6 1 Slope OBJECTIVE You will be able

6 -1: Slope OBJECTIVE: You will be able to find the slope of a line, given the coordinates of two points on the line. Initial terms: • slope - the “steepness” of a line • rise - the numerator portion of the ratio describing slope • run - the denominator portion of the ratio describing slope To simply say that “a line is steep, ” or “that line is going up” does not describe the behavior of a line. We need a mathematical representation of how steep a line is. 6. 1. 1 DEFINITION OF SLOPE The slope m of a line is the ratio of the change in the y-coordinates to the corresponding change in the x-coordinates. © William James Calhoun, 2001

6 -1: Slope The mathematical definition of slope can be written like this: In the picture on the right, the line runs through the origin and the point (5, 4). To find the slope, you first see how far the line rose going from (0, 0) to (5, 4). To get from the origin to (5, 4), you must go 4 units upward. Second, how far did the line run to the right? You must go 5 units to the right. Therefore, the slope is © William James Calhoun, 2001

6 -1: Slope When finding the slope of a line from a graph, look at two points (reading from left to right) and ask: (1) how far did the line go up or down from left to right, and (2) how far did the line go to the right. The slope is the first number over the second number. Look at the lines in the following two graphs: In one of the graphs, the line goes up from left to right. In the other, the line goes down from left to right. In math, we need a way to say whether the line is rising up or down. Lines with positive slopes go up from left to right. (rise is positive) Lines with negative slopes go down from left to right. (rise is negative) © William James Calhoun, 2001

6 -1: Slope There is a formula that we use to find the slope of any line if we know two points on that line. 6. 1. 2 DETERMINING SLOPE GIVEN TWO POINTS Given the coordinates of two points, (x 1, y 1) and (x 2, y 2), on a line, the slope m can be found as follows: , where x 1 x 2. You must utilize and/or memorize the formula to make it through the rest of graphing. The formula simply says what has been said before: Slope is how much the line went up/down by (reading from left to right) over how much the line went to the right by. Remember this: “Rise over Run - and always run to the right. ” © William James Calhoun, 2001

6 -1: Slope EXAMPLE 1: Determine the slope of each line. A. B. C. D. We do not need the formula for these, although the formula will work. How much did y change? up 3 How much did x change? right 2 Write the fraction of changes in y over change in x: m= How much did y change? up 4 How much did x change? left 3, so -3 Write the fraction of changes in y over change in x: m= How much did y change? no up or down, so 0 How much did x change? right 2 Write the fraction of changes in y over change in x: m= How much did y change? up 3 How much did x change? no right/left, so 0 Write the fraction of changes in y over change in x: =0 m= (!!!) I do not like this example from the book. You should always run to the right and make the y-value be the value to turn negative. © William James Calhoun, 2001

6 -1: Slope That last part of Example 1 gave us a slope of. This fraction has a zero in the denominator. We know that you can never divide by zero. You cannot break something up into no pieces. Therefore, lines like those in EX 1 D are said to have “no slope” and are always vertical lines. In the first example we saw the four general slopes of lines: (1) Positive - the line goes up and to the right. (2) Negative - the line goes down and to the right. Again, always think of running to the right which is not the way the book did it in the example. (3) Zero-slope - the line is horizontal. These lines will always be of the form y = #. (4) No-slope - the line is vertical. These lines will always be of the form x = #. The lines we will see in this class will always be in one of the four categories above. © William James Calhoun, 2001

6 -1: Slope EXAMPLE 2: Determine the slope of the line that passes through (2, -5) and (7, -10). x 1 y 1 x 2 y 2 First step should be to label your points as x 1, x 2, y 1, and y 2. Second, plug those values into the formula for slope: -10 - -5 = -10 + 5 = 7 -2 7 - 2 = -5 = -1 5 Finally, simplify the expression. Now for a new type of problem that uses the slope formula, but not to find the slope. © William James Calhoun, 2001

6 -1: Slope EXAMPLE 3: Determine the value of r so the line through (r, 6) and 3 (x 1, y 1) (10, -3) has a slope of –. 2 (x , y ) m 2 2 Label what you know from the problem. Plug the known values into the slope formula. Simplify. -3 – 6 -3 = 2 10 – r Cross-multiply. Distribute. Solve for r. -3(10 - r) = 2(-9) -30 + 3 r = -18 +30 3 r = 12 3 3 r=4 © William James Calhoun, 2001