Chapter 1 Skills Points and Linear Equations

Example 1. Solve We have Chapter 1 Skills Points and Linear Equations t = 3 t t = 3 t for t. = ( t)( t) = ( t) = 3( t) = 4 4t = 6 3t = = t t = 3 ( t)( t) t Example. Solve We have = A + Bt A Bt for t. = A + Bt A Bt = (A Bt) = A + Bt A Bt A Bt 1 = (A Bt) = A + Bt = A Bt = A + Bt = A = 3Bt = A 3B = t. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 011 1

Example 3. Write A C + Bt + B D Ct as a single fraction. A C + Bt + B D Ct = = A C + Bt D Ct D Ct + B D Ct C + Bt C + Bt A(D Ct) + B(C + Bt) (C + Bt)(D Ct) = AD ACt + BC + B t. (C + Bt)(D Ct) Example 4. Solve 4x + 3y = 11 x y 3 = 0 First, we solve for y in the nd equation: for x and y. x y 3 = 0 = x = y 3 = 6x = y Now, substituting 6x into the first equation for y, we obtain 4x + 3(6x) = 11 = 4x + 18x = 11 = x = 11 = x = 1. Therefore, since y = 6x, we have y = 6 1 = 3. Our final answer is therefore x = 1/ and y = 3. Developed by Jerry Morris

Section 1.1 Functions and Function Notation Definition. A function is a rule that takes certain values as inputs and assigns to each input value exactly one output value. Example. Let y = 1 + x. x 0 1 3 1 y 1 3 y as a function of x: f(0) =, f(1) = 1, f() = /3, f(1/) = 3 x as a function of y: g() = 0, g(1) = 1, g(/3) =, g(1/) = 3 Example. t = time (in years) after the year 000 w = number of San Francisco 49er victories t 0 1 3 4 5 6 7 8 9 w 6 1 10 7 4 7 5 7 8 Observations: 1. w is a function of t: f(0) = 6 means that in 000, the 49ers won 6 games f(1) = 1 means that in 001, the 49ers won 1 games f() = 10 means that in 00, the 49ers won 10 games.. t is not a function of w, since the input "7 wins" would have three outputs, 3, 6, and 8. 3. In general, for a function, the input variable is called the independent variable and the output variable is called the dependent variable. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 011 3

Example. Which of the graphs below represent y as a function of x? Graph 1 y Graph y Graph 3 y x x x 1. In Graph 1, y is a function of x because every input value of x has exactly one corresponding output value, y.. In Graph, y is not a function of x, because if it were, we would have f(0) = and f(0) =, which is impossible. Graphically, this occurs because Graph fails the vertical line test. 3. In Graph 3, y is a function of x because every input value of x has exactly one corresponding output value, y. Example. A woman drives from Aberdeen to Webster, South Dakota, going through Groton on the way, traveling at a constant speed for the whole trip. (See map below). 0 miles 40 miles Aberdeen Groton Webster a. Sketch a graph of the woman s distance from Webster as a function of time. d Comment: Note that the function touches the t-axis; that is, d = 0, when the woman reaches Webster, not at t = 0. The constant slope of the graph indicates that her speed is constant for the duration of the trip. Woman reaches Webster t b. Sketch a graph of the woman s distance from Groton as a function of time. d Woman reaches Groton Woman reaches Webster t Comment: Note that the function crosses the t-axis; that is, d = 0, when the woman reaches Groton. Also, d is positive both when the woman is in Aberdeen and when the woman is in Webster, since both cities lie a positive distance away from Groton. 4 Developed by Jerry Morris

Section 1. Rates of Change Preliminary Example. The table to the right shows the temperature, T, in Tucson, Arizona t hours after midnight. t (hours after midnight) 0 3 4 T (temp. in F) 85 76 70 Question. When does the temperature decrease the fastest: between midnight and 3 a.m. or between 3 a.m. and 4 a.m.? Even though the temperature drops more between midnight and 3 a.m. than it does between 3 a.m. and 4 a.m., the temperature decreases fastest between 3 a.m. and 4 a.m., as the calculations below demonstrate: Interval T t 0 t 3 9 F 3 hours 3 t 4 6 F 1 hour T t 9 3 = 3 F per hour 6 1 = 6 F per hour Thus, the temperature decreases at an average rate of only 3 F per hour between midnight and 3 a.m., but it decreases at an average rate of 6 F per hour between 3 and 4 a.m. Graphical Interpretation of Rate of Change Definition. The average rate of change, or just rate of change of Q with respect to t is given by Change in Q Change in t = Q t. Q f(a) f f(b) a b t Alternate Formula for Rate of Change: The average rate of change of a function Q = f(t) on the interval a t b is given by the following formula: f(b) f(a) b a Note: Graphically, the average rate of change given by the above formula is just the slope of the line segment in the above diagram. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 011 5

Examples and Exercises 1. Let f(x) = 4 x. Find the average rate of change of f(x) on each of the following intervals. (a) 0 x (b) x 4 (c) b x b (a) We have f() f(0) 0 = (4 ) (4 0 ) = 0 4 =, so the average rate of change of f is over the interval 0 x. (b) Similarly, we have f(4) f() 4 = (4 4 ) (4 ) = 1 0 = 6. (c) Similarly, we have f(b) f(b) b b = (4 (b) ) (4 b ) b = 4 4b 4 + b b = 3b b = 3b.. To the right, you are given a graph of the amount, Q, of a radioactive substance remaining after t years. Only the t-axis has been labeled. Use the graph to give a practical interpretation of each of the three quantities that follow. A practical interpretation is an explanation of meaning using everyday language. f(1) Q (grams) f(3) 1 3 t (yrs) a. f(1) f(1) represents the amount of the substance present, in grams, after 1 year. b. f(3) c. f(3) represents the amount of the substance present, in grams, after 3 years. f(3) f(1) 3 1 This is the average rate at which the substance is decaying, in grams per year, between 1 and 3 years. 6 Developed by Jerry Morris

3. Two cars travel for 5 hours along Interstate 5. A South Dakotan in a 1983 Chevy Caprice travels 300 miles, always at a constant speed. A Californian in a 009 Porsche travels 400 miles, but at varying speeds (see graph to the right). d (miles) 400 300 California Car 00 100 South Dakota Car 1 3 4 5 t (hours) (a) On the axes above, sketch a graph of the distance traveled by the South Dakotan as a function of time. (b) Compute the average velocity of each car over the 5-hour trip. For the Chevy Caprice, we have d t = 300 0 300 miles = = 60 miles per hour, 5 0 5 hours so the Chevy Caprice has an average velocity of 60 miles per hour. For the Porsche, we have d t = 400 0 400 miles = = 80 miles per hour, 5 0 5 hours so the Porsche has an average velocity of 80 miles per hour. (c) Does the Californian drive faster than the South Dakotan over the entire 5 hour interval? Justify your answer! No. For example, the Californian has a velocity of 0 between t = hours and t =.5 hours, because the slope of the graph corresponding to the Porsche is 0 on this time interval. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 011 7

Sections 1.3 & 1.4 Linear Functions and Their Formulas Preliminary Example. The cost, C, of your monthly phone bill consists of a $30 basic charge, plus $0.10 for each minute of long distance calls. (a) Complete the table below, and sketch a graph. 30 15 t (minutes) 0 30 60 90 10 150 C ($) 30 33 36 39 4 45 45 C t 30 60 90 10 150 (b) Compute the average rate of change of C over any time interval. C 33 30 = t 30 0 = 3 dollars = 0.10 dollars per minute. 30 minutes Note that this rate of change is the charge per minute of calling. (c) Find a formula for C in terms of t. so our answer is C = 30 + 0.10t. ( ) $.10 C = $30 + (number of minutes talked) minute = 30 + 0.10t, (d) If your bill is $135, how long did you talk long distance? We have C = 135 = 135 = 30 + 0.10t = 105 = 0.10t = 1050 = t, which means that you talked 1050 minutes, or 17 hours and 30 minutes. 8 Developed by Jerry Morris

Notes on Linear Functions: 1. If y = f(x) is a linear function, then y = mx + b, where m = slope (rate of change of y with respect to x) b = y-intercept (value of y when x = 0.). If y = f(x) is linear, then equally spaced input values produce equally spaced output values. Different forms for equations of lines: 1. y = mx + b (m = slope, b = vertical intercept). y y 0 = m(x x 0 ) (m = slope, (x 0, y 0 ) is a point on the line) 3. Ax + By + C = 0 (A, B, and C are constant) Example. Find the slope and the y-intercept for each of the following linear functions. (a) 3x + 5y = 0 Rewriting, we have 3x + 5y = 0 = 5y = 3x + 0 = y = 3x + 0 5 = y = 3 5 x + 4, so the slope is 3/5 and the y-intercept is 4. (b) x y 5 = Rewriting, we have x y 5 = = 5 x y = 5 5 = x y = 10 = y = x 10, so the slope is 1 and the y-intercept is 10. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 011 9

Examples and Exercises 1. Let C = 0 0.35t, where C is the cost of a case of apples (in dollars) t days after they were picked. (a) Complete the table below: t (days) 0 5 10 15 C (dollars) 0 18.5 16.50 14.75 (b) What was the initial cost of the case of apples? so the initial cost is $0. t = 0 = C = 0 0.35(0) = 0, (c) Find the average rate of change of C with respect to t. Explain in practical terms (i.e., in terms of cost and apples) what this average rate of change means. Since the cost function is linear, the rate of change of C with respect to t is constant, so we can choose any interval to do this calculation. Therefore, the average rate of change is C 18.5 0 = = 0.35 cents per day. t 5 0 This means that for each day that goes by after the picking day, the price of the apples decreases by 35 cents.. In parts (a) and (b) below, two different linear functions are described. Find a formula for each linear function, and write it in slope intercept form. (a) The line passing through the points (1, ) and ( 1, 5). (b) C 10 15 0 5 F 50 59 68 77 (a) First we calculate the slope: m = y x = 5 1 1 = 3 Therefore, we know that the equation of the line looks like y = 3 x + b. To obtain a final answer, we need to calculate b. Since (1, ) is a point on the line, we can substitute x = 1 and y = into our formula to obtain y = 3 x + b = = 3 1 + b = + 3 = b, which means that b = 4 + 3 = 7. Our final answer is therefore y = 3 x + 7. (b) Letting F be the output variable and C be the input variable, we first calculate the slope: m = F 59 50 = C 15 10 = 9 5 Therefore, we know that the equation of the line looks like F = 9 5C + b. To obtain a final answer, we need to calculate b. Since (10, 50) is a point on the line, we can substitute C = 10 and F = 15 into our formula to obtain F = 9 5 C + b = 50 = 9 10 + b = 50 18 = b, 5 which means that b = 50 18 = 3. Our final answer is therefore F = 9 5 C + 3. 10 Developed by Jerry Morris

3. According to one economic model, the demand for gasoline is a linear function of price. If the price of gasoline is p = $3.10 per gallon, the quantity demanded in a fixed period of time is q = 65 gallons. If the price is $3.50 per gallon, the quantity of gasoline demanded is 45 gallons for that period. (a) Find a formula for q (demand) in terms of p (price). First, we calculate the slope: m = q 45 65 0 gallons = = = 50 gallons per dollar p 3.5 3.1 0.4 dollars Therefore, we have q = 50p + b, so we need to find b. Since q = 65 when p = 3.1, we have q = 50p + b = 65 = 50(3.1) + b = 155 + 65 = b, so b = 0. Our final answer is therefore q = 50p + 0. (b) Explain the economic significance of the slope in the above formula. In other words, give a practical interpretation of the slope. The slope of -50 gallons per dollar indicates that, for every dollar increase in the price, the demand for gasoline goes down by 50 gallons. (c) According to this model, at what price is the gas so expensive that there is no demand? We have q = 0 = 0 = 50p + 0 = 50p = 0 Thus, at a price of $4.40 per gallon, there is no demand. = p = 4.4. (d) Explain the economic significance of the vertical intercept of your formula from part (a). The vertical intercept occurs at the point (0, 0). This indicates that, at a selling price of $0 per gallon, the demand for gasoline would be 0 gallons. In other words, if the gas station gave the gas away for free, 0 gallons would be the demand. 4. Look back at your answer to problem (b). You might recognize this answer as the formula for converting Celsius temperatures to Fahrenheit temperatures. Use your formula to answer the following questions. (a) Find C as a function of F. Since F = 9 5 C + 3 = F 3 = 9 5 C = C = 5 (F 3), our final answer is 9 C = 5 (F 3). 9 (b) What Celsius temperature corresponds to 90 F? Using part (a), we have F = 90 = C = 5 9 (90 3) = 5 9 58 = 3., so 3. C is our answer. (c) Is there a number at which the two temperature scales agree? The temperature scales will agree if and only if F = C, so we have F = C = F = 5 (F 3) 9 = 9F = 5F 160 = 4F = 160 = F = 40, so the scales agree at 40. In other words, 40 F and 40 C are equal temperatures. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 011 11

Section 1.5 Geometric Properties of Linear Functions Example 1. You need to rent a car for one day and to compare the charges of 3 different companies. Company I charges $0 per day with an additional charge of $0.0 per mile. Company II charges $30 per day with an additional charge of $0.10 per mile. Company III charges $70 per day with no additional mileage charge. (a) For each company, find a formula for the cost, C, of driving a car m miles in one day. Then, graph the cost functions for each company for 0 m 500. (Before you graph, try to choose a range of C values would be appropriate.) Total Cost = Daily Charge + (Charge Per Mile)(Miles Driven) C = 0 + 0.0m Company I C = 30 + 0.10m Company II C = 70 Company III C III I II m 1 m (b) How many miles would you have to drive in order for Company II to be cheaper than Company I? From the graph in part (a), we see that Company II is cheaper if we drive more than m 1 miles. We therefore begin by finding the point of intersection of lines I and II: 0 + 0.0m = 30 + 0.10m = 10 = 0.10m = 100 = m, Thus, m 1 = 100 miles, and we conclude that Company II is cheaper if we drive more than 100 miles. 1 Developed by Jerry Morris

Example. Given below are the equations for five different lines. Match each formula with its graph to the right. y A B C f(x) = 0 + x g(x) = 0 + 4x h(x) = x 30 u(x) = 60 x v(x) = 60 x B A C D E D E x Facts about the Line y = mx + b 1. The y-intercept, b (also called the vertical intercept), tells us where the line crosses the y-axis.. If m > 0, the line rises left to right. If m < 0, the line falls left to right. 3. The larger the value of m is, the steeper the graph. Parallel and Perpendicular Lines Fact: Two lines (y = m 1 x + b 1 and y = m x + b ) are... 1.... parallel if m 1 = m (slopes are the same)..... perpendicular if m 1 = 1 m (slopes are negative reciprocals of one another). Activities to accompany Functions Modeling Change, Connally et al, Wiley, 011 13

1. Consider the lines given in the figure to the right. Given that the slope of one of the lines is, find the exact coordinates of the point of intersection of the two lines. ( Exact means to leave your answers in fractional form.) First, note that the slope of the line having a positive slope goes through the points (0, ) and (, 0), so its slope is m = 0 ( ) 0 = 1, and we can see that the y-intercept is ; therefore, this line has equation y = x. We are given that the slope of the other line is, and we can see that its y-intercept is 3; therefore, its equation is given by y = x+3. 3 y x To solve the problem, we must find the point of intersection of the lines y = x and y = x + 3. Thus, we have x = x + 3 = 3x = 5 = x = 5 3, and so y = x = 5 3 = 1 3. Our final answer is therefore (5 3, 1 3 ).. Parts (a) and (b) below each describe a linear function. Find a formula for the linear function described in each case. (a) The line parallel to x 3y = that goes through the point (1, 1). First, note that x 3y = = 3y = x = y = 3 x 3, so the given line has slope /3. Therefore, using point-slope form for the equation of a line, we have y 1 = 3 (x 1) = y = 3 x 3 + 1 (b) The line perpendicular to x 3y = that goes through the point (1, 1). As in part (a), the given line x 3y = has slope /3. Therefore, the slope of a perpendicular line is 3/. Once again, using point-slope form, we have y 1 = 3 (x 1) = y = 3 x + 3 + 1 = y = 3 x + 5, so our final answer is y = 3 x + 5. = y = 3 x + 1 3. Thus, our final answer is y = 3 x + 1 3. 14 Developed by Jerry Morris