2. The SLOPE of a Line Section 4.4

3. In the real world, the roofs of houses are “pitched” differently Some have a shallow, flat tilt Others have a steep, large tilt

4. In algebra, graphs of lines have a special name for the “tilt” associated with them -2-4 2 4 -2 -4 2 4 The SLOPE (m) of a line is defined as the following: *the triangle is the “DELTA” function, which means “the change in” or “the difference between” Today we want to be able to calculate the slope of a line

5. WHAT DOES THAT MEAN ??? -2-4 2 4 -2 -4 2 4 It means we only need TWO POINTS on the line to find the slope. Let’s use (-2,-3) and (0,1) as our points

6. (-2,-3) and (0,1) The formula for slope requires us to break apart the x and y values of their coordinates in order to substitute and solve. I like to use the “taller” point as the “subscript 2’s” and the “lower” point as The “subscript 1’s”. The subscripts simply help identify the coordinates but have no mathematical value themselves (x 2, y 2 ) (x 1, y 1 ) Write formula 1 st ! Sub-in the components For y 2, y 1, x 2, and x 1 : simplify:

7. So what does a slope of 2 mean ? -2-4 2 4 -2 -4 2 4 It means the line follows A pattern of rising up 2 Units for every 1 unit it travels to the right. Sometimes we refer to the slope as the “RISE” “RUN”

8. What would a slope of mean ? -2-4 2 4 -2 -4 2 4 It means the line follows A pattern of falling down 2 units for every 3 units it travels to the right. Looking left to right, negative slopes travel DOWNWARD Remember, the negative moves to the numerator, making it

9. How about a slope of -5 ? -2-4 2 4 -2 -4 2 4 It means the line follows A pattern of falling down 5 units for every 1 unit it travels to the right. Integers can always be placed over a “1”

10. How about a horizontal line ? -2-4 2 4 -2 -4 2 4 Although the points have different x components, the “y” values NEVER change…so since the graph doesn’t “RISE”… Like the equation y = -3 … The slope of a horizontal line is ZERO (m = 0)

11. How about a vertical line ? -2-4 2 4 -2 -4 2 4 Although the points have different y components, the “x” values NEVER change…so since the graph doesn’t “RUN” side to side…and our formula would have zero in the denominator… Like the equation x = 4 … The slope of a vertical line is UNDEFINED

12. In general… -2-4 2 4 -2 -4 2 4 As lines rotate counterclockwise thru quadrant I, their slopes increase until they Are vertical and undefined. As lines rotate clockwise down thru quadrant IV, their slope becomes more NEGATIVE until they are vertical and undefined.

13. Lastly, find the missing piece… A line travels thru (-2, 1) and (4, k). Find the value of “k” if the line’s slope is -2/3. So, k = -3. The true Coordinate was (4, -3)