Chapter 2: Functions and Models. 2.1 The Language of Functions. Definitions

Transcription

1 Chapter 2: Functions and Models 2.1 The Language of Functions Definitions Relation any set of ordered pairs o Often times the second number depends on the first number in some way o When this happens, the first number is called the and the second number is called the Function a set of ordered pairs (x, y) in which each first component (x) is paired with second component (y) o Ex: o In other words, a function is a correspondence between two sets A and B in which each element in A corresponds to element of B. Domain the set of all first components (x s) of a function o Ex: Range the set of all second components (y s) of a function o Ex: Real Functions functions whose domain and range are sets of numbers o Unless the domain of a function is explicitly stated, you may assume that it is the set of all real numbers for which the function is define Member/element if x is in a set A, then x is said to be a or an of A o Ex: Sets of numbers 1. Integers 2. Real Numbers 3. Rational Numbers 4. Natural Numbers

2 Ex 1: A bakery charges $2.00 per muffin. Customers get a $2.00 discount for every 6 muffins purchased. a. Which statement is true: the cost c is a function of the number m of muffins or the number m of muffins is a function of the cost c? b. Identify the independent and dependent variables of the function c. State the domain and range of the function Description of Functions Finding Domain and Range Linear Functions Quadratic Functions

3 Rational Functions Square Root Finding Domain and Range from a Graph Ex 3: A rule for the function graphed at the right is y = 2 x 4. State the domain and range of the function. Vertical Line Test Naming Functions and Their Values

4 Ex 4: Suppose f is the function defined by the rule f(x) = 4 ( 1 2 )x for all real numbers x. a. Evaluate f(5) b. Does f( 2 + 3) = f( 2) + f(3)? c. Evaluate f(q + 1) Ex 5: Suppose g(x) = x 2 2x + 1 a. Evaluate g( 2) b. Evaluate g(x 4) 2.2 Linear Models Definitions Slope and Point Slope Formulas

5 Ex 1: Jewelers emphasize that the price of a diamond is determined by the cut, carat weight, color and clarity. The table at the right gives carat weights and approximate prices in U.S. dollars for twenty diamond rings sold at a recent auction in Singapore. All rings are of the same quality gold and contain a single diamond. The data are graphed below. The linear model is based on the weight and diamond used. Although the data is not collinear, the line through (0.18, 600) and (0.32, 1400) seems close to the points. It has been added to the graph. An equation of this graphed line is one model for these data. Is the size alone a good predictor for the price? a. Find an equation of the graphed line which related weight and price. b. Interpret the slope of the line in the context of the problem. c. Use the model to estimate the price of a 0.3 carat diamond ring. d. Why is the set of data not a function? Prediction of Values using a Linear Model

6 Measuring How Well a Line Models Data Observed values data collected from sources such as or o Ex: o To the right, the scatterplot shows the data along with the graph of the line with equation y = 2400x Is this the best model for the data??? How do we find out?? Predicted values the values predicted by the model Residual the observed value minus the predicted value o Residuals can be positive or negative o Ex: Sum of Squared Residuals If you compare two lines, the one with the sum of squared residuals is not as good a model as the one with the sum of squared residuals Ex 2: Calculate the sum of squared residuals for the above data and model.

7 Ex 3: Let s collect our own data. Let s record everyone height (in inches) vs their shoe size (in men s sizes). a. Find an equation which relates height to shoe size. b. Find the predicted values using this model. c. Find the sum of squared residuals for our model.

8 2.3 Linear Regression and Correlation Line of Best Fit Three Important Properties How to Find the Line of Best Fit in your Calculator How to Find the Sum of Squared Residuals in your Calculator Ex 1: Use the data to answer the following a. Use your calculator to find the line of best fit b. According to the regression line, how much will a 0.5-carat diamond ring cost? c. Verify that the center of mass (0.212, $793.65) is on the line d. Find the sum of squared residuals for the linear regression

9 Correlation coefficient a value that describes the nature of a set of data the more closely the data fit a line, the closer the correlation coefficient,, is to. Correlation Coefficient (r value) Ex 2: Look at the following data a. Estimate a correlation coefficient for each WITHOUT using a calculator b. Choose a data set to place in your calculator. Draw a scatterplot for the data. Do you think your estimate is accurate? c. Calculate the line of best fit and correlation coefficient for your data set.

10 2.4 Exponential Functions Ex 1: The town of Centerburg is suffering from a decline in population. A demographer has predicted that, under current conditions, the current population of 28,500 will decrease by 2% each year over the next decade. However, a manufacturer claims that a new factory will reverse the trend and cause the population to grow by 4% annually instead. a. Create equations to describe the population of Centerburg as a function of time under these two conditions: b. Compare the projected populations after 10 years Ex 2: The population of Winnemucca, Nevada, can be modeled by P=6191(1.04)t where t is the number of years since What was the population in 1990? By what percent did the population increase by each year? What would the population be today? Another Form of Exponential Growth

11 Ex 3: $2500 is invested in an account with a 5.3% annual yield. a. What is the yearly growth factor b. What is the equation for the balance A after t years? c. What is the balance after 7 years? d. What value is associated with t = -2? What does t = -2 mean? Ex 2: Without a graphing utility, graph y = 16( 1 4 )x **in all cases, the x-axis is a horizontal asymptote (a line that the function approaches but will never touch) of the function Ex 4: For each function characteristic compare and contrast he linear function f(x) = 2x + 8 with the exponential function g(x) = 8 4 x a. Domain and range b. Y-intercept and x-intercept c. Asymptote d. Increasing or decreasing

12 2.5 Exponential Models Finding an Exponential Function Using a System of Equations Ex 1: Huntley, Illinois had been a small farming town. But when a large housing development was built, the population growth pattern changed. Two special censuses gave village planners the data in the table at the right. a. Find an exponential model for the data. Let p(t) be the population t years after b. Predict the population of Huntley in the year Ex 2: The population of a certain cell type was observed to be 100 on the second day and 2700 on the fifth day. Assuming the growth is exponential, find the number of cells present initially, and find the number of cells expected on the seventh day. Exponential Regression

13 Ex 2: Bald eagles were once threatened with extinction. In the 48 contiguous states, their numbers were at an all-time low of 417 in But protection programs helped them rebound. In 2007, they were removed from the list of endangered species kept by the U.S. Fish and Wildlife Services. a. Use your graphing calculator to draw a scatterplot for the data to the right. Let x be years after b. Use your graphing calculator to find a model for the data. c. Place the graph of the model in the same plane as the data and describe how well the exponential curve fits the data. d. Identify the initial amount and the growth factor and explain their meanings. e. Find the residuals for the models predicted values for 2000 and Half-Life and Exponential Decay Ex 3: Detectives in the Litvinenko investigation found polonium on a cup in a hotel that he had visited. Suppose that 4 micrograms were found, and it had been 30 days since Litvinenko was there. (The half-life of polonium is 138 days.) a. Find how much polonium was on the cup originally b. Derive a model for this situation

14 2.6 Quadratic Models Properties of Quadratic Functions Ex 1: Consider the function f with equation f(x) = 2x 2 3x 2 a. Find the x- and y- intercepts of its graph b. Tell whether the parabola has a maximum or minimum point and find its coordinates c. Graph the parabola

15 Ex 2: Consider the function f with equation f(x) = 3x 2 + 9x a. Fins the x- and y- intercepts of its graph b. Tell whether the parabola has a maximum or minimum point and find its coordinates c. Graph the parabola Using Quadratic Models Ex 3: A ball is thrown upward from a height of 15m with initial velocity 20 m/sec. a. Find the relation between height h and time t after the ball is released b. How high is the ball after 3 seconds? c. When will the ball hit the ground?

16 Quadratic Regression Ex 4: Use the data to do the following a. Use your graphing calculator to create a scatterplot of the data b. What type of regression would best fit the data? c. Find a regression equation and correlation coefficient. d. Estimate the percent weight gain for 12 pellet dosage.