Chapter 1: Functions and Graphs

1. Chapter 1:Functions and Graphs Section 1-1: Modeling and Equation Solving

2. Objectives • You will learn about: • Numerical models • Algebraic models • Graphical models • The zero factor property • Problem solving • Grapher failure and hidden behavior • A word about proof • Why: • Numerical, algebraic, and graphical models provide different methods to visualize, analyze, and understand data.

3. Vocabulary • Mathematical model • Mathematical modeling • Numerical model • Algebraic model • Graphical model • Solve graphically • Solve numerically • Solve algebraically • Root (or solution) • Equation • Zero • X-intercept • Graph • Supported numerically • Grapher failure • Hidden behavior • Deductive reasoning

4. Example 1:Tracking the Minimum Wage • The numbers in the table show the minimum hourly wage from 1955 to 2005. The table also shows the MHW adjusted to the purchasing power of 1996 dollars. • In what five- year period did the actual MHW increase the most? • In what year did a worker earning minimum wage enjoy the greatest purchasing power? • A worker on minimum wage in 1980 was earning nearly twice as much as a worker on minimum wage in 1970, and yet there was greater pressure to raise minimum wage again. Why?

5. Example 2:Analyzing Prison Populations • The growth in the number of prisoners incarcerated in state and federal prisons from 1980 to 2000 is shown. Is the proportion of female prisoners over the years increasing?

6. Look at the percentage of female prisoners:

7. Example 3:Comparing Pizzas • A pizzeria sells a rectangular 18” by 24” pizza for the same price as its large round pizza (24” in diameter). If both pizzas are the same thickness, which option gives more pizza for the money?

8. Example 4:Visualizing Galileo’s Gravity Experiments Galileo spent a good deal of time rolling balls down inclined planes carefully recording the distance they traveled as a function of elapsed time. His experiments are commonly repeated in physics classes today, so it is easy to reproduce a typical table of Galilean data.

9. Example 4:Visualizing Galileo’s Gravity Experiments Create a scatter plot with the data. Enter: time in L1 distance in L2 Does the graph appear to be linear or quadratic? We have a parabola, so we will use a quadratic model. General Form: d = kt2 Use the point (1, 0.75) to form the specific equation. Does the equation satisfy the rest of the data?

10. Example 5:Fitting a Curve to Data • Recall Example #2. • Model the growth graphically and use the graphic model to suggest an algebraic model.

11. Example 6:Solving an Equation Algebraically • Find all real numbers x:

12. Example 7:Solving an Equation: Comparing Methods • Solve the equation using the quadratic formula:

13. Fundamental Connection • If a is a real number that solves the equation f(x)=0, then these three statements are equivalent: • The number a is a root (or solution) of the equation f(x)=0 • The number a is a zero of y=f(x) • The number a is an x-intercept of the graph y=f(x). The point (a, 0) is the x-intercept.

14. A Problem Solving Process • Step 1: Understand the problem. • Step 2: Develop a mathematical model of the problem. • Step 3: Solve the mathematical model and support or confirm the solution. • Step 4: Interpret the solution in the problem setting.

15. Example 8:Applying the Problem Solving Process • The engineers at an auto manufacturer pays students $0.08 per mile plus $25 per day to road test their new vehicles. • How much did the auto manufacturer pay Sally to drive 440 miles per day? • John earned $93 test driving a new car in one day. How far did he drive?

16. Example 9:Seeing Grapher Failure • Look at the graph of y=3/(2x-5). • Is there an x-intercept?

17. Example 10:Not Seeing Hidden Behavior • Solve Graphically:

18. Example 11:Proving a Peculiar Fact • Prove that 6 is a factor of n3 – n for every positive integer n. • Start by setting up a table of values.