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ca_m1_handouts - MAC 1105 Module 1 Introduction to...

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Unformatted text preview: MAC 1105 Module 1 Introduction to Functions and Graphs Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Recognize common sets of numbers. Understand scientific notation and use it in applications. Find the domain and range of a relation. Graph a relation in the xy-plane. Understand function notation. Define a function formally. Identify the domain and range of a function. Identify functions. Identify and use constant and linear functions. Interpret slope as a rate of change. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 2 Learning Objectives 11. 12. Write the point-slope and slope-intercept forms for a line. Find the intercepts of a line. 13. Write equations for horizontal, vertical, parallel, and perpendicular lines. 14. Write equations in standard form. 15. 16. 17. 18. Identify and use nonlinear functions. Recognize linear and nonlinear data. Use and interpret average rate of change. Calculate the difference quotient. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 3 1 Introduction to Functions and Graphs There are four major topics in this module: - Functions and Models - Graphs of Functions - Linear Functions - Equations of Lines Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 4 Let’s get started by recognizing some common set of numbers. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 5 What is the difference between Natural Numbers and Integers? •Natural Numbers (or counting numbers) are numbers in the set N = {1, 2, 3, ...}. are •Integers are numbers in the set I = {… −3, − 2 , − 1, 0, 1, 2, 3, ...}. 3, 2, Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 6 2 What are Rational Numbers? Rational Numbers are real numbers which can be expressed as the ratio of two integers p/ q where q ≠ 0 ratio Examples: 0.5 = ½ 3 = 3/1 3/1 − 5 = − 10/2 0.52 = 52/100 0 = 0/2 0.333… = 1/3 Note that: • Every integer is a rational number. E very • Rational numbers can be expressed as decimals R ational which either terminate (end) or repeat a sequence of digits. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 7 What are Irrational Numbers? • Irrational Numbers are real numbers which are not rational numbers. • Irrational numbers Cannot be expressed as the ratio of two Cannot integers. integers. • Have a decimal representation which does not does terminate and does not repeat a sequence of digits. terminate does Examples: Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 8 Classifying Real Numbers Classify each number as one or more of the following: Classify natural number, integer, rational number, irrational number. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 3 Let’s Look at Scientific Notation A real number r is in scientific notation real scientific when r is written as c x 10n, where when and n is an integer. and Examples: The distance to the sun is 93,000,000 mi. In scientific notation for this is 9.3 x 107 mi. The size of a typical virus is .000005 cm. In scientific notation for this is 5 x 10−6 cm. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 10 What is a Relation? What are Domain and Range? A relation is a set of ordered pairs. relation is If we denote the ordered pairs by (x , y) The s et of all x − values is the DOMAIN. The The s et of all y − values is the RANGE. The Example The relation {(1, 2), (− 2, 3), (− 4, − 4 ), (1, − 2), (− 3 ,0), (0, − 3)} The r elation 4, 4), 3,0), h as domain D = {− 4 , − 3 , − 2, 0, 1} has d omain 4, 3, a nd range R = {− 4 , − 3, − 2, 0, 2, 3} and range 4, 3, Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 11 How to Represent a Relation in a Graph? The relation {(1, 2), (− 2, 3), (− 4, − 4 ), (1, − 2), (− 3 , 0), (0, − 3)} The r elation 4, 4), 3, has the following graph: Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 12 4 Is Function a Relation? Recall that a relation is a set of ordered pairs (x , y) . Recall set If we think of values of x as being inputs and values of y If and as being outputs, a function is a relation such that as outputs function for each input there is exactly one output. for input exactly o utput. This is symbolized by output = f (input) or o utput y = f (x ) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 13 Function Notation y = f( x) – Is pronounced “ y is a function of x.” Is – Means that given a value of x (input), there is exactly one Means ( input), e xactly corresponding v alue of y (output). corresponding (output). – – Rev.S08 x is called the independent variable as it represents is i ndependent iinputs, a nd y is called the dependent variable as it nputs, dependent represents outputs. represents outputs Note that: f(x ) iis NOT f multiplied by x . f is NOT a s NOT Note is variable, but the name of a function (the name of a variable, t he relationship between variables). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 14 What are Domain and Range? The set of all meaningful inputs is called the DOMAIN The inputs DOMAIN of the function. The set of corresponding outputs is called the RANGE The outputs RANGE of the function. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 15 5 What is a Function? A function is a relation in which each element of the function domain corresponds to exactly one element in the range. The function may be defined by a set of ordered pairs, a table, a graph, or an equation. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 16 Here is an Example • Suppose a car travels at 70 miles per hour. Let y be the Suppose distance the car travels in x hours. Then y = 70 x. distance • Since for each value of x (that is the time in hours the car Since travels) there is just one corresponding value of y (that is the travels) distance traveled), y is a function of x and we write distance y = f ( x ) = 70x • Evaluate f(3) and interpret. – f(3) = 70(3) = 210. This means that the car travels 210 miles in 3 hours. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 17 Here is Another Example • Given the following data, is y a function of x ? Given – Input x 3 4 8 Input – Output y 6 6 −5 Output • Note: The data in the table can be written as the set of ordered pairs {(3, 6), (4 ,6), (8, − 5)}. • Yes, y is a function of x, because for each value of x , there is because jjust one corresponding value of y . Using function notation we ust write f(3) = 6 ; f( 4) = 6; f ( 8) = − 5. write Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 18 6 One More Example • • Undergraduate Classification at Study-Hard University (SHU) Undergraduate at iis a function of Hours Earned. We can write this in function s H ours notation as C = f ( H). notation Why is C a function of H? Why For each value of H there is exactly one corresponding For exactly value of C. – In other words, for each input there is exactly one In e ach exactly corresponding output. corresponding o utput – Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 19 One More Example (Cont.) • Here is the classification of students at SHU (from catalogue): • No student may be classified as a sophomore until after earning at least 30 semester hours. • No student may be classified as a junior until after earning at least 60 hours. • No student may be classified as a senior until after earning at least 90 hours. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 20 One More Example (Cont.) • Remember C = f (H) Remember • Evaluate f ( 20), f ( 30), f (0 ), Evaluate ), ), ), – – – – • • f ( 20) and f(61): and f( 20) = Freshman Freshman f( 30) = Sophomore Sophomore f( 0 ) = Freshman F reshman f( 61) = Junior Junior What is the domain of f? What domain What is the range of f? What range Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 21 7 One More Example (Cont.) Domain of f is the set of non-negative integers n on-negative Alternatively, some individuals say the domain is the set of positive rational numbers, since technically one could earn a fractional number of hours if they transferred in some quarter hours. For example, 4 quarter hours = 2 2/3 semester hours. Some might say the domain is the set of non-negative real Some n on-negative r eal numbers , but this set includes irrational numbers. It is impossible to earn an irrational number of credit hours. For example, one could not earn hours. Range of f is {Fr, Soph, Jr, Sr} R ange http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 22 Identifying Functions Referring to the previous example concerning SHU, is hours earned a function of classification? That is, is H = f (C )? Explain why or why not. Is classification a function of years spent at SHU? Why or why not? • • • Given x = y 2, iis y a function of x? Why or why not? s Given Given x = y 2, iis x a function of y? Why or why not? s Given Given y = x 2 +7, iis y a function of x? Why, why not? s Given http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 23 Identifying Functions (Cont.) • Is hours earned a function of classification? That is, is H = Is f( C )? • That is, for each value of C is there just one corresponding That is value of H? – No. One example is • Rev.S08 if C = Freshman, then H could be 3 or 10 (or lots of other values for that matter) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 8 Identifying Functions (Cont.) • • Is classification a function of years spent at SHU? That is, is C = f(Y )? That is, for each value of Y is there just one corresponding That is value of C? – No. One example is • Rev.S08 iif Y = 4, then C could be Sr. or Jr. It could be Jr if a f student was a part time student and full loads were not taken. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 25 Identifying Functions (Cont.) • • Given x = y 2 , iis y a function of x ? s Given That is, g iven a value of x , is there just one corresponding That value of y ? value – No, if x = 4, then y = 2 or y = − 2. No, Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 26 Identifying Functions (Cont.) • Given x = y 2 , iis x a function of y ? s Given • That is, given a value of y, is there just one corresponding value of x ? value – Rev.S08 Yes, given a value of y, there is just one corresponding value of x , namely y 2. value namely http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 27 9 Identifying Functions (Cont.) • Given y = x 2 + 7, iis y a function of x ? s Given • That is, given a value of x, is there just one corresponding value of y ? value – Yes, given a value of x , there is just one corresponding Yes, value of y , namely x 2 + 7 . value namely http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 28 Five Ways to Represent a Function • • Verbally Numerically • • • Diagrammaticly Symbolically Graphically http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 29 Verbal Representation • Referring to the previous example: If you have less than 30 hours, you are a freshman. – – – – Rev.S08 If you have 30 or more hours, but less than 60 hours, you are a sophomore. If you have 60 or more hours, but less than 90 hours, you are a junior. If you have 90 or more hours, you are a senior. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 30 10 Numeric Representation H 0 1 ? ? ? ? 29 30 31 ? ? ? 59 60 61 ? ? ? 89 90 91 ? ? ? C Freshman Freshman Freshman Sophomore Sophomore Sophomore Junior Junior Junior Senior Senior http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 31 Symbolic Representation http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 Di R e ag ra p re m m sen at ta t ic io n H Rev.S08 0 1 2 • • • 29 30 31 • • • 59 60 61 • • • 89 90 91 • • • 32 C Freshman Sophomore Junior Senior http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 33 11 Graphical Representation • In this graph the domain is considered to be In domain • instead of {0,1,2,3… }, and note that inputs are typically inputs graphed on the horizontal axis and outputs are typically graphed horizontal o utputs graphed on the vertical axis. graphed vertical http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 34 Vertical Line Test • Another way to determine if a graph represents a function, simply visualize vertical lines in the xy- plane. If each vertical simply xy-plane. If line intersects a graph at no more than one point , then it is the graph of a function. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 35 What is a Constant Function? A function f represented by f ( x ) = b, function w here b is a constant (fixed number), is a where constant function. f(x ) = 2 Examples: Note: Graph of a constant function is a horizontal line . Note: constant horizontal Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 36 12 What is a Linear Function? A function f represented by f ( x ) = ax + b, function ax w here a and b are constants, iis a linear function. where constants s linear (It will be an identity function, if constant a = 1 and constant b = 0.) Examples: f(x ) = 2x + 3 Note that a f(x ) = 2 iis both a linear function and a constant function. s Note linear constant A constant function is a special case of a linear function . constant special http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 37 Rate of Change of a Linear Function x y −2 −1 −1 1 0 1 2 3 3 5 7 9 Rev.S08 Table of values for f ( x ) = 2x + 3. Table • Note throughout the table, as x increases by 1 unit, y increases by 2 units. In other words, the RATE OF units. RATE CHANGE of y with respect to x is constantly 2 throughout the table. Since the rate of change of y with respect to x is constant, the function is LINEAR. Another name for rate of LINEAR. A nother change of a linear function is SLOPE. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 38 The Slope of a Line • The s lope m of the line passing through the points (x 1 , y 1) and The of (x2, y2) is Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 39 13 Example of Calculation of Slope • Find the slope of the line passing through the Find slope points (− 2, − 1) and (3, 9). 2, (3, 9) (-2, -1) • The slope being 2 means that for each unit x increases, the The s lope corresponding increase in y is 2. The rate of change of y with corresponding rate respect to x is 2/1 or 2. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 40 How to Write the Point-Slope Form of the Equation of a Line? The line with slope m passing through the point (x 1, y 1 ) has equation The p oint y = m ( x − x1) + y1 or or y − y 1 = m ( x − x1) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 41 How to Write the Equation of the Line Passing Through the Points (−4, 2) and (3, − 5)? Through 4, To write the equation of the line using point-slope form To point-slope y = m ( x − x 1) + y 1 t he slope m and a point ( x 1, y 1 ) are needed. the slope point Let (x 1, y 1) = (3, − 5). Calculate m using the two given points. Calculate Equation is Equation This simplifies to This Rev.S08 y = − 1 (x − 3 ) + (− 5) y = −x + 3 + (− 5) y = −x − 2 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 42 14 Slope-Intercept Form The line with s lope m and y-intercept b is given by The -intercept – y=mx+b Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 43 How to Write the Equation of a line passing through the point (0,-2) with slope ½? Since the point (0, − 2) has an x -coordinate of 0, the point is a y Since 2) -coordinate iintercept. Thus b = − 2 ntercept. Using slope-intercept form Using slope-intercept y=mx+b the equation is y = (½ ) x − 2 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 44 How to Write an Equation of a Linear Function in Slope-Intercept Form? • – • What is the slope? As x increases by 4 units, y decreases by 3 units so the slope is − 3/4 What is the y-intercept? – The graph crosses the y− axis at (0,3) so the y− iintercept is 3. ntercept • – – What is the equation? Equation is f ( x ) = (− ¾ ) x + 3 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 45 15 What is the Standard Form for the Equation of a Line? ax + by = c iis standard form (or general form) for the equation of s standard general a line. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 46 How to Find x-Intercept and y-intercept? • • To find the x-intercept, let -intercept, l et y = 0 and solve for x. – 2x – 3(0) = 6 – 2x = 6 – x=3 To find the y-intercept, let -intercept, l et x = 0 and solve for y. – 2(0) – 3y = 6 2(0) – –3 y = 6 – y = –2 (0, 2) (3, 0) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 47 What are the Characteristics of Horizontal Lines? Slope is 0, since Δy = 0 and m = Δy / Δx since Equation is: y = mx + b Equation mx y = (0)x + b y = b w here b is the y-intercept Example: y = 3 (or 0x + y = 3) (or (-3, 3) Rev.S08 (3, 3) Note that regardless of the value of x, the value the of y is always 3. of http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 48 16 What are the Characteristics of Vertical Lines? • • Slope is undefined, since Δ x = 0 and m = Δy /Δ x since Example: • Note that regardless of the value of y, the value of x is always 3. of the • Equation is x = 3 (or x + 0 y = 3) Equation • Equation of a vertical line is x = k Equation where k is the x -intercept. where Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 49 What Are the Differences Between Parallel and Perpendicular Lines? • Parallel lines have the same slant, thus they have the same slopes. the s ame Rev.S08 Perpendicular lines have slopes Perpendicular slopes which are negative reciprocals (unless one line is vertical!) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 50 How to Find the Equation of the Line Perpendicular to y = -4x - 2 Through the Point (3,-1)? The The slope of any line perpendicular to y = −4x – 2 is ¼ (−4 a nd ¼ are negative reciprocals) and are Since Since we know the slope of the line and a point on the line we can use slope point point-slope form of the equation of a line: y = m (x − x 1 ) + y1 y = ( 1/4) ( x − 3) + (− 1) y = − 4x – 2 In s lope-intercept form: slope-intercept y = ( 1/4)x − (3/4) + (− 1) y = ( 1/4)x − 7/4 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. y = (1/4)x − 7/4 51 17 Example of a Linear Function The table and corresponding graph show the price y of x The tons of landscape rock. X (tons) y (price in dollars) 25 5 75 4 100 y is a linear function of x and the slope is The rate of change of price y with respect to tonage x is 25 to 1. The r ate This means that for an increase of 1 ton of rock the price increases by $25. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 52 Example of a Nonlinear Function x y 0 0 1 2 1 4 Table of values for f ( x ) = x2 Table Note that as x increases from 0 to 1, y increases by 1 unit; while as x Note iincreases from 1 to 2, y increases by 3 units. 1 does not equal 3. ncreases This function does NOT have a CONSTANT RATE OF CHANGE of This does C ONSTANT y with respect to x , so the function is NOT LINEAR. so N OT Note that the graph is not a line. Note n ot Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 53 Average Rate of Change Let (x1, y1 ) and (x2, y 2 ) be distinct points on the g raph graph of a function f. The average rate of The average c hange change of f from x 1 to x2 is Note Note that the average rate of change of f from x1 to x 2 is is the s lope of the line passing through slope (x1, y 1 ) and (x2 , y 2 ) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 54 18 What is the Difference Quotient? The difference quotient of a function f is an The d ifference expression of the form w here h is not 0. where Note that a difference quotient is actually Note difference an average rate of change. an a verage http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 55 What have we learned? We have learned to: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Recognize common sets of numbers. Understand scientific notation and use it in applications. Find the domain and range of a relation. Graph a relation in the xy-plane. Understand function notation. Define a function formally. Identify the domain and range of a function. Identify functions. Identify and use constant and linear functions. Interpret slope as a rate of change. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 56 What have we learned? (Cont.) 11. 12. Write the point-slope and slope-intercept forms for a line. Find the intercepts of a line. 13. Write equations for horizontal, vertical, parallel, and perpendicular lines. 14. Write equations in standard form. 15. 16. 17. 18. Identify and use nonlinear functions. Recognize linear and nonlinear data. Use and interpret average rate of change. Calculate the difference quotient. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 57 19 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: • Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 58 20 ...
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