Warm-up (Use own paper) Your group will be assigned a section –don’t do all! 1.Find the endpoints and state whether the interval is bounded or unbounded. a) [0,5]b) [2,  ) 2.Simplify: a) (uv 2 ) 3 /(v 2 u 3 ) b) (3x 2 y 3 ) -2 3.Solve each equation or inequality algebraically. a) 6x 2 + 7x = 3b) |4x + 1 | = 3 c) 4x 2 – 4x + 2 = 0d) |2x – 5| < 7 e) |3x + 4| > 2f) 4x x + 9 > 0 4.Solve graphically: x 3 + 2x 2 – 4x – 8 = 0

1.1 Functions and Graphs After today’s lesson you should be able to  Use numerical, algebraic, and graphical models to solve problems  Translate from one model to another

Let’s get motivated If you would like to predict the total expected rainfall in China over the next year, which type of model would be most helpful: Numerical (rainfall totals over the last 10 years) Graphical (scatter plot of rainfall totals vs year) Algebraic (formula for yearly rainfall totals) Explain your reasoning.

Definitions Mathematical model –mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior. Mathematical modeling –process of devising mathematical models

Numerical Models Numbers (or data) are analyzed to gain insights into phenomena Examples The major league baseball standings Network of interrelated numbers that measure the global economy

YearMinimum Hourly Wage Purchasing Power in 2001 Dollars a) In what five- year period did the actual minimum wage increase the most? b) In what year did a worker earning the minimum wage enjoy the greatest purchasing power? c) What was the longest period during which the minimum wage did not increase? d) A worker on minimum wage in 1980 was earning nearly twice as much as a worker on minimum in 1970, and yet there was a great pressure to raise the minimum wage again. Why? Ex 1

Algebraic Models Uses formulas to relate variable quantities associated with the phenomena being studied. What advantage does an algebraic model have over a numerical model?

Ex 2 A bakery sells a 9” by 13” panettone for the same price as an 8” diameter round panettone. If the round panettone is twice the height of the rectangular panettone, which option would you choose? Why?

Exploration 1: Finding an Algebraic Model A department store is having a sale in which everything is discounted 25% off the marked price. The discount is taken at the sales counter, and then a state sales tax of 6.5% and a local sales tax of 0.5% are added on. 1)If you only have $30, can you afford to buy a shirt marked $36.99? 2)If you have a credit card but are determined to spend no more than $100, what is the maximum total value of your marked purchases before you present them at the sales counter? Be prepared to support your answers.

Ex 3 Come up with a situation that could be modeled with this data set. Find an algebraic model for the data. Justify your choice of model. t y

Graphical Models A visible representation of a numerical model or an algebraic model that gives insight into the relationship between variable quantities. Using a graphing calculator aids in solving problems using a graphical model

Limitations of Graphical Models Graph of numerical or algebraic model is not visible (adjust screen window) Vertical Asymptotes (use the table menu on the calculator to view specific value of x) Part of graph is hidden (adjust screen window)

Solving Equations Graphically Method 1 1. Enter left-hand side as equation 1 2.Enter right-hand side as equation 2 3.Graph both equations 4.The solutions are the points of intersection Method 2 1.Set equation equal to zero. 2.Enter altered equation into calculator 3.The solutions are the x-intercepts (y = 0)

Ex 4 Solve the equation x 2 = x algebraically and graphically.

Graph y = 3/(2x – 5). Is there an x-intercept? Solve graphically: x 3 – 1.1x 2 – 65.4x = 0 The behavior of a graphical model is sometimes hidden.

Let’s summarize Which method is visual? Which method generates unknown values from known quantities? Which method is all about the numbers? Do you have a preference?