ca_m1 - MAC 1105 Module 1 Introduction to Functions and...

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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. Rev.S08 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. 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 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 What are Rational Numbers? Rational Numbers are real numbers which can be expressed as the Rational ratio of two integers p/q where q ≠ 0 Examples: 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. • Rational numbers can be expressed as decimals which either terminate (end) or repeat a sequence of digits. of 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 Irrational 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. 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 natural number. number. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 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 and n is an integer. and Examples: Examples: The distance to the sun is 93,000,000 mi. The In scientific notation for this is 9.3 x 107 mi. In 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 If we denote the ordered pairs by (x, y) If The set of all x − values is the DOMAIN. The The set of all y − values is the RANGE. The Example The relation {(1, 2), (− 2, 3), (− 4, − 4), (1, − 2), (− 3,0), (0, − 3)} The relation 4, 4), 3,0), has domain D = {− 4, − 3, − 2, 0, 1} has domain 4, 3, and 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 relation 4, 4), 3, 3)} has the following graph: has Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 12 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 outputs function for each input there is exactly one output. input exactly output. This is symbolized by output = f(input) or This output 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.” .” – Means that given a value of x (input), there is exactly one Means (input), exactly corresponding value of y (output). (output). – x is called the independent variable as it represents is independent inputs, and y is called the dependent variable as it dependent represents outputs. outputs – Note that: f(x) iis NOT f multiplied by x. f is NOT a s NOT Note variable, but the name of a function (the name of a the relationship between variables). relationship Rev.S08 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. of The set of corresponding outputs is called the RANGE The outputs RANGE of the function. of Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 15 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. range. The function may be defined by a set of ordered pairs, The a table, a graph, or an equation. table, 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. • 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 distance traveled), y is a function of x and we write y = f(x) = 70x • Evaluate f(3) and interpret. – f(3) = 70(3) = 210. This means that the car travels 210 f(3) miles in 3 hours. miles 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 Note: ordered pairs {(3,6), (4,6), (8, − 5)}. ordered • Yes, y is a function of x, because for each value of x, there is because there just one corresponding value of y. Using function notation we Using write f(3) = 6; f(4) = 6; f(8) = − 5. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 18 One More Example • • Undergraduate Classification at Study-Hard University (SHU) Undergraduate is a function of Hours Earned. We can write this in function Hours We notation as C = f(H). ). Why is C a function of H? Why – For each value of H there is exactly one corresponding For exactly value of C. value – In other words, for each input there is exactly one In each exactly corresponding output. output 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 No least 30 semester hours. least • No student may be classified as a junior until after earning at least No 60 hours. 60 • No student may be classified as a senior until after earning at least No 90 hours. 90 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), f(20) and f(61): Evaluate ), ), ), and ): – – – – • • f(20) = Freshman Freshman f(30) = Sophomore Sophomore f(0) = Freshman Freshman 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 One More Example (Cont.) Domain of f is the set of non-negative integers Alternatively, some individuals say the domain is the set of positive rational numbers, since technically one could positive since 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 non-negative numbers , but this set includes irrational numbers. numbers It is impossible to earn an irrational number of credit hours. For example, one could not earn hours. hours. Range of f is {Fr, Soph, Jr, Sr} Range Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 = y2, iis y a function of x? Why or why not? s Given • Given x = y2, iis x a function of y? Why or why not? s Given • Given y = x2 +7, iis y a function of x? Why, why not? s Given Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 value of H? value – No. One example is • if C = Freshman, then H could be 3 or 10 (or lots of other values for that matter) other Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 Identifying Functions (Cont.) • Is classification a function of years spent at SHU? That is, is Is C = f(Y)? • That is, for each value of Y is there just one corresponding That value of C? value – No. One example is • 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. taken. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 25 Identifying Functions (Cont.) • • Given x = y2, iis y a function of x? s Given That is, given a value of x, iis there just one corresponding s That value of y? – 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 = y2, iis x a function of y? s Given That is, given a value of y, is there just one corresponding That value of x? – Yes, given a value of y, there is just one corresponding Yes, value of x, namely y2. namely Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 27 Identifying Functions (Cont.) • • Given y = x2 +7, iis y a function of x? s Given That is, given a value of x, is there just one corresponding That value of y? – Yes, given a value of x, there is just one corresponding Yes, there value of y, namely x2 +7. namely Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 28 Five Ways to Represent a Function • • • • • Rev.S08 Verbally Numerically Diagrammaticly Symbolically Graphically http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 29 Verbal Representation • Rev.S08 Referring to the previous example: – If you have less than 30 hours, you are a freshman. If – If you have 30 or more hours, but less than 60 hours, If you are a sophomore. – If you have 60 or more hours, but less than 90 hours, If 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 Numeric Representation Rev.S08 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. 31 Symbolic Representation Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 32 iag Re r a p re mm s en a t ta t ic io n H Rev.S08 C 0 1 2 • • • 29 30 31 • • • 59 60 61 • • • 89 90 91 • • • Freshman Sophomore Junior Senior http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 33 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 horizontal outputs graphed on the vertical axis. vertical Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 34 Vertical Line Test • Another way to determine if a graph represents a function, Another simply visualize vertical lines in the xy-plane. If each vertical xy-plane. line intersects a graph at no more than one point, then it is line then the graph of a function. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 35 What is a Constant Function? A function f represented by f(x) = b, function where b is a constant (fixed number), is a constant function. constant f(x) = 2 Examples: Examples: Note: Graph of a constant function is a horizontal line. constant horizontal Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 36 What is a Linear Function? A function f represented by f(x) = ax + b, function ax where a and b are constants, iis a linear function. constants s linear (It will be an identity function, if constant a = 1 and constant b = 0.) (It 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 37 Rate of Change of a Linear Function xy −2 −1 −1 1 03 15 27 39 Rev.S08 Table of values for f(x) = 2x + 3. • Note throughout the table, as x Note increases by 1 unit, y increases by 2 units. In other words, the RATE OF CHANGE of y with respect to x is CHANGE 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 change of a linear function is SLOPE. change http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 38 The Slope of a Line • The slope m of the line passing through the points (x1, y1) and The of and (x2, y2) is Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 39 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) (3, (-2, -1) • The slope being 2 means that for each unit x increases, the The slope corresponding increase in y is 2. The rate of change of y with rate respect to x is 2/1 or 2. respect Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 40 How to Write the Point-Slope Form of the Equation of a Line? The line with slope m passing through the point (x1, y1) has equation The point y = m(x − x 1) + y 1 or or y − y 1 = m(x − x 1) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 41 How to Write the Equation of the Line Passing How 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 the slope m and a point (x1, y1) are needed. the slope point Let (x1, y1) = (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 Slope-Intercept Form The line with slope 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? ½? Since the point (0, − 2) has an x-coordinate of 0, the point is a ySince 2) -coordinate iintercept. Thus b = − 2 ntercept. Using slope-intercept form Using slope-intercept y=mx+b the equation is the 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 The y− axis at (0,3) so the axis 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 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 for a line. line. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 46 How to Find x-Intercept and y-intercept? • • To find the x-intercept, let -intercept, y = 0 and solve for x. – 2x – 3(0) = 6 – 2x = 6 – x=3 To find the y-intercept, let -intercept, x = 0 and solve for y. – 2(0) – 3y = 6 – –3y = 6 – y = –2 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. (0, 2) (3, 0) 47 What are the Characteristics of Horizontal Lines? Slope is 0, since Δy = 0 and m = Δy / Δx since Equation is: y = mx + b mx y = (0)x + b y = b where 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 48 What are the Characteristics of Vertical Lines? • • Slope is undefined, since Δx = 0 and m = Δy /Δx since Example: • • • Rev.S08 Note that regardless of the value Note of y, the value of x is always 3. the Equation is x = 3 (or x + 0y = 3) Equation Equation of a vertical line is x = k Equation where k is the x-intercept. 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 Parallel same slant, thus they have same thus the same slopes. same Rev.S08 Perpendicular lines have slopes which are negative reciprocals which (unless one line is vertical!) (unless 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 slope of any line perpendicular to y = − 4x – 2 is ¼ The (− 4 and ¼ are negative reciprocals) and 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: point-slope y = m(x − x1) + y1 y = (1/4)(x − 3) + (− 1) y = − 4x – 2 In slope-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 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 25 5 75 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. rate This means that for an increase of 1 ton of rock the price increases by $25. increases Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 52 Example of a Nonlinear Function x 0 1 2 y 0 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 increases from 1 to 2, y increases by 3 units. 1 does not equal 3. This function does NOT have a CONSTANT RATE OF CHANGE of does CONSTANT y with respect to x, so the function is NOT LINEAR. so NOT Note that the graph is not a line. not Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 53 Average Rate of Change Let (x1, y1) and (x2, y2) be distinct points on the graph of a function f. The average rate of graph The change of f from x1 to x2 is Note that the average rate of change of f from x1 to x2 iis the slope of the line passing through s (x1, y1) and (x2, y2) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 54 What is the Difference Quotient? The difference quotient of a function f is an The difference expression of the form where h is not 0. Note that a difference quotient is actually difference an average rate of change. average Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 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 ...
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

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