1. Section 1-4 Linear Functions and Models

2. Using Equations to Solve Problems Tony’s new Cadillac Escalade costs $420 per month for car payments (he chose the 29” spinners) and insurance. Tony estimates that gas and maintenance cost $0.25 per mile. Express Tony’s total monthly cost in relation to the number of miles driven during the month. Tony needs to determine if he can afford this high class ride and maintain his image as a high roller. What is the slope of the graph of the cost function? If Tony earns $515 per month (after taxes) how many miles can he afford to drive monthly?

3. Problem Solving Guidelines Phase 1: UNDERSTAND the problem What am I trying to find? What data am I given? Have I ever solved a similar problem? Phase 2: Develop and carry out a PLAN What strategies might I use to solve the problem? How can I correctly carry out the strategies I selected? Phase 3: Find the ANSWER and CHECK Does the proposed solution check? What is the answer to the problem? Does the answer seem reasonable? Have I stated the answer clearly?

4. Function A function describes a dependent relationship between quantities. When we looked at Tony’s monthly cost of owning the Cadillac Escalade, we saw that cost is dependent upon miles driven, or in other words, cost is a function of miles driven. Simple example: The value of the expression 3x - 5 is a function of x and is written f(x) = 3x – 5 The function notation f(x) is read “f of x” It represents the value of f evaluated at x. Thus if f(x) = 3x – 5, f(2) = 3*2 – 5 = 6 – 5 = 1

5. Zeros of Functions Let’s look at f(x) = 3x – 5 again Notice that f(5/3) = 3(5/3) – 5 = 5 – 5 = 0 We say that 5/3 is a zero of the function f. In general, if f(a) = 0, then a is called a zero of the function f.

6. Linear Functions Examples of linear functions f(x) = 3x – 5f is a linear function of x L(T) = 0.0001T + 10L is a linear function of T g(s) = -1.2s + 4.7g is a linear function of s h(t) = 3h is a linear function of t Linear functions have the form f(x) = mx + b The function h above is linear because it can be written as h(t) = 0t + 3. This special kind of linear function is called a constant function. The graphs of linear functions are straight lines or points on a line. Why are linear functions straight lines?

7. Other Characteristics of Functions Domain: The domain of a function is the set of values for which the function is defined. Domain pertains to the dependent variable (typically defined as x) Ex) Tony’s monthly Escalade bill Depends on miles driven The domain is m ≥ 0 Cannot have negative mileage (although it would be nice, for that would result in smaller monthly payments) Domain = the set of input values

8. Other Characteristics of Functions Range The range is the set of output values for a function Ex) The range of Tony’s monthly Escalade payment is C ≥ 420 420 is the minimum payment and there is no limit on the miles to drive in a month, so the Cost has no limit either Is there realistically a limit on the number of miles driven in a month? How will this effect the range?

9. Mathematical Models A mathematical model is one or more functions, graphs, tables, equations, or inequalities that describe a real-world situation. Examples include Tony’s monthly Escalade bill, Example 1 on p20, and Example 2 on p21 in your book. People in marketing use modeling to study consumer preferences, determine optimal location in product space, allocate advertising resources, design distribution systems, forecast market behavior, and study competitive strategy.

10. Homework p23: 1-12 even 13, 14, 16, 18, 19, 24