Class Journal - Math 1301 (20984)

Spring 2008

What I hear, I forget; what I see, I remember; what I do, I understand. – Kung Fu Tzu (Confucius)

One learns the thing by doing the thing; for though you think you know it,

You have no certainty until you try. – Sophocles

Session	Date	Read & Study Section	Discussion Topics	Practice Exercises	Other Info
	5 - 1		Final Exam (comprehensive) Thursday, May 1, 8:30 – 11 AM Bring a #2 pencil and a scientific calculator.		You may get information about Final Exam Review sessions and download a copy of the Final Exam Review at the UHD Algebra Student web site http://cms.uhd.edu/qep/algebra
28	4 - 24	5.3 5.4	Section 5.3&5.4 Every power of 2 is a positive number – no power of 2 equals zero and no power of 2 equals a negative number. Therefore log2(-8) is not a real number because there is no exponent on 2 that gives a negative number and the domain of the log function y=log2 (x) are the positive numbers. Note: (1) a function and its inverse undo each other, that is, f^-1(f(x))=x; (2) the inverse of the function f(x)=(b)^x is f^-1(x)= logb x and so logb(b)^x=x. Look at this last equation closely: logs can be used to bring the exponent x down. This property is helpful in solving equations where the variable is an exponent: take the log of each side to help solve the equation. Study example 12 on page 428. There are many applications of exponential and logarithm functions – study the examples in sections 5.3 and 5.4.	See below.	Have you been contacted by UHD to complete the NSSE survey? Click here for more information and a chance to win an i-pod.
27	4 - 22	5.3 5.4	Numbers are used to measure and quantify data. Functions are used to model and describe the relationship between the input quantity and output quantity. Section 5.3 If the initial output of a function is 3 and the output doubles per unit increase in x, this information is captured by the exponential function f(x)=3(2)^x. The general formula of an exponential function is f(x)=C(a)^x where the coefficient C is the initial value of the function when x=0 and the base a is the multiplier of the output y per unit increase in x. See page 400. Note: (1) when the base a>0, the function increases and the graph rises from left to right – called exponential growth; (2) when the base 0<a<1, the function decreases and the graph falls from left to right – called exponential decay. See the properties of exponential function graphs on pages 403-404. By the order of operations, exponents are done before multiplication and so f(2)=3(2)^2=3(4)=12. You should review special exponents in this section: (2)^0= 1 ; (2)^-3=1/(2^3)=1/8; (4)^3/2=(sqrt(4))^3=2^3=8; (1/2 )^-1=(2/1)^1=2; (1/2 )^-2=(2/1)^2=4. Carefully study the examples in section 5.3. See page 408: the most special base of an exponential function is the number e≈2.72. Section 5.4 Every exponential function f(x)=C(a)^x is one-to-one and so has an inverse. The inverse of the exponential function f(x)=b^x with base b is the logarithm function with base b which is written as logb x “log base b of x”. Note (1) inverse functions switch inputs and outputs; (2) the input to an exponential function is an exponent; (3) the output from a logarithm is an exponent. Let’s repeat this: logarithms are exponents. Therefore log2 8 equals the exponent on the base 2 that gives 8 and so log2 8 = 3. The most special bases of logarithm functions are base 10 called the common logarithm and base e called the natural logarithm. Study the examples in section 5.4.	Section 5.3 # 1-17 odd, 18, 19, 25, 31, 41, 43-49 odd, 55, 63, 99 Section 5.4 # 1, 3-6, 13, 19-59 odd, 69-75 odd, 111, 119, 121	MML homework for sections 5.3 and 5.4 are now available. You may get information about Final Exam Review sessions and download a copy of the Final Exam Review at the UHD Algebra Student web site http://cms.uhd.edu/qep/algebra
26	4 - 17		Test 3 (Sections 3.2, 3.3, 4.1, 4.2, 4.4, 5.1, 5.2) Bring a scientific calculator.		Here some instructions for MWMilton: Enter Table/Find Fitting Line
25	4 – 15	5.1&5.2	Section 5.1&5.2 Inverse functions have many properties. (1) If the function f is one-to-one, then it has an inverse that is written as f^-1(x), read as “f inverse of x”. (2) The function f changes x to f(x), and the function f^-1 changes f(x) back to x, that is, f^-1(f(x))=x. The reverse is also true, that is, the function f^-1 changes x to f^-1(x) and the function f changes f^-1(x) back to x, or f(f^-1(x))=x. (3) Inverse functions swap inputs and outputs. Study example 5 on page 387: f(height)=crutch length and f^-1(crutch length)=height. This means f(50) stands for the crutch length needed when a person is 50 inches tall, however, f^-1(50) stands for the height of a person who needs crutches of length 50 inches. Note in example 1b on page 383 that it is possible to think through the inverse of a function: the function rule (x-7)/2 “subtract 7 from the input and then divide by 2” has inverse “multiply the input by 2 and then add 7” or 2x+7.	See below.	MyMathLab homework assignments for Test 3 must be completed by midnight Wednesday, April 16.
24	4 – 10	4.4 5.1&5.2	Section 4.4 The solutions to a quadratic equation may be complex imaginary solutions: the solutions can be written to include the number i. Study example 2 on page 295. Note that to find the x-intercepts of a function y=f(x), you substitute 0 for y and solve an equation. Only the solutions to the equation that are real numbers will be x-intercepts on the graph – this means that if all solutions to the equation are complex imaginary then the graph will have no x-intercepts. See example 3 on page 297. Section 5.1&5.2 There are problems where the output from one function is used as the input to another function – this is called composition of functions. The notation g(f(x)), “g of f of x”, means to use the output f(x) as input to the function g. Composition is also written as g◦f, “g composition f”. Given the formulas for f and g separately, to calculate g(f(2)): 1^st, calculate the output f(2); 2^nd, input the value of f(2) into the formula for g. The formula for the composition of the functions can be found too. Study example 7 on page 370. A function f changes a number x into f(x) – is there a way to change f(x) back to x? This is the idea behind an inverse function. Only one-to-one functions (no repeated outputs) can be reversed and will have an inverse. See the steps on page 387.	Section 5.1 # 39, 53, 61, 63-71odd,79 Section 5.2 # 1, 3, 7, 8, 13, 15, 17, 19, 23, 25, 45, 51, 71, 72, 77, 103, 107, 122, 127	MML 5.1&5.2 is now available.
23	4 – 8	4.1&4.2 4.4	Section 4.1&4.2 There are four possibilities for the end behavior of any polynomial function, looking from the origin out: the ends go in opposite directions with the left falling and the right rising like f(x)=x, the ends go in opposite directions with the left rising and the right falling like f(x)=-x, the ends go in the same direction with the left rising and the right rising, like f(x)=x^2, or the ends go in the same direction with the left falling and the right falling, like f(x)=-x^2. Click here to see sketches of the four possible graphs. Note the four possible graphs correspond to: a polynomial with odd degree and positive leading coefficient, a polynomial with odd degree and negative leading coefficient, a polynomial with even degree and positive leading coefficient, and a polynomial with even degree and negative leading coefficient. If you plot a table of data, you can choose the degree of the polynomial that will model the data by inspecting the number of turning points and x-intercepts. Study problems 41-44 on page 270. Section 4.4 The complex number i equals the square root of -1 and has the property that i^2=-1. The standard form of a complex number is a+bi where a and b are real numbers. See page 293. Also, the square root of a negative number is a complex imaginary number: √(-a)=i√a. For example: √(-9)=i√9=3i. Complex numbers may be added, subtracted, multiplied and divided: study pages 293-294. It is important to remember that you should substitute -1 for i^2 when calculating with complex numbers. A quadratic equation may have a complex imaginary solution. Study example 2 on page 295.	See below.	MML for 4.4 is now available. Have you been contacted by UHD to complete the NSSE survey? Click here for more information.
22	4 – 3	3.2 4.1&4.2	Section 3.2 The domain of a function is the set of all x-values or the set of all input numbers for the function. If a number is in the domain of a function, then the matching output must be a real number. Consider problem 83 on page 203: find the domain of the function g(t)=(5-t)/( t^2-t-2). 1^st, examine the function formula: there is a variable in the denominator. 2^nd, since division by zero is not defined, set the denominator=0 to find the numbers to omit from the domain. 3^rd, the solutions to t^2-t-2=0 are t=-1 and t=2. Therefore, the domain is all real numbers except -1 and 2. Section 4.1&4.2 When a graph is rising from left to right, we say the function is increasing on the matching interval of x-numbers. When a graph is falling from left to right, we say the function is decreasing on the matching interval of x-numbers. Study example 1 on page 244. A turning point occurs whenever the graph changes from increasing to decreasing or from decreasing to increasing. If a polynomial function has degree=n, then the function graph has n or fewer x-intercepts and it has (n-1) or fewer number of turning points. Therefore, in advance of seeing the graph, you can predict the most number of x-intercepts and the most number of turning points. See problem 13 on page 269. Given the graph of a polynomial function first, you can go in reverse and determine the minimum degree of the polynomial. See problems 5 and 6 on page 268. The maximum value of a function is the y-value of the highest point on the graph of the function. The minimum value of a function is the y-value of the lowest point on the graph of the function. When the turning point is a valley point it is also called a local minimum, and when the turning point is a peak point, it is also called a local maximum. Maximum and minimum values (also called extrema) have practical importance: see problems 120 and 121 on page 255.	Section 4.4 Complex numbers # 1-23 odd Quadratic equations # 47, 49, 51, 55 Number of real zeros # 63, 64, 65
21	4 - 1	3.3 4.1&4.2	Section 3.3 Carefully study example 1 on page 207: graphically solve the quadratic inequality ax^2+bx+c<0 by using the graph of the related function f(x)= ax^2+bx+c or y= ax^2+bx+c. 1^st, rewrite the problem from “find all x so ax^2+bx+c<0” to “find all x so y<0; 2^nd, locate the x-intercepts x=-1 and x=2 where y=0; 3^rd, note that the x-intercepts break up the x-axis into three intervals and on each interval the matching y-values on the graph are always positive or always negative (when x<-1, y>0, when -1<x<2, y<0, when x>2, y>0); 4^th, altogether, y<0 when -1<x<2 gives the solution. Study the application examples: example 3 (safe speeds) and example 5 (heart rates). Section 4.1&4.2 Constant functions, linear functions and quadratic functions are also called polynomial functions; see page 242. Note that a polynomial function can be written as a sum of terms where each term has the variable with an exponent that is a nonnegative integer 0, 1, 2, ...; the variable cannot have a negative exponent (appear in a denominator) and the variable cannot have a fractional exponent (appear under a root). The highest power on the variable is called the degree of the polynomial and the corresponding coefficient of the term with the highest power of the variable is called the leading coefficient of the polynomial. See page 243 for examples. The graph of a polynomial function is a straight line or an unbroken curve with no sharp turning points.	Section 4.1 Polynomial functions # 3-9 odd Intervals where increasing or decreasing # 11 – 23 odd, 29, 31 Turning points # 39, 41, 43, 49, 55, 57 Application # 121 Section 4.2 Turning points # 1, 5, 7, 11, 12, 13, 15, Degree, end behavior # 19, 21, 26, Conjecture degree # 41 Piecewise-defined functions # 69, 71, 75, 77 Application # 81	MML for 4.1 & 4.2 is now available.
20	3 - 27	3.2 3.3	Section 3.2 Note that to find the x-intercepts of the quadratic function y= ax^2+bx+c, you set y=0 and get 0= ax^2+bx+c, a quadratic equation. This means the number of x-intercepts equals the number of solutions to the quadratic equation that are real numbers: if the parabola has two x-intercepts, then the quadratic equation has two real number solutions; if the parabola has one x-intercept, then the quadratic equation has one real number solution; if the parabola has no x-intercepts, then the quadratic equation has no real number solutions. See page 196. After you use the quadratic formula to find the x-intercepts exactly, you can use a calculator to find the x-intercepts approximately. See page 195 and 197. Problem 97, height of a baseball, was discussed in class. Section 3.3 If you take a quadratic equation and replace the = by an inequality <, >, <, >, you will have a quadratic inequality. We will solve quadratic inequalities by using only the graphing method. Carefully study example 1 on page 207.	Section 3.3 Solve inequality graphically # 1, 5, 7-17 odd, 27 - 35odd, 39, 42, 45-47, Application # 61	MML for section 3.3 is now available.
19	3 - 25	3.2	Section 3.2 A quadratic equation can be written in the form ax^2+bx+c=0 where the coefficient of the square term, a, cannot be 0. There are different methods to solve a quadratic equation: (1) solve by factoring – one side must be a product and the other side 0 like in example 1 on page 190; (2) solve by the square root property – one side must be a squared quantity like x^2 or (x+3)^2 and the other side must be a number like in example 4 on page 192; (3) solve by the quadratic formula – one side must have the form ax^2+bx+c and the other side must be 0 like in example 7 on page 194; (4) solve by graphing – study example 8 on page 195.	Section 3.2 Solve quadratic equation # 1, 3, 5, 11, 13, 15, 19, 21 Find x-intercepts # 25, 27 Solve graphically # 31, 33 Find the domain # 81, 83 Literal equations # 93, 95, Applications # 99, 105	MML for section 3.2 is now available.
18	3 - 13		Test 2 (Sections 2.2, 2.3, 2.4, 2.5, 3.1) Bring a scientific calculator.		MyMathLab homework assignments for Test 2 must be completed by midnight Wednesday, March 12.
17	3 – 11	2.5 3.1	Section 2.5 Some absolute value problems were done to review for the test. Section 3.1 The formula of a quadratic function can be written in the form f(x)=ax^2+bx+c, where a is not zero. The graph of a quadratic function has a shape that is called a parabola. The formula to find the x-coordinate of the vertex is on page 177: x = -b/2a. To find the y-coordinate of the vertex , you then substitute the x-coordinate of the vertex into the function formula. Study example 5. Note in applications of quadratic functions that the maximum or minimum output in the range occurs at the vertex. Study example 9: the flight of a baseball. Note that the scatter plot of a data set may fall more or less along a parabola: to find the equation of the parabola, you can use the MWMilton software on the CD.	See below.
16	3 - 6	2.5 3.1	Section 2.5 (1) A piecewise-defined function f(x) has more than one formula; (2) each formula has a matching interval of numbers; (3) to calculate f(x), first you find the interval of numbers that x belongs to, second, you substitute x into the matching formula. The domain is the union of all of the intervals of numbers that belong to each formula. The graph consists of pieces of graphs; each formula typically adds a distinct piece to the graph. Study example 2 on page 141. To solve an absolute value problem using algebra, it is helpful to think of the absolute value of a number as the distance from zero to the number on the number line. Study example 5 on page 146 and example 8 on page 148. Section 3.1 The formula of a quadratic function can be written in the form f(x)=ax^2+bx+c, where a is not zero. The graph of a quadratic function has a shape that is called a parabola; the parabola will open up, a U-shape, if a>0 and will open down, a ∩-shape, if a<0. The tip of the parabola is called the vertex and the vertical line that goes through the vertex is called the axis of symmetry. There is a formula to find the vertex: study page 177.	Section 3.1 Quadratic function # 1-7 odd Graph of quadratic function # 9 – 15 odd Vertex # 17, 19, 29 Sketch graph # 59-75odd Applications # 79, 83, 85-88
15	3 - 4	2.4 2.5	Section 2.3 Another example was done in class to find the equation of a fitting line by using the software MWMilton that is on the CD given to every student. Section 2.4 The strategy to solve a linear inequality is the same as the strategy to solve a linear equation: use algebra to isolate the variable on one side of the inequality. The Properties of Inequalities that you use to solve an inequality are on page 126: you may add or subtract the same number from each side of an equality; you may multiply or divide by a positive number to each side of an inequality; you may multiply or divide by a negative number to each side of an inequality but you must reverse the inequality. Study example 1 on page 126 to see how inequalities are solved. Solutions to linear inequalities are often written using interval notation: see page 125 for details. A compound inequality or 3-part inequality is a shorthand way to write two inequalities connected by the word and. The goal in a compound inequality is to isolate the variable in the middle: you must remember to add/multiply the same number to the left part, to the middle part, and to the right part of the inequality. Study example 6 on page 130 to see how this is done. Section 2.5 A piecewise-defined function is a function with more than one rule, but each rule is only used for certain x-values. Study example 2 on page 141 to start understanding how piecewise-defined functions work.	Section 2.4 Interval notation # 1-11 odd Solve the inequality # 13 - 31odd, 35 Applications # 88, 91,101 Section 2.5 Piecewise-defined function # 3, 7, 9, 11, 15, 19, 20, 22 Absolute value # 41, 45, 51, 71, 73, 77, 79, 87, Application # 103	MML homeworks 2.4, 2.5 and 3.1 are now available.
14	2 - 28	2.3	Section 2.3 More examples were done in class of solving linear equations using two methods: solve symbolically and solve graphically. Study problem 49: use the graph of f(x) to solve each equation f(x)=-1, f(x)=0, and f(x)=2. In this problem, remember that y=f(x) and to solve f(x)=-1, rewrite it as y=-1; then find the point on the graph with y=-1 and find the matching x to get the answer. Study example 3 to review how you can symbolically solve a linear equation with fractions. Sometimes a problem involves table data that is approximately linear: when you plot the table the points fall more or less along a line, but not exactly on the line. We can find a fitting line that is the best line to capture the trend of the data; this line is also called a best fitting line, trend line, least squares line and regression line. The equation of a fitting line can be found by using the computer. Each student was given a CD with freely available graphing software: GraphCalc and MWMilton. You can use MWMilton to find the equation of the fitting line: click here for step-by-step instructions.	See below.	MML homework 2.3 is now available.
13	2 - 26	2.2 2.3	Section 2.2 Study problem 88 on page 103: (1) note that this is a linear function and two points are given; (2) first, you find the equation of the line; (3) second, you interpret the slope; (4) third, you must use the equation of the line to find the hourly wage in 1990. Study problem 107: remember that a direct proportion linear function has the y-intercept b equal to 0. Section 2.3 A linear equation can be written in the form ax+b=0; see page 107. This means a linear equation has a highest power of 1 on the variable, and there can be no variable in a denominator and there can be no variable under a root √. Linear equations can be solved by using the addition property of equality and/or the multiplication property of equality. See page 107. Linear equations can be solved by hand (also called “solve symbolically” or “solve analytically”). Also, linear equations can be solved graphically (also called “intersection-of-graphs method”). Study example 5 to see how an equation is solved graphically.	Section 2.3 Is the equation linear? # 1, 3, 5 Solve symbolically (by hand) # 17 – 25 odd Solve graphically # 49, 57, 63 Applications # 84, 86, 88, 96, 112
12	2 - 21	2.2	Section 2.2 The reciprocal of 2 is ½, and the negative reciprocal of 2 is -½. Similarly, to find the negative reciprocal of any nonzero number a: 1^st, form the reciprocal 1/a, and 2^nd, change the sign, -1/a. Carefully study example 6 – find the equation of a line parallel to a given line and passing through a given point, and example 7 – find the equation of a line perpendicular to a given line and passing through a given point. Remember that to find the slope-intercept form of a line: 1^st, find the slope m, and 2^nd, find the y-intercept b. There are two special cases of a linear function f(x)=mx+b: (I) when the slope is 0, the formula is f(x)=b which is called a constant function; (II) when the y-intercept is 0, the formula is f(x)=mx which is called a direct proportion (direct variation) function. In a direct proportion function, it is common to use k for the slope and to call it the constant of proportionality. Note that for a direct proportion function, only one point different from the origin is needed to find the slope/constant of proportionality, because you already know that the line goes through the origin (0,0). Study pages 96 and 97.	See below.
11	2 - 19	2.2	Section 2.2 To find any x-intercept, let y=0 in the equation and solve for x. To find any y-intercept, let x=0 in the equation and solve for y. See page 91. The equation of a horizontal line can be written in the form y = b, where b is the y-intercept. The equation of a vertical line can be written in the form x=k where k is the x-intercept. See page 92. The most import form for the equation of a line is the slope-intercept form, but there are other forms such as the point-slope form and standard form. Carefully study the examples: example 1 – find the equation of the line passing thru two given points; example 2 – find the equation of a line with a given slope and passing thru a given point; example 6 – find the equation of a line parallel to a given line and passing through a given point; example 7 – find the equation of a line perpendicular to a given line and passing through a given point. Note: parallel lines that are not vertical have equal slopes, see page 92; and perpendicular lines have slopes whose product is -1 if the lines have nonzero slopes, see page 93.	See below.	MML homework 2.2 is now available.
10	2 - 14		Test 1 (Sections 1.1, 1.2, 1.3, 1.4, 2.1) Bring a scientific calculator.		MyMathLab homework assignments for Test 1 must be completed by midnight Wednesday, February 13.
9	2 - 12	2.2	There was discussion about the sections to be covered on test 1 and some of the types of questions that could be on the test. Section 2.2 In this section, you get more practice with writing the equations/formulas of linear functions. The variable name “m” is often used for the slope of a line. This means the formula of a linear function can be written as f(x)=mx+b = (slope)x+(y-intercept). This is called the slope-intercept form of a line. See page 89. To use this formula in a problem, you must be given enough information to find the slope and the y-intercept. There is a common key step in many of these problems: if a point (x,y) is on the graph of a line, then substituting for x and y into the formula gives a true equation. Carefully study the examples in section 2.2.	Section 2.2 Find the equation of a line # 1-31odd, 35, 39- 55 odd Determine the intercepts # 61, 63, 75 Applications # 81, 84, 87, 99, Direct proportion # 101,103-107
8	2 - 7	1.4 2.1	Section 1.4 One special case of a linear function f(x)=ax+b is when a=0; the formula is then of the form f(x)=b where b is a real number. This type of linear function is called a constant function. For example f(x)=1 is a constant function; the output is constantly 1. This means f(-2)=1, f(0)=1, f(a)=1, etc. The graph of a constant function is a horizontal line. The slope of a constant function is zero. Study pages 46-47. Section 2.1 Remember that for a function y=f(x), that is, the output f(x) is the y-value when you graph the function. To find the x-intercept of a function, set the output y=0; and to find the y-intercept of a function, set the input x=0. Note that this means f(x)=ax+b=(slope)*x+(y-intercept) and so you can just read off the y-intercept by visual inspection of the function formula. For example, the linear function f(x)=2x+1 has y-intercept=1. In application problems, the intercepts may have practical meaning – study example 4 on page 76. If the output in a word problem has a constant rate of change, this means the problem involves a linear function: f(x)=(constant rate of change)*x+(initial amount); the sign of the constant rate is positive if the output is increasing and the sign of the constant rate is negative if the output is decreasing. Study example 5 on page 77. Algebra review: To solve the equation (2/3)x=6 you may multiply both sides by 3/2, the reciprocal of 2/3. Get-out-of-class activity 3 was done in class today.	See below.
7	2 - 5	1.4 2.1	Section 1.4 The slope of a line gives geometric information about the line: (1) a line with positive slope rises from left to right; (2) a line with negative slope falls from left to right; (3) a line with zero slope is horizontal; (4) a vertical line has undefined slope. Another interpretation of slope is rise/run, that is, the slope gives the directions in two steps to get from one point on the line to another point on the line, where rise gives the amount of up-down movement and the run gives the amount of left-right movement. The slope of a line can be interpreted as a rate of change: the slope tells you how y (output) changes per unit increase in x (input). For example, the linear function f(x)=3x+4 has slope 3: This means y increases 3 units per unit increase x OR The line rises 3 units per unit increase in x. In applications, the slope units are the “y units” per “the x unit.” For example, Sheila’s weekly salary in dollars (output unit is dollars) is a function of the number of clients she has (input unit is number of clients), and so the slope units are dollars per client. In applications, to interpret the slope means to write a sentence that gives the practical meaning of the slope. In problem 30, the number of banks (output) in year t (input) is given by the linear function N(t)=-458t + 973769, from 1987 to 1997. The practical meaning of the slope is “The number of banks is decreasing by 458 banks per year during this time” or “The number of banks is decreasing by 458 for each additional year during this time.” Read ahead about constant functions. Section 2.1 Note in a linear function f(x)=ax+b that f(0)=a(0)+b=0+b=b and so we say b is the initial amount, of the output. In applications, the formula of a linear function can be remembered as f(x) = ax+b = (constant rate of change)*x + (initial amount). For example, Sheila’s base salary is $100 (initial amount) and her salary increases by 8 dollars per client (constant rate of change), and so the function formula for her salary is f(x)= (constant rate of change)*x + (initial amount)=(8)x+(100)=8x+100. Alltogether then, in any function problem where the output quantity changes at a constant rate, we can model the problem with a formula that has the form f(x)=ax+b. Study the examples in section 2.1.	Section 2.1 Is the table linear, exactly or approximately? # 1, 2, 3 Determine the slope, intercepts and write a formula for the linear function # 5, 8, Graph the linear function # 13, 15, 21, Write a formula for the linear function # 25, 27, 30, Application # 37-40, 41, 44, Write a formula for the linear function # 49, 51, Approximately linear data # 53,55
6	1 - 31	1.3 1.4	Section 1.3 For the functions we are currently studying, if the function formula has no variable in a denominator and the function formula has no variable inside a square root, for now, you may assume the domain is the set of all real numbers. Section 1.4 A rate of change between two quantities relates how the quantities change in comparison to each other: 60 miles per hour, 3 dollars per gallon, 5 points per hour studied, etc. A linear function has many key properties: (1) the formula can be written in the form f(x)=ax+b; (2) in a table, if the inputs are equally spaced, then the outputs are equally spaced; (3) the function has a constant rate of change. The constant rate of change is also called the slope of the function. The slope equals the coefficient of x in the formula f(x)=ax+b. The slope can also be found from any two points on the graph of a linear function: slope = (y2-y1)/(x2-x1) = Δy/Δx where Δy denotes the change in y and Δx denotes the change in x. Study all the examples in section 1.4. Get-out-of-class activity 2 was done in class today.	Section 1.4 Calculate slope from points # 3, 5, 13, Find slope from function formula # 17, 19, 21-25 odd Find slope from function graph # 27 Interpret slope # 29 Is the table linear # 33, 34, 35 Is the function linear # 37, 41-49 odd, 53, 55, Write a formula # 62-64, 66, 67, Curve sketching # 69 Writing problem #104	MML Homework 1.4 is now available.
5	1 - 29	1.3 1.4	Section 1.3 To find the domain of a function that has a variable under a square root, you set the expression under the square root (the radicand) greater than or equal to 0. Hence, to find the domain of f(x)=√(x-1), you solve x-1>0, and so the domain is x>1. For a function graph: (1) a number on the x-axis is in the domain of the function if a vertical line through that number touches a point on the graph; (2) a number on the y-axis is in the range of the function if a horizontal line through that number touches a point on the graph. Section 1.4 The most important special type of function is a linear function. Read ahead: How can you tell from a graph if the function is linear? How can you tell from a formula if the function is linear? How can you tell from a table if the function is linear? How can you tell from a word problem if the function is linear?	See below.
4	1 - 24	1.3	Section 1.3 These are equivalent statements: (1) y = f(x); (2) y is a function of x; (3) the value of y depends on the value of x; (4) there is exactly one y-value for each x-value. Note that in a function, when we start with an x-value (quantity at end of sentence), then we can find exactly one y-value (quantity at beginning of sentence). It is correct to say that the time of day is a function of a person’s location on earth since given the location there is exactly one matching time of day. It is not correct to say that the location on earth is a function of the time of day since given the time of day there is not exactly one matching location on earth. For a function graph, y=f(x): (1) to find f(0) you know x=0 and you must find the y-value of the matching point ; (2) to find x where f(x)=0 means you know y=0 and you must find the x-value of the matching point. For a function formula (symbol rule), a number is included in the domain if the matching output is a real number. So the function f(x)=1/x includes x=5 in the domain since the matching output f(5)=1/5 is a real number. However, f(0)=1/0 is not defined so x=0 is not in the domain. To find the domain of a function that has a variable in a denominator, you should set the denominator equal to 0, solve this equation and exclude these numbers from the domain. Study example 3 on page 34. Read ahead in example 5 on how to find the domain and range of a function graph. Get-out-of-class activity 1 was done in class today.	See below.	MML Homework 1.3 (part 1 and part 2) is now available.
3	1 - 22	1.3	Section 1.3 Some relations are stronger than others – if each x-value is matched with exactly one y-value, the relation is also called a function. For a graph relation, if each vertical line touches the graph at most once, this means each x-value is matched with exactly one y-value and so the graph is a function. But, if some vertical line touches the graph more than once, this means some x-value has more than one y-value and so the graph is not a function. Function notation: if the function f matches the input x with the output y, we write y=f(x) read as “y equals f of x.” Hence f(2)=3 means the x-value 2 is matched with the y-value 3. Note too that f(x) stands for the output when the input is x. Functions, Points and Graphs on page 32: If f(a)=b then the point (a,b) lies on the graph of f. Conversely, if the point (a,b) lies on the graph of f, then f(a)=b. A function formula such as f(x)=2x+3 can be written with empty parentheses as f( )=2( )+3 to remind you that the same value is substituted all the way across the equation. That is, f(0)=2(0)+3=3 and f(a)=2(a)+3=2a+3. You must practice learning to identify functions and to use function notation.	Section 1.3 Functions and points # 1, 3 Graph by hand 5, 9, 13, 15, 21, Evaluate function notation # 23-33, odd Determine the domain # 37-49 odd Find all x so f(x)=0# 53, 55, 57 Verbal, graphical, numerical # 63, 64 Write as set of ordered pairs # 71 Interpret # 75 Determine if the graph is a function # 79, 81 Determine if S is a function # 87-99 odd	MML Homework 1.2 is now available. Free tutoring help for our course is available in the Math Lab.
2	1 - 17	1.1 1.2	Section 1.1 A calculator can be used to calculate any root, by first rewriting the root with exponents: the nth root of x equals x to the power (1/n). For example, to calculate the cube root of 5, enter 5^(1/3) in your calculator. Negative exponents indicate reciprocal, i.e., 2^-3 = 1/2^3 = 1/8. The formula to find the percent change in a quantity from value P1 to value P2 is on page 4; also, see example 4 on page 4. Section 1.2 A relation can be written as a set of ordered pairs (x,y). The domain of a relation is the set of all x values. The range of a relation is the set of all y values. A relation can also be written as a table, and a relation can also be graphed. The graph of a table is called a scatterplot. See examples 3, 4 and 5 on pages 16-18. A viewing rectangle or viewing window tells you the smallest number (Xmin) and the largest number (Xmax) on the x-axis and the distance between tick marks on the axis (Xscl); it also tells you the smallest number (Ymin) and the largest number (Ymax) on the y-axis and the distance between tick marks on the axis (Yscl). This is written altogether as [Xmin, Xmax, Xscl] by [Ymin, Ymax, Yscl]. See pages 22 – 23.	See below and Section 1.2 Write the table as a set relation # 9, 11 Find the domain and range, and plot the relation (draw a scatterplot) # 61, 63, 65, 67, By hand, draw the viewing rectangle (viewing window) # 69, 71 Draw a scatterplot # 81, 83
1	1 - 15	1.1	You should be able to describe the following sets of numbers and recognize when a given number belongs to one of the sets: (1) the set of natural numbers; (2) the set of integers; (3) the set of rational numbers; (4) the set of irrational numbers; (5) the set of real numbers. Study the descriptions on page 2. Note that every natural number is an integer; every integer is a rational number; every rational number is a real number; every irrational number is a real number; every real number can be plotted on the number line. An important application of sets of numbers is to state the set of numbers that is most appropriate to describe the possible values of a given measureable quantity. Study exercises 13 – 18 on page 11.	Section 1.1 Classify the number # 1, 2, 3, 5, 7, 11, 13, 15, 17 Find the percent change # 19, 21, 95c Use a calculator # 69, 70	Please register in MyMathLab as soon as possible. MML Homework 1.1 is now available.