Unit 2: Linear Functions

Transcription

1 Unit 2: Linear Functions 2.1 Functions in General Functions Algebra is the discipline of mathematics that deals with functions. DEF. A function is, essentially, a pattern. This course deals with patterns that we encounter in the real world, particularly in the disciplines of business, life sciences, and social sciences (history, psychology, sociology, etc.) There are four main was that a pattern can be expressed 1. With words 2. A table 3. An equation (mathematically) 4. A graph or mapping Recall the Math Lab we did the cubes and the stickers: Words: One cube has six stickers, then as we add cubes to the rod four stickers are added with each cube. Table: #cubes #stickers Equation: Graph:

2 Example Let s examine some techniques of showing how Fahrenheit measurements are related to Celsius measurements. One method is to create a table listing some Celsius measurements and the corresponding Fahrenheit measurement. We can call the measurements data. Table. Celsius ( C) Fahrenheit ( F) Define these values as a set of ordered pairs: Create a scatter plot, then a regression line on your calculator. Then draw it on the graph below. Equation for this function: DOMAIN: RANGE: Characteristics of Function: 1) 2) 3) 4)

3 For each of the following graphs determine whether it is a function, then find the DOMAIN and RANGE. If it is a function, please state whether it is continuous or discrete.

4 Vertical Line Test What is the purpose of the Vertical Line Test? Check out this function: y = x 2 + 4x Function Notation We use function notation In particular, the point (a, f(a)) lies on the graph of y = f(x) for any number a in the domain of the function. We can also say that So, for example Example If f(x) = 4x 2 2x + 3. Solve algebraically and by graphing: a) f(3) = b) f( 1) = Example If f(x) = 2x 3 + 5x 2 28x 15. Solve algebraically and by graphing: a) f( 2) = b) f(4) =

5 Problems: Mathematical Modeling The process of translating real-world information into a mathematical form so that it can be applied and then interpreted in the real-world setting is called modeling. Example Consider the following data table (can be found on pg. 18). It illustrates the number of drinks and the resulting blood-alcohol percent for a 90-pound woman. The drinks represent one 12 ounce beer. Number of Drinks Blood Alcohol Percent On your calculator create a scatter plot and regression line. What is the equation for this line? What can we do with this information?

6 Alinging Data When finding a model to fit a set of data it is often easier to use rather than the. Aligned inputs are simpy input values that have been converted to smaller numbers by subtracting the from each input. For instance, instead of using t as the actual year, it might be more convenience to use an aligned input that is the of years after Example Public health care expenditures for the period of can be modeled by the function E(t) = 738.1(1.065) t where E(t) is in the billions of dollars and t is the number of years after a) What value of t represents 2010? b) Approximate the public health care expenditures from c) Use the model to estimate the public health care expenditures for Can we be sure that this estimate is accurate?

7 2.2 Graphs of Functions Graphing an equation by plotting points. Example Graph the equation f(x) = x 2 by plotting points and drawing a smooth curve through the points determined by the integer values between -3 and 3. x f(x)=x 2 A graph is complete if it shows the basic shape of the graph and important points on the graph (including points where the graph crosses the axes and points where the graph turns) and suggests what the unseen portions of the graph will be. Example Graph the equation f(x) = x 3 3x by plotting points and drawing a smooth curve through the points determined by the integer values between -3 and 3. Use the fact that the graph has at most two turning points. x f(x)=x 2 Determine the ZEROS by factoring:

8 Graphing on TI-84. Using restrictions. Example Cost Benefit. Suppose that the cost C of removing p% of the pollution from drinking water is given by the model C = 5350p 100 p dollars. a) Use the restriction on p to determine the limitation on the horizontal-axis values (which are the x-values on the calculator) b) Graph the function on the viewing window [0,100] by [0, 50,000]. Why is it reasonable to graph this model on a viewing window with the limitation C 0? c) Find the point on the graph that corresponds to p= 90. Interpret the coordinates of this point. Determining the Viewing Window. Example Aging Workers. The millions of Americans who are working full time jobs at selected ages from can be modeled as a function of their age, x. A model is f(x) = x x x a) What would be the appropriate viewing window for the x-values of this function? Graph it. b) What would be the appropriate viewing window for the y-values of this function? c) Use the model to estimate the number of Americans age 55, and age 64 who are working full time. 2.3 Linear Functions (y= mx + b) Def. A linear function is a function whose graph is a line. Linear functions can be written in the form f(x) = ax + b, or y = ax + b, where a and b are constants.

9 Problems: Determine which of the following functions are linear. Circle the linear ones: a. 0 = 2t s + 1 b. y = 5 c. xy = 2 Now determine the domain and range for each of the above functions. Intercepts The points where the graph crosses or touches the x-axis and the y-axis are called the x-intercepts and y-intercepts, respectively. The x-intercept occurs on the graph at the point when. The y-intercept occurs on the graph at the point when. Example We can find the x and y intercepts graphically and algebraically. Find the x and y intercepts for 2x 3y = 12 both graphically and algebraically. Solve graphically: Solve algebraically: 2x 3y = 12 is the same as y = 2 3 x 4

10 How do I do this on my calculator? Another name for the x-intercept is the. KEY STROKES FOR FINDING THE ZERO Step One: Step Two: Step Three: Step Four: Another name for the y-intercept is the. KEY STROKES FOR FINDING THE Y-INTERCEPT Step One: Step Two: Step Three: Example Loan Balance. A business property is purchased with a promise to pay off a $60,000 loan plus the $16,500 interest on this loan by making 60 monthly payments of $1275. The amount of money, y, remaining to be paid on $76,500 (the loan plus interest) is reduced by $1275 each month. Although the amount of money remaining to be paid changes every month, is can be modeled by the linear function y = 76, x where x is the number of monthly payments made. We recognize that only the integer value of x from 0 to 60 apply to this application. a. Find the x-intercept and the y-intercept of the graph of this linear equation. b. Interpret the intercepts in context of this problem situation. c. How should x and y be limited in this model so that they make sense in the application?

11 d. Use the intercepts and the results from part c to sketch the graph of the given equation. SLOPE Def. The slope of the graph of the equation y = mx + b is m, and the y-intercept of the graph is b. So the graph crosses the y-axis at (0, b). Problems: Find the slope of the following linear equations: a. y = 7x 12 b. 2x 3y = 12 Slope formula: To find the slope of a line between two points (x1, y1) and (x2, y2), use the following formula: Practice: Refer back to the example 2.2.6, the Loan Balance problem. What is the slope of that line? What does that slope tell us in terms of the situation?

12 2.4 Constant Rate of Change Constant Rate of Change Constant Rate of Change: The rate of change of the linear function y=mx+b is the constant m, the slope of the graph of the function. The rate of change is a ratio that compares how much one quantity changes relative to the change in another quantity. If x is the independent variable, then the rate of change change in y y =. This is sometimes referred to as change in x x. Example College Enrollment. The total fall enrollment in 4-year state institutions is given by y = 0.014x , where x is the number of years after 1990 and y is the millions of students. a) Is this a linear function? b) Why or why not? c) What is the independent variable? d) What is the rate of change? e) What is the year at t = 0? f) What was the total fall enrollment in 2000, according to this model? Boiling Water Lab*** Example Chemistry. The table shows the temperature of a solution after it has been removed from a heat source. Time (min) Temperature ( )

13 a) Is this a linear function? b) Find the rate of change in the temperature for the solution. c) Write a linear equation, which would be an accurate model for this data. d) According to the model, what would be the temperature of the water at 14 minutes? e) According to the model, when would the temperature of the water be room temperature? Revenue, Cost, Profit (pg. 52) The profit that a company makes on its product is the difference between the amount received from sales (revenue) and the production and sales costs. If x units are produced and sold, we can write where P(x) = R(x) C(x) P(x) = profit from the sale of x units R(x) = total revenue from the sale of x units In general, revenue is found from the equation C(x) = total cost of the production and sale of x units Revenue = (price per unit) (number of units) The total cost is composed of two parts: fixed costs and variable costs. Fixed costs (FC), such as depreciation, rent, and utilities, remain constant regardless of the number of unites produced. Variable Costs (VC) are those directly related to the number of units produced. This, the total cost, often simply called the cost, is found by using the equation cost = variable cost + fixed cost

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