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Slope

Learn about positive, negative, zero, and undefined slopes, the slope formula, midpoint calculation, and how to find endpoints. Practice plotting points, finding slopes, and exploring graphs with varying slopes. Interactive activities and examples provided.

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Slope

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  1. Slope

  2. Traditional Slopes: Positive Slope: This is a slope that increases as you move from left to right on a coordinate plane…think of riding a ski lift up a mountain. Negative Slope: This is a slope that decreases as you move from left to right on a coordinate plane…think of downhill skiing.

  3. Zero Slope: This is a slope of zero which forms a horizontal line…think of cross-country skiing. Undefined Slope: This is a vertical line with a non-existent slope…think of extreme skiing.

  4. Slope Formula: The y2 and y1 represent the 2 y-values in the ordered pairs, and x2 and x1 represent the x-values. Remember, because slope is “rise over run” the y-values need to be in the numerator and the x-values go in the denominator!

  5. Example #1: Plotting points and finding slopes Find the slope of the segment with endpoints at (2, 3) and (8, 6) by plotting the points and counting the rise over the run. Over 6 Up 3

  6. Example #1: Plotting points and finding slopes Using these same points, let’s plug them into the slope formula to see what we get: (2, 3) and (8, 6)

  7. Lots More Practice! • Using the slope formula, find the slopes for the following sets of points: • (6, 12) and (5, 4) • (3, -1) and (-3, 2) • (7, -2) and (3, -2) • (7, 12) and (7, 8)

  8. So What Do These Slopes Look Like Side-by-Side? We are going to take a short trip to the National Library of Virtual Manipulatives to check out some graphs and see their differences in slopes! http://nlvm.usu.edu/en/nav/frames_asid_109_g_4_t_2.html?open=activities&from=category_g_4_t_2.html Once you arrive at the page click on the “functions” tab and enter in different equations for f(x), g(x), and h(x) and hit “graph” after each one. All three graphs will appear so we can notice some differences with different slopes!

  9. Midpoint What is the Midpoint? • A midpoint is the point on a line segment that is the same distance from both endpoints. • (It’s the exact middle of the line segment) A B C

  10. Coordinate Midpoint • You can find the coordinates of the midpoint using this formula: How do you find the midpoint of a segment? Midpoint =

  11. Find the coordinates of the midpoint (6,13) (-2,-5)

  12. Find the Midpoint • Ex. 1) Find the midpoint of the segment with the endpoints (-2,-5) & (6,13). x1 y1 x2 y2 The midpoint between (-2, -5) and (6, 13) is (2, 4)

  13. Find the Midpoint (6,13) (2, 4) (-2,-5)

  14. Ex. 2) Find the midpoint of the segment with the endpoints (14,-7) & (3,18). The midpoint between (14, -7) and (3, 18) is (17/2, 11/2)

  15. Ex #3 Find the midpoint of (5,4) and (3,6)

  16. What if we have the midpoint and one endpoint and want to find the other endpoint. Mx = x value of the midpoint And My = y value of the midpoint

  17. Ex 1 Midpoint (0, -6) Endpoint (7, -12)Find the other endpoint (?, ?) (0, -6) (7, -12)

  18. Midpoint (0, -6) Endpoint (7, -12) For the x value: Step 1: plug values Step 2: Multiply both sides by 2 -7 -7 Step 4: Solution Step 3: Solve for x X = -7 Now do the same for the y value. So the other endpoint is: (-7, 0) Y = 0

  19. Midpoint (0, -6) Endpoint (7, -12)So the other endpoint is (-7, 0) (-7, 0) (0, -6) (7, -12)

  20. Ex 2) Find the other endpoint Midpoint: (7,4) Endpoint: (2,4) so so X = 12 so so Y = 4 The other endpoint is (12, 4)

  21. Ex 3) Find the other endpointMidpoint (-9, 7) Endpoint (15, -7) Answer: (-33, 21) X = -33 Y = 21

  22. Summary: Write 3 things about slope that you learned today from this lesson.

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