CHAPTER 6 Notes: Functions A mathematical model is an equation or formula in two variables that represents some real-world situation. The key building blocks of mathematical models are functions and their properties. To have a function there must first be a rule, which describes how to match each input with an output. Such a rule also results in a set of ordered pairs and is called a relation. A function is a special type of relation such that each input corresponds to EXACTLY ONE output. Inputs are also called independent variables; outputs are also called dependent variables. When graphing, the input names the horizontal axis and the output names the vertical axis. The set of all inputs that result in a real number output is called the domain. The set of all outputs that results from using all inputs is called the range. A reasonable domain refers to only those inputs that "make sense" in the context of the situation being described. The first step in creating or using a mathematical model is to determine which of the variables represents the input and, if appropriate, decide on a name for the rule or function. In general, the output depends on the input OR the output is a function of the input OR if the function is called f, then output = f(input) read "output equals f of the input". The following example and exercises practice these ideas. EXAMPLE 1: Suppose a ten inch candle is lit and is then extinguished 60 minutes later when the candle is only 1 inch in length. Use L for length and t for minutes since the candle was lit. i) Express the function in words. The length L depends on or is a function of t. ii) Identify the independent variable and the dependent variable. time t is the independent variable (input); length L is the dependent variable (output). iii) Express the function using an ordered pair. (t, L) iv) Make a rough sketch of the function. Label the axes using the given quantities. The key points are the vertical intercept (0, 10) and the point (60,1). L inches Note: An appropriate domain would be 0 t 60
Chapter 6 Additional Exercises For each situation described in 1 & 2 below, i) Express the function in words. Include the words: one quantity "depends on" or "is a function of" the second quantity. ii) Clearly identify the independent variable and the dependent variable. ii) Express the function using an ordered pair. iv) Make a rough sketch of the function. Label the axes using the given quantities. 1. As the slope of a hill increases, the speed of a person on a skateboard increases. 2. As the price of a product increases, the demand for that product decreases. For 3 to 5 below, i) Write a short statement that expresses a possible function between the quantities. Include the words: one quantity "depends on" or "is a function of" the second. ii) Identify the independent and dependent variables. 3. (weight of a bag of apples, price of the bag) 4. (tax rate, revenue collected) 5. (distance from Earth, strength of gravity) For each function described in 6 to 9, use your intuition or, if necessary, additional research to do the following. a) Describe an appropriate domain for graphing the function. b) Describe the range. c) Make a rough sketch of the function. Explain any assumptions you used when making your graph. 6. (altitude, temperature) 7 (day of year, high temperature) 8. (mortgage interest rate, number of homes sold) 9. (time of day, traffic flow) on a busy stretch of highway 1 in Santa Cruz ================================================================================== Chapter 6 Exercises: Possible Answers 1. i) the speed depends on the slope OR the speed is a function of the slope. ii) input/ind var: slope; output/dep var: speed iii) (slope, speed) iv) will give in class 2. i) the demand depends on the price OR the demand is a function of the price. ii) input/ind var: price; output/dep var: demand iii) (price, demand) iv) will give in class 3. price is a function of weight; ind var: weight; dep var: price 4. revenue is a function of tax rate; ind var: tax rate; dep var: revenue 5. strength is a function of distance; ind var: distance; dep var: strength of gravity. #s 6-9: Answers may vary. Some possible graphs will be given in class. Possible domains & ranges could be: 6. a) poss domain: -300 to 20,000 ft. b) poss range: -50 to 150 degrees F 7.a) the days of a year b) -50 to 150 degrees F 8.a) 0% to 15% b) 0 to 1000 (or more) 9. a) any 24 hour period b) 0 to 20,000 cars (or more).
Chapter 6.1 Notes: Linear Models A linear model is a linear equation or linear function in two variables. The emphasis will be: 1) writing the linear equation or function from given information 2) interpreting the meaning of the slope in the context of the problem 3) interpreting, in the context of the problem, the meaning of the horizontal and vertical intercepts as well as any other points of interest. Recall: To find the vertical intercept, set input = 0. To find the horizontal intercept, set output = 0. EXAMPLE 2: Interpret the meaning of the slope and, when appropriate, the intercepts. 1) Let s be the speed of a car in miles per hour (mph) and C be the cost of a speeding ticket in dollars. C = 3s - 65 slope m = 3 = 3 change in output change in $s of the fine = = 1 change in input change in mph for every 1 mph over the speed limit, the fine increases $3. 2) Let v be the value of a car and a be the age of the car in years. v = -4000a + 42000 i) slope m = -4000 = -4000 = 4000 change in output = 1-1 change in input every year the value decreases by $4000 = change in $s of value change in years of age ii) the vertical intercept is 42000 initially (a = 0) the value is $42000 iii) the horizontal intercept is (10.5, 0) after 10.5 years the car has no value (v = $0). =========================================================================== Chapter 6.1 Linear Models: Additional Exercises 1) The number N chirps per minute made by a cricket is given by N(t) = 7.2t - 32 where t is the temperature in degrees Celsius. a) Find the number of cricket chirps per minute when t = 5! C. b) Sketch the graph of this equation. Label the axes using N and t (not x & y!). c) Using specific numbers and the context of this model, describe in words the
For each of the following, do each part, using the given letters for the variables, not x & y! 2) When he first joined a health club, Clay could lift 80 pounds. Ten weeks later, he could lift 120 pounds. a) Construct a linear equation relating he amount the could lift (w) with the length of time (t) he belonged to the club. [Hint: use the two points (0,80) and (10,120).] b) Using specific numbers and the context of this model, describe in words the c) Using specific numbers and the context of this model, describe in words the meaning of the y-intercept. 3) When Sue and Jim were first married, they spent about 4 hours a night talking to each other. Two years later, they averaged 3 hours a night. a) Write a linear function relating hours spent talking (h) to number of years (Y) married. [Hint: use the two points (0,4) and (2,3).] b) Using the model from part a), when do Sue and Jim stop talking to each other? c) What is a reasonable domain for this function? d) Using specific numbers and the context of this model, describe in words the e) Using specific numbers and the context of this model, describe in words the meaning of the y-intercept. 4) When Joe was 9 years old, he weighted 64 pounds. At 13 years, he weighed 109 pounds. a) Find a linear model to relate Joe's age (A) and weight (w). b) Use your model to estimate the age at which he will weigh 130 pounds. c) What is a reasonable domain for this function? d) Using specific numbers and the context of this model, describe in words the ================================================================================== chapter 6.1: Possible Answers (Graphs will be discussed in class) 1. a) N = 4 chirps b) The vertical axis is N and horizontal axis t. The horizontal intercept is 32 7.2, 0 and the line is increasing. c) For every increase of 1 degree Celsius there will be 7.2 more chirps OR every increase of 10 degrees means there will be 72 more chirps. 2. a) w = 4t + 80 b) Every week Clay can lift 4 more pounds c) At the beginning (t = 0) Clay could lift 80 pounds. 3. a) h = -1 Y + 4 b) t = 8 years c) 0 Y 8 2 d) Every 2 years they talk one hour less e) At first (Y = 0) they talked 4 hours. 4. a) w = 45 4 A - 149 b) A = 14.87 years or about 14 years 10.4 months c) using common knowledge about 4 ages for boys' growth, about ages A = 8 or 9 years to A = 16 or 18 years. d) Joe gains 45 pounds every 4 years
Chapter 6.2 Notes: Quadratic Models A quadratic model is a quadratic function in two variables. Only those quadratic equations in two variables that are functions will be considered here. The graphs of all quadratic functions are parabolas that open upward or downward. The key facts about a quadratic function are the (two or fewer) horizontal intercepts, the vertical intercept and the input and output of the vertex. Finding these numbers will be reviewed in class. The emphasis will be: 1) To find the key facts about a quadratic function. There will be at most 5 key numbers or facts. 2) To determine which number or key fact answers a question in an application. EXAMPLE 3: 1) Draw a rough graph of a concave up quadratic function C = C(t) that has no horizontal intercepts. Label the vertical intercept (0, A) and the vertex (B, D). 2) Assume C is the number of milligrams of cholesterol-consumed daily by people in the U.S. t years after 1980. Thus, the horizontal axis is named t and the vertical axis is the C axis. 3) Determine whether the y-intercept, the input and the output of the vertex represents milligrams of cholesterol or years after 1980. 4) For the following, determine if A, B, or D would answer the question. a) How many milligrams of cholesterol were consumed daily in 1980? Answer: 1980 is the initial year of the study or when t = 0, so A milligrams is the answer. b) What is the minimum number of milligrams of cholesterol-consumed daily? Answer: The output of the vertex in milligrams or D milligrams is the answer. c) When is the number of milligrams of cholesterol-consumed daily a minimum? Answer: The input of the vertex is years since 1980 so the answer is B years.
EXAMPLE 4: The equation of the graph in Example 3 could be C(t) = 0.11t 2-4.04t + 445. 1) Find the following. For each include appropriate units. If necessary, round to one decimal place. a) vertical intercept. b) the input of the vertex. c) the output of the vertex. 2) Why are there no horizontal intercepts. Solution: 1a) Set input t = 0 and find output C = C(0) = 0.11(0) 2-4.04(0) + 445 = 445 milligrams. b) the input of the vertex is t = b 2a = -4.04 = 4.04 2(0.11) 0.22 = 18.3636... 18.4 years since 1980 or in the year 1980 + 18.4 = 1998.4. c) The output of the vertex is found by finding C when t = 18.3636... C = C(18.3636... ) = 0.11(18.3636... ) 2-4.04(18.3636... ) + 445 = 407.90545... 407.9 milligrams of cholesterol daily. 2) The vertex is above the x-axis and the parabola opens upward so the graph cannot cross the horizontal axis, OR, setting the output C = 0 and using the quadratic formula to find t results in a negative under the radical which is not a real number. Note: Use your calculator's memory function and substitute-unrounded values. Round your final answer only at the end of the calculation. =============================================================================== Chapter 6.2 Quadratic Models: Additional Exercises If appropriate, round to one decimal place. 1) A person standing on a balcony 30 feet above the ground throws a tennis ball directly up with an initial velocity of 25 feet per second. The function that describes the height h of the tennis ball with respect to the number of seconds t since the ball was released is given by: a) What was the initial height of the ball? h(t) = -16t 2 + 25t + 30. b) How long does it take for the tennis ball to reach its maximum height? c) What is the maximum height of the ball? d) How long does it take for the tennis ball to return to the ground? 2) The revenue R for a bus company depends on the number n of unsold seats on the bus. Suppose this relationship is given by: R(n) = 1280 + 48n - 2n 2. a) Find the vertical intercept and interpret its meaning in the context of this problem. b) What number of unsold seats will maximize revenue? c) What is the maximum profit? d) Find the horizontal intercepts. In the context of this problem interpret the meaning of each or explain why one or both are unreasonable.
3) The number of miles M that a certain car can travel on one gallon of gas when traveling at a speed of v miles per hour is given by: M = - 1 30 v2 + 5 v for 0 < v < 70. 2 a) At what speed will the number of miles per gallon be the largest? b) What is the largest number of miles per gallon of gasoline? 4) Methane is a gas produced by landfills, natural gas systems and coal mining that contributes to the greenhouse effect and global warming. Methane emissions in the US can be modeled by the quadratic function G = -0.74t 2 + 8.66t +159.07 where G is the amount of methane produced in million metric tons and t is the number of years after 1990. In what year were methane emissions in the US at their maximum? 5) The number of inmates in custody in US prisons and jails can be modeled by the quadratic function p = -716.2t 2 + 87,453.7t + 1,148,702 where p is the number of inmates and t is the number of years after 1990. a) Will this function have a maximum or a minimum? How can you tell? b According to this model, when will the number of prison inmates in custody in the US be at its maximum/minimum? c) What is the number of inmates predicted for that year. Round your answer to the nearest hundred inmates. Chapter 6.2: Possible Answers 1.a) 30 feet b) 25 0.78 second c) 39.8 feet d) t = 2.4 seconds 2.a) R = 1280 dollars, the revenue 32 if no seats are unsold (n=0) b) n = 12 seats unsold c) R = $1568 d) n = 40 seats unsold means no revenue; n = -16 doesn't make sense in this context. 3.a) v = 37.5 mi/hr b) M = 46.875 mi/gal 4) 1996 5.a) maximum b) 2051 c) 3,818,400 inmates