hit traffic and could only drive 40 miles per hour? 17.) Mrs. Gladysz was filling the fishtank in her basement, which holds a total of 00 gallons. It took her 3 hours to fill the first 75 gallons. How long will it take to finish filling the tank? 18.) If x varies inversely with y, and x = 10 when y = 5, what is x when y = 0? 19.) If x varies directly with y, and x = 10 when y = 5, what is x when y = 0? p67