1. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 2 Linear Functions and Equations

2. 2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. More Modeling with Functions ♦ Model data with a linear function ♦ Evaluate and graph piecewise-defined functions ♦ Evaluate and graph the greatest integer function ♦ Use direct variation to solve problems 2.4

3. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Modeling with Linear Functions To model a quantity that is changing at a constant rate with f(x) = ax + b, the following formula may be used. f(x) = (constant rate of change)x + initial amount The constant rate of change corresponds to the slope of the graph of f, and the initial amount corresponds to the y-intercept.

4. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 A 100-gallon water tank, initially full of water, is being drained at a rate of 5 gallons per minute. (a)Write a formula for a linear function f that models the number of gallons of water in the tank after x minutes. (b)How much water is in the tank after 4 minutes? (c)Graph f. Identify the x- and y-intercepts and interpret each. (d)Discuss the domain of f. Example: Finding a symbolic representation

5. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 (a)Water in tank decreasing at 5 gal/min, so constant rate of change is –5. The initial amount of water is 100 gal. (b)After 4 min the tank contains: Solution Example: Finding a symbolic representation

6. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 (c) Slope: –5 y-intercept: 100 - the number of gallons of water initially in the tank x-intercept: 20 - time in min to empty the tank Example: Finding a symbolic representation

7. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 (d)From the graph: the domain of f must be restricted to 0 ≤ x ≤ 20. It makes sense: can’t have negative time, f(21) = –5(21) + 100 = –5 tank can’t hold –5 gallons of water Example: Finding a symbolic representation

8. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Piecewise-Defined Functions A function f that models data and is defined on pieces of its domain is called a piece-wise function. If each piece is linear, the functions is a piece-wise linear function.

9. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Fujita Scale - Intensity of Tornadoes An F1 tornado has winds speeds between 40 and 72 miles per hour. Here is the piece-wise function for the F-scale.

10. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Use f(x) to complete the following: (a)What is the domain of f ? (b)Evaluate f(–3), f(2), f(4), and f(5). (c)Sketch a graph of f. (d)Is f a continuous function on its domain? Example: Evaluating and graphing a piecewise-defined function

11. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 Solution (a)f is defined for –4 ≤ x < 2 or 2 ≤ x ≤ 4 domain is D = {x |–4 ≤ x ≤ 4}, or [–4, 4] (b)f(–3) = –3 – 1 = –4 f(2) = –2 2 = –4 f(4) = –2 4 = –8 f(5) is undefined (5 is not in the domain) Example: Evaluating and graphing a piecewise-defined function

12. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 (c)Sketch a graph of f. (d)f is not continuous because there is a break at x = 2 From –4 up to 2, graph y = x – 1, open circle at 2 From 2 to 4, graph y = –2x Example: Evaluating and graphing a piecewise-defined function

13. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Greatest Integer Function The greatest integer function is defined as follows. is the greatest integer less than or equal to x.

14. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 Direct Variation Let x and y denote two quantities. Then y is directly proportional to x, or y varies directly with x, if there exists a nonzero number k such that y = kx k is called the constant of proportionality or the constant of variation.

15. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Hooke’s Law Hooke’s law states that the distance that an elastic spring stretches beyond its natural length is directly proportional to the amount of weight hung on the spring, as illustrated in the Figure.

16. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Hooke’s Law This law is valid whether the spring is stretched or compressed. The constant of proportionality is called the spring constant. Thus if a weight or force F is applied and the spring stretches a distance x beyond its natural length, then the equation F = kx models this situation, where k is the spring constant.

17. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 A 12-pound weight is hung on a spring, and it stretches 2 inches. (a)Find the spring constant. (b)Determine how far the spring will stretch when a 19-pound weight is hung on it. Solution Let F = kx: F = 12 pounds, x = 2 inches F = k(2) so k = 6 The spring constant equals 6 Example: Working with Hooke’s law

18. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 (b)F = 19 Example: Working with Hooke’s law

19. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 Solving a Variation Problem When solving a variation problem, the following steps can be used. STEP 1:Write the general equation for the type of variation problem that you are solving. STEP 2:Substitute given values in this equation so the constant of variation k is the only unknown value in the equation. Solve for k.

20. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 Solving a Variation Problem STEP 3:Substitute the value of k in the general equation in Step 1. STEP 4:Use this equation to find the requested quantity.

21. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 Let T vary directly with x, and suppose that T = 33 when x = 5. Find T when x = 31. Solution STEP 1: Direct variation is T = kx STEP 2: Substitute 33 for T, 5 for x, solve for k Example: Solving a direct variation problem

22. Copyright © 2014, 2010, 2006 Pearson Education, Inc. 22 STEP 3: STEP 4: When x = 31, we have: Example: Solving a direct variation problem