Math A-100 IUSB Sec. 3.3 detailed notes – Week 5 Slope In our basic linear equation, y = mx + b, the “m” stood for “the slope”. The slope is the relationship of how much the y value goes up or down compared to how the x value increases. Slope is written as a number, somethings as a integer (3 or -4), sometimes as a fraction (-4/7 or 5/2), or a decimal (-.2 or .75). When the slope is positive, that means you have a positive correlation (or direction) and when the slope is negative, you have a negative correlation (or direction). (See Sec 3.2 notes) Often in word questions, we don’t use the term “slope”. You will see “average rate of change”, “grade” or “gradient” (when talking about roads), “pitch” (when talking about roofs), etc. used instead. They all mean the same thing!! Finding the Slope We have several ways to find the slope: a) Reading it directly from a graph – pick two points. Go from the point on the left to the point on the right. Count how many places you go up or down and how many places you move from the left to the right. Moving down gives you a negative number. Put those numbers into a fraction (up or down on top, left to right on bottom) and reduce if possible Example 1: you move up 6 place and moved to the right 4 places. The Slope = 6/4 which reduces to 3/2 Example 2: you move down 7 places and to the right 4 places. The Slope = -7/4 b) Using the formula m = (y2 – y1) /(x2 – x1) – pick two points. Subtract the y values and make that the top number. Subtract the x values and make that the bottom number. The reduce the fraction if possible. Make sure you are consistent – if you start with the left point when subtracting the y values, then start with the left point when subtracting the x values. Example 1: you have the points (2, 3) and (8, 12). Subtracting the y values gives you 12 – 3 = 9. Subtracting the x values gives you 8 – 2 = 6. So, Slope = 9/6 which reduces to 3/2 Example 2: you have the points (-3, 5) and (4, 1). Subtracting the y values gives you 5 - 1 = 4. Subtracting the x values gives you -3 - 4 = -7. So, Slope = 4/-7 which is -4/7 NOTE: it does not matter which point you start with, as long as you are consistent: Revisiting Example 1 just above: 3 – 12 = -9 and 2 – 8 = -6. Slope = -9/-6 = 3/2 Revisiting Example 2 just above: 1 – 5 = -4 and 4 - -3 = 7. Slope = -4/7 NOTE: if you have a single number for the slope, turn it into a fraction by putting 1 on the bottom. For example: slope = 4. Change 4 into 4/1, which means your y value went up by 4 while your x value moved to the right by 1 place. Interpreting the slope: If your slope is 3/2, it means that your y increased by 3 units for every 2 units your x increased If your slope is -4/7, it means your y decreased by 4 units for every 7 units your x increased Special Slopes We have two special slopes: a) horizontal lines – in a horizontal line, the y value does not change, so when you subtract them you get 0. It does not matter how much your x value went up, because 0/ any number is still 0. For example: you have the points (-4, 6) and (5,6). Subtracting the y values gives you 6 – 6 = 0. Subtracting the x values give you 5 - -4 = 9. Putting them in the fractions, Slope = 0/9 = 0 NOTE: the y values of both points are THE SAME when you have a horizontal line, so the Slope = 0 b) vertical lines – in a vertical line, the x value does not change, so when you subtract you get 0. It does not matter how much your y value went up or down, because dividing by 0 gives you the answer “UNDEFINED”. For example: you have the points (-4, -3) and (-4,6). Subtracting the y values gives you 6 – -3 = 9. Subtracting the x values give you -4 - -4 = 0. Putting them in the fractions, Slope = 9/0 = UNDEFINED NOTE: the x values of both points are THE SAME when you have a vertical line, so the Slope = UNDEFINED Using the slope Once you have found the slope, you can use it to find the next point on the line. To find the next point on the line, apply the parts of the slope to the x and y values of the point. For example, you are at point (3, 5) and your slope is 7/4. The top part of the slope means the y value went up by 7, so your next y value on the line is 5 + 7 = 12. The bottom part means your x value went up by 4, so your next x value is 3 + 4 = 7. Put them together to get your next point: (x, y) = (7, 12) Videos that may help: Finding Slope given two points http://www.youtube.com/watch?v=ZW7rQa8SJSU