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  1. Announcement • If any student enrolled in the class late, they should email Rekha (rekha@math.ksu.edu) so that she can adjust your iClicker scores appropriately.

  2. Special Points • All points on a graph are equally important • But some points are easier to calculate and draw than others • Two of these points are called “y-intercept” and “x-intercept”

  3. Intercepts • y-intercept is where the graph crosses the y-axis • Easy to calculate because x is 0. • x-intercept is where the graph crosses the x axis • Easy to calculate because y is 0 • Zero is a useful number to calculate with because 0a=0 and a+0=a

  4. Intercepts y x-intercept (-2,0) x-intercept (2,0) x y-intercept (0,-4)

  5. Consider the graph given here. Which set of statements is completely correct? • Has y-intercept at (0,4), x-intercept at (-3,0) • Has y-intercept at (0,-4), x-intercept at (3,0) • Has y-intercept at (0,3), x-intercept at (-4,0) • Has y-intercept at (0,-3), x-intercept at (4,0) • None of the above

  6. Consider the graph given here. Which set of statements is completely correct? • Has y-intercept at (0,4), x-intercept at (-3,0) • Has y-intercept at (0,-4), x-intercept at (3,0) • Has y-intercept at (0,3), x-intercept at (-4,0) • Has y-intercept at (0,-3), x-intercept at (4,0) • None of the above

  7. Average Rate of Change

  8. I drive from work to home. Along the way, I have to stop for stop lights, I speed up and a slow down because of traffic, I spend a few minutes on the highway going really fast. At the end I have to slow down and find a place to park. When I get home, my average speed was 18 miles per hour (this is actually true) What does it mean that my average speed was 18 mph? Why do you care?

  9. I drive from work to home. Along the way, I have to stop for stop lights, I speed up and a slow down because of traffic, I spend a few minutes on the highway going really fast. At the end I have to slow down and find a place to park. When I get home, my average speed was 18 miles per hour (this is actually true) What does it mean that my average speed was 18 mph? Why do you care? Knowing what something means tells you how to calculate it. (This makes the formula easier to remember or figure out)

  10. Wrong Answers • I drove 18 mph the whole way • I sped up and slowed down. • 18 mph was my most common speed. • My most common speeds were 0mph, 30mph, and 50mph • If you add up all my speeds and divide my how many there are, you will get 18. • The reason this is wrong takes a little more explanation.

  11. Here’s me accelerating from 0 to 50 How many speeds did I go? What would happen if you tried to add them all up?

  12. Speeds I drove between 0 and 50 • 5 mph • 10 mph • 20 mph • 25 mph • 20.1 mph • 20.003 mph • 20.00000547234543784732678432 mph • All of them

  13. Mean Speed doesn’t work • If I added up all my speeds and divided them by how many there were, I’d have to divide infinity by infinity. • I don’t know what ∞/∞ is. • Nobody does.

  14. Mean is only one kind of average

  15. Wrong Answers • I drove 18 mph the whole way • I sped up and slowed down. • 18 mph was my most common speed. • My most common speeds were 0mph, 30mph, and 50mph • If you add up all my speeds and divide my how many there are, you will get 18. • The reason this is wrong takes a little more explanation.

  16. I drive from work to home. Along the way, I have to stop for stop lights, I speed up and a slow down because of traffic, I spend a few minutes on the highway going really fast. At the end I have to slow down and find a place to park. When I get home, my average speed was 18 miles per hour (this is actually true) What does it mean that my average speed was 18 mph? I did not drive 18mph the whole time. If my friend Lazaro started next to me and drove at 18mph, he would arrive at my home at the same time.

  17. Average Speed • is the constant speed needed to travel the same distance in the same amount of time. • If I know (or pretend) that I traveled a constant speed,I can find speed using (distance traveled) / (time elapsed) • So how do I find distance traveled and time elapsed?

  18. Distance Traveled and Time Elapsed • I started my trip with my odometer reading 6273.1 • I ended my trip with the odometer reading 6276.7 • Distance traveled 6276.7-6263.1=3.6 miles • I started my trip with the clock reading 6:30 • I ended my trip with the clock reading 6:42 • Time elapsed: 42-30 = 12 minutes (0.2 hours)

  19. Average Speed • Constant speed needed to travel the same distance in the same amount of time. • If I know (or pretend) that I traveled a constant speed,I can find speed using (distance traveled) / (time elapsed) • Distance traveled (end distance – start distance) • Time elapsed is (end time – start time)

  20. Average Speed

  21. Average Rate of Change If I extend this idea to two changing variables y and x instead of just distance (d) and time (t), I get the formula for average rate of change. Some people call these y2 and y1 instead of yend and ystart. I don’t care what you do.

  22. Average Rate of Change If I extend this idea to two changing variables y and x instead of just distance (d) and time (t), I get the formula for average rate of change. We choose Δy/Δx instead of Δx/Δy because x is traditionally the independent variable (the one that controls y). Just like time goes on whether my distance is changing or not, x is imagined to go on without y. t is on the bottom, so x is on the bottom.

  23. Average Rate of Change If I extend this idea to two changing variables y and x instead of just distance (d) and time (t), I get the formula for average rate of change. When working with a graph of a line, the average rate of change is also called “the slope.”

  24. Find the slope of the line connecting the points (2,3) and (4,8). • 5/2 • -5/2 • 2/5 • -2/5 • None of the above

  25. Find the slope of the line connecting the points (2,3) and (4,8). I picked (2,3) for “begin” and (4,8) for “end”, but it doesn’t matter which A

  26. Lines

  27. What is a line? • A line is the graph of two changing variables with a constant rate of change • This means that no matter what points you pick, the average rate of change (slope) between those points is always the same.

  28. Slope is always the same Δy/Δx=3/6=0.5

  29. Slope is always the same Δy/Δx=3/6=0.5 Δy/Δx=1/2=0.5

  30. Slope is always the same Δy/Δx=3/6=0.5 Δy/Δx=2.25/4.5=0.5

  31. Slope is always the same Δy/Δx=3/6=0.5 Δy/Δx=0.05/0.1=0.5

  32. Slope is always the same Line Δy/Δx=3/6=0.5 Δy/Δx=0.05/0.1=0.5

  33. Ex: Slope is NOTalways the same Δy/Δx=1/1=1

  34. Ex: Slope is NOTalways the same Δy/Δx=1/1=1 Δy/Δx=1/1=1

  35. Ex: Slope is NOTalways the same Δy/Δx=1/1=1 Δy/Δx=0.728/0.2=3.64

  36. Ex: Slope is NOTalways the same NOT a Line Δy/Δx=1/1=1 Δy/Δx=0.728/0.2=3.64

  37. What is a line • Imagine if you pick a point (a,b), and run through every point (x,y) on the line. • The slope between (x,y) and (a,b) always stays the same. • And I’ve covered every point.

  38. Point slope form • The equation of a line with slope m through point (a,b) is

  39. Point slope form • The equation of a line with slope m through point (a,b) is • If you don’t know the slope,know two points (a1,b1) and (a2,b2), then the slope m is just the slope formula for those points.

  40. Slope intercept form • The equation of a line with slope m through point (a,b) is • The “easiest” point to use is the y-intercept (0,b). If we plug that in, we get slope-intercept form

  41. Slope intercept form • Slope intercept form is the “simplest” form of a line • “Simplify” means put in slope intercept form

  42. Find the equation of the line below. • y = (4/3)x-3 • y = (3/4)x-3 • y= (4/3)x+3 • y= (3/4)x+3 • None of the above

  43. Find the equation of the line below. Two points are (0,-3) and (4,0) Going from 04 gives Δx=4 Going from -30gives Δy=3 m=Δy/Δx=3/4 y-intercept=(0,-3) Slope intercept form: y=(3/4)x-3 B