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1.5   Geometric Properties of Linear Functions

1.5   Geometric Properties of Linear Functions. Interpreting the Parameters of a Linear Function The slope-intercept form for a linear function is y  =  b  +  mx , where b is the y -intercept and m is the slope. The parameters b and m can be used to compare linear functions.

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1.5   Geometric Properties of Linear Functions

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  1. 1.5   Geometric Properties of Linear Functions

  2. Interpreting the Parameters of a Linear Function The slope-intercept form for a linear function is y = b + mx, where b is the y-intercept and m is the slope. The parameters b and m can be used to compare linear functions. Page 35

  3. With time, t, in years, the populations of four towns, PA, PB, PC and PD, are given by the following formulas: PA=20,000+1,600t, PB=50,000-300t, PC=650t+45,000, PD=15,000(1.07)t (a)  Which populations are represented by linear functions? (b)  Describe in words what each linear model tells you about that town's population. Which town starts out with the most people? Which town is growing fastest? Page 35 Example 1

  4. PA=20,000+1,600t, PB=50,000-300t, PC=650t+45,000, PD=15,000(1.07)t (a) Which populations are represented by linear functions? Page 35

  5. PA=20,000+1,600t, PB=50,000-300t, PC=650t+45,000, PD=15,000(1.07)t (a) Which populations are represented by linear functions? Which of the above are of the form: y = b + mx (or here, P = b + mt)? Page 35

  6. PA=20,000+1,600t, PB=50,000-300t, PC=650t+45,000, PD=15,000(1.07)t (a) Which populations are represented by linear functions? Which of the above are of the form: y = b + mx (or here, P = b + mt)? A, B and C Page 35

  7. With time, t, in years, the populations of four towns, PA, PB, PC and PD, are given by the following formulas: PA=20,000+1,600t, PB=50,000-300t, PC=650t+45,000, PD=15,000(1.07)t (b)  Describe in words what each linear model tells you about that town's population. Which town starts out with the most people? Which town is growing fastest? Page 35

  8. PA = 20,000 + 1,600t when t=0: 20,000 grows: 1,600 / yr b m PB = 50,000 + (-300t) when t=0: 50,000 shrinks: 300 / yr b m PC = 45,000 + 650t when t=0: 45,000 grows: 650 / yr b m PD = 15,000(1.07)t when t=0: 15,000 grows: 7% / yr Page 35

  9. Let y = b + mx. Then the graph of y against x is a line. • The y-intercept, b, tells us where the line crosses the y-axis. • If the slope, m, is positive, the line climbs from left to right. If the slope, m, is negative, the line falls from left to right. • The slope, m, tells us how fast the line is climbing or falling. • The larger the magnitude of m (either positive or negative), the steeper the graph of f. Page 36 Blue Box

  10. (a)  Graph the three linear functions PA, PB, PC from Example 1 and show how to identify the values of b and m from the graph. (b)  Graph PD from Example 1 and explain how the graph shows PD is not a linear function. Page 36 Example 2

  11. Note vertical intercepts. Which are climbing? Which are climbing the fastest? Why? Page 36

  12. A Page 36

  13. Intersection of Two Lines To find the point at which two lines intersect, notice that the (x, y)-coordinates of such a point must satisfy the equations for both lines. Thus, in order to find the point of intersection algebraically, solve the equations simultaneously. If linear functions are modeling real quantities, their points of intersection often have practical significance. Consider the next example. Page 37

  14. The cost in dollars of renting a car for a day from three different rental agencies and driving it d miles is given by the following functions: C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d (a)  Describe in words the daily rental arrangements made by each of these three agencies. (b)  Which agency is cheapest? Page 37 Example 3

  15. C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d (a)  Describe in words the daily rental arrangements made by each of these three agencies. C1 charges $50 plus $0.10 per mile driven. C2 charges $30 plus $0.20 per mile. C3 charges $0.50 per mile driven. Page 37

  16. What are the points of intersection? A C1 C2 C3 Page

  17. C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d Page 37

  18. C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d Page 37

  19. C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d Page 37

  20. A C1 C2 C3 Page 38

  21. Thus, agency 3 is cheapest up to 100 miles. A C1 C2 C3 Page 38

  22. Agency 1 is cheapest for more than 200 miles. A C1 C2 C3 Page 38

  23. Agency 2 is cheapest between 100 and 200 miles. A C1 C2 C3 Page 38

  24. Equations of Horizontal and Vertical Lines An increasing linear function has positive slope and a decreasing linear function has negative slope. What about a line with slope m = 0? Page 38

  25. Equations of Horizontal and Vertical Lines An increasing linear function has positive slope and a decreasing linear function has negative slope. What about a line with slope m = 0? If the rate of change of a quantity is zero, then the quantity does not change. Thus, if the slope of a line is zero, the value of y must be constant. Such a line is horizontal. Page 38

  26. Plot the points: (-2, 4) (-1,4) (1,4) (2,4) Page 38

  27. The equation y = 4 represents a linear function with slope m = 0. A Page 38

  28. Plot the points: (-2, 4) (-1,4) (1,4) (2,4) Let's verify by using any two points- Let's use (-2,4) & (1,4): Page 38

  29. Plot the points: (-2, 4) (-1,4) (1,4) (2,4) Let's verify by using any two points- Let's use (-2,4) & (1,4): Page 38

  30. Plot the points: (-2, 4) (-1,4) (1,4) (2,4) Let's verify by using any two points- Let's use (-2,4) & (1,4): Page 38

  31. Now plot the points: (4,-2) (4,-1) (4,1) (4,2) Page 38

  32. A The equation x = 4 does not represent a linear function with slope m = undefined. Page 38

  33. Now plot the points: (4,-2) (4,-1) (4,1) (4,2) Let's verify by using any two points- Let's use (4,-2) & (4,1): Page 38

  34. Now plot the points: (4,-2) (4,-1) (4,1) (4,2) Let's verify by using any two points- Let's use (4,-2) & (4,1): Page 38

  35. Now plot the points: (4,-2) (4,-1) (4,1) (4,2) Let's verify by using any two points- Let's use (4,-2) & (4,1): Page 38

  36. For any constant k: The graph of the equation y = k is a horizontal line and its slope is zero. The graph of the equation x = k is a vertical line and its slope is undefined. Page 39 Blue Box

  37. Slope of Parallel (||) & Perpendicular ( ) lines: || lines have = slopes, while lines have slopes which are the negative reciprocals of each other. So if l1 & l2 are 2 lines having slopes m1 & m2, respectively. Then: these lines are || iff m1 = m2 these lines are iff m1 = - 1 / m2 Page 39 Blue Box

  38. So if l1 & l2 are 2 lines having slopes m1 & m2, respectively. Then: these lines are || iff m1 = m2 these lines are iff m1 = - 1 / m2 Page 39

  39. Are these 2 lines || or or neither? y = 4+2x, y = -3+2x Page N/A

  40. Are these 2 lines || or or neither? y = 4+2x, y = -3+2x Parallel (m = 2 for both) Page N/A

  41. Are these 2 lines || or or neither? y = -3+2x, y = -.1 -(1/2)x Page N/A

  42. Are these 2 lines || or or neither? y = -3+2x, y = -.1 -(1/2)x : m1 = 2, m2 = -(1/2) Page N/A

  43. End of Section 1.5.

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