Any questions on the Section 3.3 homework?

1. Any questions on the Section 3.3 homework?

2. Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note-taking materials.

3. Section 3.4: The Slope of a Line Slope of a line: Informally, slope is the tilt of a line. It is the ratio of vertical change to horizontal change, or

4. Example Find the slope of the line through (4, -3) and (2, 2). If we let (x1, y1) be (4, -3) and (x2, y2) be (2, 2), then Note:If we let (x1, y1) be (2, 2) and (x2, y2) be (4, -3), then we get the same result.

5. Given the graph of a line, how do you find the slope? Find 2 points on the graph, then use those points in the slope formula.

6. Which points do you use? It’s your choice, but it’s much easier if you pick points whose x- and y-coordinates are both integers. (2, 2) Slope = -4 – 2 = -6 = 3 = 3 0 – 2 -2 1 (0, -4)

7. Slope-intercept form of a line • y = mx + b • has a slope of m and has a y-intercept of (0, b). • This form is useful for graphing, since you have a point and the slope readily visible.

8. Example Find the slope and y-intercept of the line –3x + y = -5. • First, we need to solve the linear equation for y By adding 3x to both sides, y = 3x – 5. • Once we have the equation in the form of y = mx + b, we can read the slope and y-intercept.  slope is 3 y-intercept is (0,-5)

9. Example y = x – 2 (divide both sides by –6) Find the slope and y-intercept of the line 2x – 6y = 12. • First, we need to solve the linear equation for y. -6y = -2x + 12 (subtract 2x from both sides) • Since the equation is now in the form of y = mx + b, • slope is 1/3 • y-intercept is (0,-2)

10. Example from today’s homework:

11. What is the slope of a horizontal line? For any 2 points, the y values will be equal to the same real number. The numerator in the slope formula = 0 (the difference of the y-coordinates), but the denominator  0 (two different points would have two different x-coordinates). So the slope = 0.

12. What is the slope of a vertical line? For any 2 points, the x values will be equal to the same real number. The denominator (x2 – x1) in the slope formula = 0 so the slope is undefined (since you can’t divide by 0).

13. Example from today’s homework: N

14. Summary of relationship between graphs of lines and slope • If a line moves up as it moves from left to right, the slope is positive. • If a line moves down as it moves from left to right, the slope is negative. • Horizontal lines have a slope of 0. • Vertical lines have undefined slope (or no slope).

15. Two lines that never intersect are called parallel lines. • Parallel lines have the same slope • unless they are vertical lines, which have no defined slope. • Vertical lines are also parallel to each other, even though their slope is undefined.

16. Two lines that intersect at right angles are called perpendicular lines. • Two nonvertical perpendicular lines have slopes that are negative reciprocals of each other. • The product of their slopes will be –1. • Horizontal and vertical lines are perpendicular to each other.

17. Example y = x + 1 (divide both sides by 5) The first equation has a slope of 5 and the second equation has a slope of , so the lines are perpendicular. Determine whether the following lines are parallel, perpendicular, or neither. -5x + y = -6 and x + 5y = 5 • First, we need to solve both equations for y. • In the first equation, y = 5x – 6 (add 5x to both sides) • In the second equation, 5y = -x + 5 (subtract x from both sides)

18. Example y = x – 1 (divide both sides by 2) y = x – (divide both sides by 4) Both lines have a slope of , so the lines are parallel. Determine whether the following lines are parallel, perpendicular, or neither. -x + 2y = -2 and 2x = 4y + 3 • In the first equation, 2y = x – 2 (add x to both sides) • In the second equation, • 4y = 2x – 3 (subtract 3 from both sides)

19. Example from today’s homework:

20. Web site that shows the effect of changing slope and y-intercept: http://www.mathsisfun.com/graph/straight_line_graph.html

21. A quick way to check if you goofed on a negative sign: Graph the two points and see if the slope should be positive or negative. -3/2

22. Watch out for this problem: -0.27 Hint: To start this problem, convert both distances to the same units. Also, make sure you type a 0 before decimals < 1. (e.g. type 0.08, not just .08.)

23. Practical application of slope to construction: Web site that shows how to calculate the pitch of a roof: http://roofgenius.com/ Roof-Pitch-xamples.asp Pitch of a roof = rise/run Just make sure you measure both rise and run in the same units (e.g. feet, meters, inches)

24. Problem: Is it safe to walk on this roof?

25. Note: You will be given sheet containing formulas to use on quizzes/tests. You can print off a copy of the formula sheet to use while you do your homework and practice quizzes/tests by clicking on the “Formula sheet” menu button. It is a good idea to print a copy of this sheet (or keep a yellow copy that we’ve handed out in class) to use while you do your homework assignments and practice quizzes/tests so you can get used to the location of the formulas and know which ones will be available to you on the sheet. (You will be given a clean copy of the sheet to use at each quiz and test, so don’t plan to rely on any notes you might write on your copy of the formula sheet...)

26. Note: The homework for this section shouldn’t take too long, so you might want to look ahead at section 3.5 on slope-intercept equations of lines. (The homework for that section is a longer assignment so you might want to get a head start…)

27. Reminder: This homework assignment on Section 3.4 is due at the start of next class period.

28. You may now OPEN your LAPTOPS and begin working on the homework assignment.