[latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\n
where [latex]\\Delta y[\/latex] is the change in output and [latex]\\Delta x[\/latex] is the change in input. In function notation, [latex]{y}_{1}=f\\left({x}_{1}\\right)[\/latex] and [latex]{y}_{2}=f\\left({x}_{2}\\right)[\/latex] so we could write:<\/p>\n
[latex]m=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\n
The graph below indicates how the slope of the line between the points, [latex]\\left({x}_{1,}{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2,}{y}_{2}\\right)[\/latex], is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.<\/p>\n
![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18223100\/CNX_Precalc_Figure_02_01_005n2.jpg\")
<\/center>You've just learned that slope is calculated as the change in the vertical distance (rise) divided by the change in the horizontal distance (run). A quick way to remember this is the phrase \"Rise Over Run.\"<\/p>\n
When you're looking at a graph, you can quickly identify two points and use \"Rise Over Run\" to calculate the slope right there. To do so pick two points on the line. Count how much you have to go up or down to get from one point to the other\u2014that's your rise. Then count how much you have to go left or right\u2014that's your run. The slope is simply the rise divided by the run.<\/p>\n
If you go down instead of up, the rise will be negative. If you go left instead of right, the run will be negative. A negative slope means the line is going downhill as you read from left to right.<\/p>\n<\/section>\n The slope, or rate of change, [latex]m[\/latex], of a function can be calculated according to the following:<\/p>\n <\/p>\n [latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\Rightarrow \\dfrac{f(x_2)-f(x_1)}{x_2 - x_1}[\/latex]<\/p>\n <\/p>\n where [latex]{x_1}[\/latex] and [latex]x_2[\/latex] are input values and [latex]{f(x_1)}[\/latex] and [latex]f(x_2)[\/latex] are output values.<\/p>\n<\/div>\n<\/section>\n [latex]m=\\frac{\\text{change in output}}{\\text{change in input}}=\\frac{1-\\left(-2\\right)}{8 - 3}=\\frac{3}{5}[\/latex]<\/p>\n\nWe could also write the slope as [latex]m=0.6[\/latex]. The function is increasing because [latex]m>0[\/latex].\n\n [ohm2_question hide_question_numbers=1]13522[\/ohm2_question]<\/p>\n<\/section>\n [latex]\\dfrac{\\text{change in distance}}{\\text{change in time}} = \\dfrac{180\\text{ miles}-57\\text{ miles}}{6.35\\text{ hours} - 4.15\\text{ hours}} = \\dfrac{123\\text{ miles}}{2.1\\text{ hours}} = \\dfrac{123}{2.1}\\text{ miles per hour} \\approx 58.6 \\text{ mph}[\/latex]<\/p>\n\n[\/hidden-answer]<\/section>\n Q&A<\/strong><\/p>\n Are the units for slope always [latex]\\frac{\\text{units for the output}}{\\text{units for the input}}[\/latex] ?<\/strong><\/p>\n Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.<\/em><\/p>\n<\/section>\n [latex]\\frac{4,400\\text{ people}}{4\\text{ years}}=1,100\\text{ }\\frac{\\text{people}}{\\text{year}}[\/latex]<\/p>\n\nSo the population increased by [latex]1,100[\/latex] people per year.\n\n [ohm2_question hide_question_numbers=1]13523[\/ohm2_question]<\/p>\n<\/section>\n","rendered":" In the examples we have seen so far, we have had the slope provided for us. However, we often need to calculate the slope<\/strong> given input and output values. Given two values for the input, [latex]{x}_{1}[\/latex] and [latex]{x}_{2}[\/latex], and two corresponding values for the output, [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] which can be represented by a set of points, [latex]\\left({x}_{1}\\text{, }{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2}\\text{, }{y}_{2}\\right)[\/latex], we can calculate the slope [latex]m[\/latex], as follows:<\/p>\n [latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\n where [latex]\\Delta y[\/latex] is the change in output and [latex]\\Delta x[\/latex] is the change in input. In function notation, [latex]{y}_{1}=f\\left({x}_{1}\\right)[\/latex] and [latex]{y}_{2}=f\\left({x}_{2}\\right)[\/latex] so we could write:<\/p>\n [latex]m=\\dfrac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/p>\n The graph below indicates how the slope of the line between the points, [latex]\\left({x}_{1,}{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2,}{y}_{2}\\right)[\/latex], is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.<\/p>\n You’ve just learned that slope is calculated as the change in the vertical distance (rise) divided by the change in the horizontal distance (run). A quick way to remember this is the phrase “Rise Over Run.”<\/p>\n When you’re looking at a graph, you can quickly identify two points and use “Rise Over Run” to calculate the slope right there. To do so pick two points on the line. Count how much you have to go up or down to get from one point to the other\u2014that’s your rise. Then count how much you have to go left or right\u2014that’s your run. The slope is simply the rise divided by the run.<\/p>\n If you go down instead of up, the rise will be negative. If you go left instead of right, the run will be negative. A negative slope means the line is going downhill as you read from left to right.<\/p>\n<\/section>\n The slope, or rate of change, [latex]m[\/latex], of a function can be calculated according to the following:<\/p>\n <\/p>\n [latex]m=\\dfrac{\\text{change in output (rise)}}{\\text{change in input (run)}}=\\dfrac{\\Delta y}{\\Delta x}=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \\Rightarrow \\dfrac{f(x_2)-f(x_1)}{x_2 - x_1}[\/latex]<\/p>\n <\/p>\n where [latex]{x_1}[\/latex] and [latex]x_2[\/latex] are input values and [latex]{f(x_1)}[\/latex] and [latex]f(x_2)[\/latex] are output values.<\/p>\n<\/div>\n<\/section>\ncalculating slope<\/h3>\n
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Analysis of the Solution<\/h4>\n\nAs noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value or [latex]y[\/latex]-coordinate used corresponds with the first input value or [latex]x[\/latex]-coordinate used. [\/hidden-answer]<\/section>\n
Analysis of the Solution<\/h4>\n\nBecause we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable. [\/hidden-answer]<\/section>\n
Calculate and Interpret Slope<\/h2>\n
<\/div>\ncalculating slope<\/h3>\n
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