Class Journal - Math 1301 (20489)

What I hear, I forget; what I see, I remember; what I do, I understand. - Kung Fu Tzu

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	Suggested Practice Problems	Other Info
29	5 – 10		Final Exam on Monday, May 10, from 10 am to 12:30 pm Bring a #2 pencil and a calculator (no cell phone calculators are allowed).
28	5 – 3	5.3 5.4 5.5	There was a review of logarithms. By hand, we graphed the exponential function f(x)=2^x and the logarithm function f(x)=log2x and noted their properties. f(x)=2^x has a horizontal asymptote (y=0); domain (-∞,∞) and range (0, ∞); and it is always increasing. f(x)=log2x has a vertical asymptote (x=0); domain (0, ∞) and range (-∞,∞); and it is always increasing. Remember that log2x is undefined if x=0 or if x is a negative number. In exercise 99 on page 416: the function W has input t which stands for the thickness of a runway in inches and output W which stands for the weight of an airplane in 1000s of pounds; so W(thickness)=weight. Then W(12)=350 can be interpreted as: If the runway is 12 in. thick, then planes of weight 350 thousand pounds can be accommodated. For the inverse, the inputs and outputs are switched: W^-1(thickness)=weight. So W^-1(350)=12 can be interpreted as: If a plane weights 350 thousand pounds, then the runway must be at least 12 in. thick to accommodate it. Information about the final exam was discussed.	See below.
27	4 - 28	5.2 5.3 5.4 5.5	Section 5.2 Remember if f(a)=b then f^-1(b)=a. This means f(x) and f^-1(x) undo each other and so when you compose f(x) and f^-1(x), you just get x. Section 5.3 Note that the exponential function f(x)=C(b^x) is an increasing function whenever the base b>1, but f(x)=C(b^x) is a decreasing function when 0<b<1. The domain of an exponential function is all real numbers (-∞,∞) and the range is all positive real numbers (0,∞). See figure 5.37 on page 401. The function f(x)=3(2^x) is an exponential function. If you make a table for this function where the inputs x are equally spaced, then the outputs y change by a constant factor of 2. For this reason, some people call the base 2 a multiplier. The most important exponential function has base the Euler number e≈2.72. See page 408. There is an “e” key on scientific calculators. Section 5.4 Since an exponential function is one-to-one, it has an inverse function: a logarithm function. Logarithms are exponents: logbx=y means y is the exponent on the base b that gives x, that is, b^y=x. Study example 11 on page 427 to see how to evaluate logs by hand. There are two special logs: (1) the base 10 log, also called the common log, is written without a base: log x= log10x. There is a “log” key on scientific calculators. (2) the base e log, also called the natural log, is written as ln x= logex. There is an “ln” key on scientific calculators. Section 5.5 Properties of logs: (1) logb x= undefined if x is 0 or if x is a negative number; (2) logs change products to sums – see rule 2 on page 436; (3) logs change quotients to differences – see rule 3 on page 436; (4) logs change exponents to coefficients – see rule 4 on page 436.	Section 5.4 Evaluate logs # 1, 3-6, 11 Use logs to solve equations # 49-65 odd Section 5.5 Properties of logs # 7 – 19 odd
26	4 - 26	5.2 5.3	Section 5.2 Only a one-to-one function has an inverse function. For a one-to-one function: (1) each y has exactly one matching x; (2) different inputs always result in different outputs. The graph of a one-to-one function passes the horizontal line test: each horizontal line touches the graph of the function at most once since this means that each y-value is used only once. Study page 385. A one-to-one function f(x) has an inverse function f^-1(x). How do these functions compare? (1) the inputs and outputs are switched, that is, if f(a)=b then f^-1(b)=a; (2) if the point (a,b) is on the graph of f(x), then the point (b,a) is on the graph of f^-1(x); (3) given the function formula for f(x), you may be able find the function formula for f^-1(x) by 1^st, substitute y for f(x); 2^nd, switch x and y in this equation; 3^rd, solve the resulting equation for y. In application problems, given the meaning of the inputs and outputs to f(x), you can reverse these to find the meaning of the inputs and outputs for f^-1(x). In example 5 on page 387, f(height)=crutch length and so switching the inputs and outputs you get f^-1(crutch length)=height. Section 5.3 The function f(x)=3(2^x) is called an exponential function, see page 400. To evaluate f(x), by the order of operations, the exponent is done first and then the multiplication: f(0)= 3(2^0)=3(1)=3. The graph of this function is always rising and so the function is always increasing; the graph is completely above the x-axis; the x-axis is a horizontal asymptote; the function is one-to-one. See figure 5.37 on page 401.	Section 5.3 Exponential expressions # 1 – 17 odd, 18 Graphs #43 – 49 odd, 53, 55, 57
25	4 - 21		Test 3 The test will cover: Section 3.1 - General form of a quadratic function; vertex-form of a quadratic function; vertex formula; axis of symmetry; graph a parabola by hand; applications. Section 3.2 - Solve a quadratic equation by factoring, square root property or quadratic formula; find x- and y-intercepts of a parabola by hand; find domain by solving a quadratic equation; applications. Section 3.3 - Solve quadratic inequality graphically. Section 4.1&4.2 - Formula of a polynomial function; degree and leading coefficient of a polynomial; shape of graph of a polynomial; number of x-intercepts of a polynomial; number of turning points of a polynomial; absolute maximum or minimum of a function; the x-intervals where the function is increasing or decreasing; end behavior of a polynomial; piecewise-defined polynomial functions. Section 4.4 - Complex numbers a+bi; add, subtract and multiply complex numbers; the quadratic formula and complex number solutions. Section 4.5 - Inverse variation. Section 5.1 - Composition of functions. Please bring a pencil and calculator to the test. Sharing of calculators is not allowed, and no cell phone calculators are allowed. Also, all electronic devices (cell phones, ipods, etc.) should be put away and not be visible during the test. The MyMathLab homework and quizzes are due by Tuesday, April 20. You can get help for the test from me in S707, from our SI leader Tanu in S405, and in the Math Lab N925.
24	4 - 19	5.1 5.2	Section 5.1 In composition of functions, we may use a function f(x) as the input to a function g(x), and this is written as: (g◦f)(x)=g( f(x) ). Study example 6 on page 369. Composition values may be done from graphs – study example 9 on page 371. Composition values may be done from tables – study example 10 on page 372. Section 5.2 Only a one-to-one function has an inverse function. For a one-to-one function: (1) each y has exactly one matching x; (2) different inputs always result in different outputs. Study page 385.	Section 5.2 Is the function one-to-one? # 13-21 odd Find a symbolic representation (formula) for the inverse function # 41, 42, 43, 45, 46
23	4 - 14	4.4 4.5 5.1	Section 4.4 The solutions to a quadratic equation may be complex imaginary numbers a+bi. Study example 2 on page 295. Section 4.5 In some problems, we have that “y varies inversely as x” or “y is inversely proportional to x.” This means the function formula can be written as y=k÷x or y=k/x where k is the constant of proportionality. Note that when y varies inversely as x, then an increase in x results in a decrease in y, and vice versa. Given one matching x-y pair, we can substitute into y=k/x to find k and then we can find the value of y for any other value of x. Problem 123 on page 324 was solved. Section 5.1 In some problems, we perform a sequence of tasks one after the other. A similar idea is used in composition of functions, where we use one function formula after another. For example, we may use a function f(x) as the input to a function g(x), and this is written as: (g◦f)(x)=g( f(x) ). Study example 6 on page 369. Composition values may be done from graphs – study example 9 on page 371. Composition values may be done from tables – study example 10 on page 372.	Section 5.1 Composition of functions # 61-71odd,79, 87, 89, 91, 101, 103, 105
22	4 - 12	4.2 4.4	Section 4.1/4.2 The largest possible y-value for a function, if it exists, is called the absolute maximum value of the function; the smallest possible y-value for a function, if it exists, is called the absolute minimum value of the function. Study examples 3 and 4 on page 246 – we will skip local maximum and local minimum. The graphs of the following four polynomials can be used to describe the end behavior of all polynomials. 1. y=f(x)=x has degree = 1 = odd, leading coefficient = 1 = positive, the left end of the graph falls, the right end of the graph rises; 2. y=f(x)=-x has degree = 1 = odd, leading coefficient = -1 = negative, the left end rises, the right end falls; 3. y=f(x)=x^2 has degree = 2 = even, leading coefficient = 1 = positive, the left end rises, the right end rises; 4. y=f(x)=-x^2 has degree = 2 = even, leading coefficient = -1 = negative, the left end falls, the right end falls. Click on “end behavior” in the last column for a graphical summary of end behavior for all polynomials. Study example 3 on page 262. A function with more than rule/formula is called a piecewise-defined function. When each rule is a polynomial, the function is called a piecewise-defined polynomial function. Each rule is used only for the specified interval of x-numbers that is given. Study example 4 on page 263: f(x)=x^2-x only if -5<x<-2, etc. Section 4.4 The complex imaginary number i has the properties: i=√(-1) and i^2=-1. Study page 293. Then the square root of any negative number can be written as a complex imaginary number: √(-16) = i√(16) = 4i. See page 293. The standard form of a complex number is a+bi. See page 293. Complex numbers may be added, subtracted and multiplied. Study the examples on pages 293-294.	Section 4.4 Complex numbers # 1-25 odd Quadratic equations # 45, 47, 49, 51, 55, 59	Click on the link for a summary of the end behavior of polynomial functions.
21	4 - 7	4.1 4.2	Section 4.1/4.2 The degree n of a polynomial is the highest power of the variable, when the polynomial is written in standard form. See page 258. Starting with the formula of polynomial function, we can predict in advance that (1) the graph of a polynomial is a straight line or an unbroken curve with no sharp turning points; (2) the number of x-intercepts is at most the degree n; (3) the number of turning points is at most n-1, i.e., the degree n less 1. See page 262. Starting with the graph of a polynomial, we can find the minimal degree of the polynomial formula by: 1^st, counting the number of x-intercepts; 2^nd, counting the number of turning points; 3^rd, then the minimal degree equals the larger of the number of x-intercepts and one more than the number of turning points. Study page 262. We say a function is increasing on an interval of x-numbers, if the graph rises over the interval. We say a function is decreasing on interval of x-numbers, if the graph falls over the interval. Study pages 243-244. Class activity #10 was done in class today.	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 - 17 odd Degree, end behavior # 19 - 25 odd Conjecture degree # 41 Piecewise-defined functions # 69, 71, 75, 77 Application # 81, 83
20	4 - 5	3.2 3.3	Section 3.2 In application problems, we may have to solve a quadratic equation to find the solution to the given question. Study example 10 on page 197 – “after how many seconds did the projectile strike the ground?” – a quadratic equation is solved to find the answer. We have to set up and solve a quadratic equation to find the solution to part d of problem 71 on page 238. Section 3.3 When we solve an equation by hand, we call this a symbolic or analytical solution method. When we solve an equation by using a graph, we call this a graphical solution method. Remember that to find the x-intercepts of a parabola y= ax^2+bx+c, we substitute 0 for y and then solve the resulting equation 0=ax^2+bx+c. We can reverse this process. This means that the solutions to the equation ax^2+bx+c=0 are the same as the x-intercepts of the graph of y=f(x)=ax^2+bx+c; and we say we are “graphically solving the equation.” Study example 3 on page 191. We can also graphically solve an inequality. To find the solutions x so that ax^2+bx+c>0, we think of this as find the numbers x so that y>0 where y= ax^2+bx+c. Then 1^st, we graph y=f(x)= ax^2+bx+c; 2^nd, we locate the x-intercepts; 3^rd, we note that the x-intercepts break up the x-axis into intervals of x-numbers. 4^th, on each interval of x-numbers, the matching y’s of the points on the parabola are all above the x-axis, which means y>0, or the matching y’s of the points on the parabola are all below the x-axis, which means y<0. Finally, the solution set for ax^2+bx+c>0 is then all of the intervals of x-numbers that have y>0. Study example 1 on page 207 and example 2 on page 208.	Section 3.3 Solve inequality graphically # 1-17 odd, 27 – 39 odd, 43, 47, Applications # 61, 63
19	3 - 31	3.2	Section 3.2 To find the x-intercepts of a parabola y= ax^2+bx+c, we substitute 0 for y and get 0=ax^2+bx+c, which is a quadratic equation. There are several solution methods to solve a quadratic equation. To solve a quadratic equation by factoring, we write the quadratic equation as a “product=0” and apply the zero-product property. Study example 1 on page 190. To solve a quadratic equation by the square root property: the equation x^2=k has the solutions x=√k and x=-√k. Study example 4 on page 192. To solve a quadratic equation by the quadratic formula, we first check that it has the form ax^2+bx+c=0 and then we substitute into the quadratic formula. Study the formula and example 7 on page 194. Since division by zero is not defined, to find the domain of the function f(x)=1/(x-4), we solve x-4=0 and omit the solution x=4 from the domain. So the domain is all real numbers except 4. Similarly, to find the domain of the function f(x)=1/(x^2-5), we solve x^2-5=0 and omit the solutions x=+√5 from the domain. So the domain is all real numbers except +√5.	Section 3.2 Solve quadratic equation # 1, 3, 5, 7,11, 13, 15, 19, 21, 23 Find x-intercepts # 25, 27, 29 Solve graphically # 31, 33 Find the domain # 81, 83 Literal equations # 93, 95, Applications # 97, 99
18	3 – 29	3.1	Section 3.1 The vertex of a parabola can be found by hand (analytically). (1) If the formula is written in vertex form y=f(x)=a(x-h)^2+k, then the vertex is (x,y)=(h,k). See page 175. (2) If the formula is written in general form y=f(x)=ax^2+bx+c, then the vertex has x=-b/2a. To find y, just substitute for x back into the original function formula. See page 177. In application problems, a quadratic function has a maximum or a minimum value at the vertex. Study examples 8 and 9 on pages 179-180.	Section 3.1 Quadratic function # 1-7 odd Graph of quadratic function # 9 – 15 odd Vertex # 17, 19, 27, 29, 35 Write formula # 51, 55 Sketch graph # 59-75odd Applications # 79, 81, 83, 85-88
17	3 - 24		Test 2 will be on Wednesday, March 24. The test will cover: Section 2.1 - Write the formula for a linear function/model. Section 2.2 - Point-slope form of a line; slope-intercept form of a line; find intercepts from the equation of the line; horizontal, vertical, parallel and perpendicular lines; direct variation (proportion). Section 2.3 - Recognize a linear equation; solve a linear equation, some have fractions or decimals; solve a linear equation for x when you are given y. Section 2.4 - Use properties of inequalities to solve linear inequalities and write the solution set in interval notation; solve compound inequalities and interval notation. Section 2.5 - Evaluate a piecewise-defined function, find its domain and recognize its graph; evaluate an absolute value function, recognize its graph and solve absolute value equations. Please bring a pencil and calculator to the test. Sharing of calculators is not allowed, and no cell phone calculators are allowed. Also, cell phones should be put away in silent mode and not be visible during the test. The MyMathLab homework and quizzes are due by Wednesday, March 24. You can get help for the test from me in S707, from our SI leader Tanu in S405, and in the Math Lab N925.
16	3 – 22	3.1	Some problems from sections 2.2, 2.3 and 2.5 were discussed to review for Test 2. Section 3.1 The formula of a quadratic function can be written in the form f(x)=ax^2+bx+c where the leading coefficient a cannot equal zero, that is, a quadratic function is a polynomial and it must have a square term. Study example 1 on page 173. The graph of a quadratic function is a parabola, a U-shape graph, which opens up when the leading coefficient is a positive number, and which opens down when the leading coefficient is a negative number. See page 173. The highest or lowest point on a parabola is called the vertex of the parabola. The vertical line that goes through the vertex and divides the parabola in-half is called the axis of symmetry. Study figures 3.4, 3.5, and 3.6 on page 173. You can find the (x,y) coordinates of the vertex from the function formula – study page 177.	Section 3.1 Quadratic function # 1-7 odd Graph of quadratic function # 9 – 15 odd Vertex # 17, 19, 27, 29, 35 Write formula # 51, 55 Sketch graph # 59-75odd Applications # 79, 81, 83, 85-88
15	3 - 10	2.5	Section 2.5 The graph of a piecewise-defined function consists of distinct pieces; and there are as many pieces in the graph as there are formulas in the function. Study example 2 on page 141: 1^st, the function uses the rule f(x)=x-1 if -4<x<2 and the function uses the rule f(x)=-2x if 2<x<4. Since there are 2 formulas/rules in this function, the graph will have 2 distinct pieces. 2^nd, the first piece of the graph is the piece of the line y=x-1 that is bounded by the lines x=-4 and x=2. This is a rising line that begins at the closed dot (x,y)=(-4,-3) and ends at the open dot (x,y)=(2,1). 3^rd, the second piece of the graph is the piece of the line y=-2x that begins at the closed dot (x,y)=(2,-4) and ends at the closed dot (x,y)=(4,-8). See figure 2.58 on page 142. 4^th, the domain is the set of x-values for the function. You combine the x’s for each rule to get the domain: combine -4<x<2 with 2<x<4 and you get -4<x<4 or [-4,4] for the domain. The absolute value function f(x)=|x| can be written as a piecewise-defined function with 2 rules: f(x)=-x if x<0 and f(x)=x if x>0. From this, we can draw the graph and we get a v-shape graph. See figure 2.63 on page 144. Note that the equation |x|=4 has two solutions x=-4 and x=4. In general, an absolute value equation |f(x)|=n has two solutions: solve f(x)=n and solve f(x)=-n. Study example 5 on page 146. Class Activity #9 was done in class today.	See below.
14	3 - 8	2.4 2.5	Section 2.4 Carefully study the properties of inequalities on page 126: (1) the same number may be added to both sides of an inequality to help solve it; (2) both sides of an inequality may be multiplied by a positive number to help solve it; (3) both sides of an inequality may be multiplied by a negative number to help solve it, but the direction of the inequality must be reversed. Study example 1 on page 126. The compound inequality or 3-part inequality 40<x<70 is shorthand for “40<x AND x<70” that is x is larger than 40 and x is smaller than 70. When we solve a compound inequality, our goal is to isolate the variable in the middle. Study example 6 on page 130. Section 2.5 It is common for a utility company to use a different formula that depends on the amount of energy used in order to calculate the amount due. This is an example of a piecewise-defined function – a function that uses different formulas for different intervals of numbers in its domain. Carefully study example 2 on page 141. Note that the graph consists of distinct pieces; and that there are as many pieces in the graph as there are formulas in the function.	Section 2.5 Piecewise-defined function # 1, 3, 7, 9, 11, 15, 19, 20, 22 Absolute value # 29, 31, 41, 43, 45, 47, 51, 87	Here is an example of a piecewise-defined function in everyday life: the water rates in Houston, click here.
13	3 - 3	2.3 2.4	Section 2.3 Suppose the problem is to solve an equation containing fractions. To eliminate the equation of fractions: (1) identify each denominator in the equation; (2) find the least common multiple of all the denominators in step 1 and multiply each side of the equation by this number; or multiply all of the denominators in step 1 and multiply each side by this number; (3) solve the resulting equation from step 2 to complete the problem. Study example 3 on page 109. Suppose the variables in a function formula are x and V. For example, V=6500x-180000. To find the value of x when V=219000, we substitute 219000 for V and then we solve the resulting equation for x. Study example 8 on page 112. Section 2.4 In a linear inequality, the highest power of the variable is one, and it can be written in the form ax+b<0 where a cannot equal zero, and the inequality may also by <, >, or >. See page 124. The solution to an inequality is often an interval of numbers along the number line and interval notation may be used to write the solution set. Study the examples of interval notation in table 2.12 on page 125. Class Activity #8 was done in class today.	Section 2.4 Interval notation # 1-11 odd Solve the inequality symbolically # 13 - 31odd, 35 Applications # 91, 93, 101
12	3 - 1	2.2 2.3	Section 2.2 When the y-intercept equals zero, the equation of the line can be written more simply as y=mx. For example y=3x is a line with y-intercept zero. Note for the line y=3x that when x=4 doubles to x=8, then the matching y doubles from 12 to 24. And for the line y=3x, when x is halved from x=4 to x=2, then the matching y is halved from 12 to 6. Therefore, we say that y varies directly as x, or y is directly proportional to x. In general, y varies directly as x means: (1) this is a linear function that has y-intercept zero and so the line goes thru the origin (0,0); (2) the slope is called the constant of proportionality and it is often written with the letter k; (3) the formula of the linear function is y=mx+0 or more simply y=mx. Study pages 96-97. Suppose a table of data is used to write a linear function formula. If an x-value is substituted into the formula and that x-value falls between two x-values in the table, then we say interpolation was used to find the matching y. But if an x-value is substituted into the formula, and that x-value is larger or smaller than all x-values in the table, we say extrapolation was used to find the matching y. Study page 95. Section 2.3 In a linear equation, the highest power of the variable is one, and it can be written in the form ax+b=0 where a cannot equal zero. To solve an equation, we can add the same number to both sides (addition property of equality) and we can multiply both sides by the same nonzero number (multiplication property of equality). To solve an equation with fractions or decimals by hand, it is recommended that the first step is to eliminate the equation of fractions or decimals. Study example 3 on page 109. Class Activity #7 was done in class today.	Section 2.3 Is the equation linear? # 1, 3, 5 Solve symbolically (by hand) # 7, 9, 11, 17, 21, 25 Solve graphically # 49, 57, 63 Applications # 79, 83, 84, 88, 96, 105, 109, 112
11	2 - 24	2.2	Section 2.2 Two lines are parallel if they have the same slope. So the lines y=3x+4 and y=3x-1 are parallel since they both have slope 3. Study example 6 on page 92. Two lines are perpendicular if the product of the slopes of the lines equals negative one; this means the slopes are negative reciprocals of each other. So the lines y=3x+4 and y=-1/3*x+6 are perpendicular since the product of the slopes equals (3)(-1/3)=-1. Study example 7 on page 93. You can find the intercepts of a line from any equation of the line: (1) to find the x-intercept, substitute 0 for y in the equation; (2) to find the y-intercept, substitute 0 for x in the equation. Study example 4 on page 91. There are two special cases of the standard form of a line ax+by=c: (1) when a=0, you get an equation like y=5 and the graph is a horizontal line that has slope zero; (2) when b=0, you get an equation like x=8 and the graph is a vertical line that has undefined slope. Study example 5 on page 92.	See below.
10	2 - 22	2.1 2.2	Section 2.1 When we create a function formula from an application problem (word problem), the function formula is called a model and if the application problem includes a table of (x,y)-values, then: (1) we say the function models the data exactly if the calculated y from the function formula exactly matches the y in the table for all of the data; and (2) we say the function models the data approximately if not all of the calculated y-values from the function formula match exactly the y in the table of data. Study example 1 on page 74. A linear function has a constant rate of change which equals the slope and the function formula can be written as f(x)=(constant rate of change)*x+(initial amount). Study examples 3 – 5 on pages 76 and 77. Note that an x-intercept of a function is also called a zero of the function because the matching output of x equals zero, that is, f(x)=0. Study page 75. Section 2.2 Equations of lines can be written in many different forms. The line with slope m that passes through the point (x1,y1) can be written in the point-slope form y=m(x-x1)+y1 Study example 1 on page 88. The line with slope m and y-intercept b can be written in slope-intercept form y=(slope)*x+(y-intercept)=mx+b Study example 2 on page 89.	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, 10 Graph the linear function # 13, 15, 21 Write a formula for the linear function # 25, 27, 29, 30 Applications # 37-40, 41, 43, 44, 47 Write a formula for the linear function # 49, 51 Approximately linear data # 53, 55 Section 2.2 Find the equation of a line # 1-31 odd, 35, 39-55 odd Determine the intercepts # 61, 63, 75 Applications # 81, 84, 87, 99 Direct proportion # 101, 103-107
9	2 - 17		Test 1 (sections 1.1, 1.2, 1.3, 1.4) Please bring a pencil and calculator to the test. The MyMathLab homework for these sections and the quizzes are due by Tuesday, February 16.
8	2 - 15	1.4	Section 1.4 The slope of a linear function can be found (1) from the formula f(x)=mx+b=(slope)x+(y-intercept), that is, the slope is the coefficient of x; (2) by choosing two points on the line and using the slope formula on page 53; (3) by choosing two points on the line, measure the run from the first point to the second point and measure the rise from the first point to the second point, then slope=rise/run. The slope m can be interpreted as a rate of change: for each unit increase in x, y changes by m units. In problem 29 on page 58, the input t is the year between 1970 and 2010, the output A(t) is the median age of the U.S. population, and the formula is A(t)=0.243t-450.8. The slope=0.243 can be interpreted as: in each year, the median age increases by 0.243 years. To graph by hand a linear function f(x)=mx+b=(slope)*x+(y-intercept), Solution 1: make a table of values by picking 3-values for x; calculate the matching y=f(x) values; plot the 3 points; and last draw the straight line thru the points. Solution 2: since slope=rise/run, start by plotting the y-intercept; then use the run and rise separately to get a second point on the graph; and last draw the straight line thru the points.	See below.
7	2 - 10	1.4	Section 1.4 A linear function has three key properties: (1) the graph is a nonvertical straight line; (2) the formula can be written in slope-intercept form as f(x)=mx+b=(slope)*x+(y-intercept); (3) in a table, for each unit increase in x, the value of y changes by a regular constant amount, or if x changes by a constant amount then y changes by a constant amount. There are problems where you must identify which functions are linear functions. Study pages 50-52. The slope of a linear function can be found (1) from the formula f(x)=mx+b=(slope)*x+(y-intercept), that is, the slope is the coefficient of x; (2) by choosing two points on the line and using the slope formula on page 53; (3) choose two points on the line, measure the run from the first point to the second point and measure the rise from the first point to the second point, then slope=rise/run. The rate of change of a linear function tells us what the steady change in y is when there is a steady change in x. This means the slope equals the rate of change because the slope tells us how y changes for each unit increase in x. So if the slope of a linear function is 2=2/1=(change in y)/(change in x), then we can say the line rises 2 units for each unit increase in x. And if a line has slope -5/3, we can say the line falls 5 units for each 3-unit increase in x. Study example 1 on page 49. Class Activity #6 was done in class today.	See below.
6	2 - 8	1.3 1.4	Section 1.3 For a function formula f(x), a real number x is in the domain only if f(x) is a real number. Study the paragraph above example 3 on page 34. For example, f(x)=1/(x-1) does not include x=1 in the domain since f(1)=1/0=undefined. To find the domain of a function formula with a variable in the denominator, solve the equation “denominator=0” and omit the solutions from the domain. So to find the domain of f(x)=1/(x-1), we solve the equation “x-1=0” and get x=1, and omit this number from the domain. Therefore the domain of f(x)=1/(x-1) is all real numbers except 1. Study example 3 on page 34. As another example: f(x)=√(x-1) does not include x=0 in the domain since f(0)= √(-1) which is not a real number. To find the domain of a function formula with a variable in the radicand of a square root, we solve the inequality “radicand>0”. So to find the domain of f(x)=√(x-1), we solve the inequality “x-1>0” and get x>1 as the domain. Study example 5 on page 35. For a function formula with no variable in a denominator and no variable in the radicand of a square root, we will assume the domain includes all real numbers, for now. So g(x)=x^2-2x has domain “all real numbers” – study example 4 on page 35. In application problems, the input x is has a unit of measure and the output f(x) has a unit of measure. Then we can write a sentence to interpret y=f(x). For example, suppose x is the number of years since 2000, and f(x) is the savings account balance in $1000’s. Then f(2)=3 means: In 2002, the savings account balance is $3000. Here, 2002 is 2 years after 2000, and $3000 is 3 times 1000 in dollars. Study example 7 on page 37: a person is of height x inches and f(x) is the appropriate crutch length in inches. Section 1.4 For a constant function, the output y is the same no matter the input x. The formula of a constant function can be written in the form f(x)=b where b can be any real number. The graph of a constant function is a straight horizontal line. Study page 46.	Section 1.4 Calculate slope from points # 1, 3, 5, 13, Find slope from function formula # 17, 19, 21, 23, 25 Find slope from function graph # 27 Interpret slope # 29 Is the table linear # 31, 33, 34, 35 Is the function linear # 37, 41-49 odd, 53, 55, Write a formula # 61, 63, 67 Curve sketching # 69 Writing problem #104
5	2 - 3	1.3	Section 1.3 If (a,b) is a point on the graph of the function f then f(a)=b, that is, the x-coordinate is the input to the function and goes inside the parentheses and the y-coordinate is the output and goes on the other side of the equals. See page 32. This rule is helpful to answer exercises 1, 3, 53, 55, and 57. The domain of a function is the set of x-values. If a point is on the graph of a function, we can find the x-coordinate of the point by drawing a vertical line from the point to the x-axis. Then the number on the x-axis is the x-coordinate of the point and this number belongs to the domain of the function. This observation gives us a strategy to find the domain of a function graph: (1) find the leftmost vertical line x=a that bounds the graph; (2) find the rightmost vertical line x=b that bounds the graph; (3) if each vertical between the left bound and right bound touches the graph then the domain includes all real numbers between x=a and x=b. Study exercise 81 on page 45: the left bound is the vertical line x=-4, the right bound is the vertical line x=4, and every vertical line between these touches the graph. Therefore the domain is all real numbers between -4 and 4, inclusive, that is, domain = {x | -4<x<4}. Similarly, study exercise 80 on page 44: the graph is bounded on the left by the vertical line x=-5; the graph is bounded on the right by the vertical line x=5, and every vertical line between them touches the graph. Therefore the domain is all real numbers between -5 and 5, inclusive, that is, domain = {x | -5<x<5} = [-5,5]. We follow a similar strategy to find the range from a graph. If a point is on the graph of a function, we can find the y-coordinate of the point by drawing a horizontal line from the point to the y-axis. Then the number on the y-axis is the y-coordinate of the point and this number belongs to the range of the function. Therefore to find the range of a function graph: (1) find the lowest horizontal line y=c that bounds the graph; (2) find the highest horizontal line y=d that bounds the graph; (3) if each horizontal line between the lower bound and upper bound touches the graph then the range includes all real numbers between y=c and y=d. Study exercise 81 on page 45: the lower bound is the horizontal line y=0, the upper bound is the horizontal line y=4, and every horizontal line between these touches the graph. Therefore the range is all real numbers between 0 and 4, inclusive, that is, range = {y | 0<y<4} = [0,4]. Class Activity #5 was done today.	See below.
4	2 - 1	1.3	Section 1.3 A function is a relation with a special property: each input x matches exactly one output y. A function can have different representations (forms): (1) a set of ordered pairs; (2) a table of xy-values; (3) a graph of points in the xy-plane; (4) a verbal description; (5) a formula (symbol rule). Study the example on page 33. It takes practice to learn to identify a function. Study the examples on pages 38-40 where a relation is given and you must determine if it is a function. In particular, to determine whether a graph is a function: (1) if each vertical line intersects the graph at most once, then each x matches exactly one y and the graph is a function; (2) if some vertical line intersects the graph more than once, then some x matches more than one y and the graph is a not a function. Function notation: When there is exactly one y for each x, we say y is a function of x, and we can write y=f(x) which is read as “y equals f of x”. See the explanation on page 30. Evaluating a function formula: For a function formula f(x), to evaluate f(3) means to substitute 3 for x in the formula and then to calculate the result. A special property of function formulas is that the input may be an expression such as “a+1”. Study example 3 on page 34. To graph a function formula by hand, it is important to note that a point (x,y) is on the graph if y=f(x), that is, the y-coordinate of the point is what you get when you substitute the x-coordinate into the formula. Study page 32.	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 Determine verbal, graphical, numerical forms # 63, 64 Write as set of ordered pairs # 71 Interpret # 75 Determine if the graph is a function # 79, 81 Determine if the relation S is a function # 87-91 odd Write a function formula # 99, 101
3	1 - 27	1.1 1.2	Section 1.1 You should practice simplifying by hand some arithmetic problems with scientific notation (see exercises 51, 55, 57, 61 in section 1.1). Section 1.2 You should learn the mathematical meaning of (1) a relation (set of ordered pairs (x,y)); (2) the domain of a relation (set of x-values); (3) the range of a relation (set of y-values); (4) the minimum and maximum of the x-values; (5) the minimum and maximum of the y-values. Study page 16. A relation can have different forms (representations): it can be written as a set of ordered pairs, or it can be written as a table, or it can be a graph that is called a scatterplot. Study pages 18-19. The viewing window (rectangle) specifies the part of the xy-plane that can be viewed; it is written in the form [Xmin, Xmax, Xscl] by [Ymin, Ymax, Yscl]. Study pages 22-23. Class Activity #4 was done in class today.	See below.
2	1 - 25	1.1 1.2	Section 1.1 You should be able to: (1) evaluate special exponents by hand, such as the zero exponent 5^0=1, a negative exponent 5^-2=1/25, and a rational exponent 4^3/2=8; (2) convert a decimal number from standard form to scientific notation, and vice versa; (3) calculate the percent change in a quantity from c1 to c2; (4) choose the most appropriate set numbers for a quantity (see exercises 13 – 18 in section 1.1). Please study (1) the review of special exponents at the end of the book in sections R.3 Integer Exponents and R.7 Radical Notation and Rational Exponents; (2) the examples of percent change on pages 4-5; (3) the examples of scientific notation on pages 5-7. Section 1.2 You should be able to: (1) write a relation as a set of ordered pairs; (2) write a relation as a table of values; (3) make a scatterplot of a relation. Class Activity #3 was done in class today.	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, 85, 87abc, 89abc	The deadline for Class Activity #2 (Register in MyMathLab) has been extended to Tuesday, January 26.
1	1 - 20	1.1	Section 1.1 You should be able to recognize when a given number belongs to one of the following 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. Note that (1) every natural number is an integer; (2) every integer is a rational number; (3) every rational number can be written as a decimal number that is finite or repeats; (4) every irrational number can be written as an infinite, nonrepeating decimal number; (5) every real number can be written as a decimal number and it can be plotted on the number line. You should study: (1) the descriptions of each set of numbers on page 2; (2) example 1 on page 3 that involves “classifying numbers.” You should be able to convert a decimal number in standard form to scientific notation, and vice versa. On page 5, study the definition and example 5.	Section 1.1 Classify the number # 1, 2, 3, 5, 7, 11 Choose most appropriate set of numbers # 13, 15, 17 Find the percent change # 19, 21, 79, 95ac Evaluate by hand # 51, 55, 57, 61 Use a calculator # 69, 70, 73, 75