Slope

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Key Questions

  • Answer:

    Slope is the change in the y values divided by the change in the x values

    Explanation:

    #"slope"="rate of change in y"/"rate of change in x" ="rise"/"run"#

    #(color(blue)(x_1),color(blue)(y_1))#

    #(color(red)(x_2),color(red)(y_2))#

    #color(green)m =(color(red)(y_2)-color(blue)(y_1))/(color(red)(x_2)-color(blue)(x_1))#

    Is is often expressed as rise over run.

  • The slope is a number that tells you how much y changes when x changes.
    For example: a slope of 5 means that for each change in x of 1 unit (for example between 6 and 7) the correspondig y changes of 5 units.
    enter image source here
    This is for a positive slope, so that your value of y is getting...bigger!!!

    The negative slope is the opposite, it tells you of how much y decreases for each increas of 1 unit in x.
    A slope of -5 tells you that the value of y decreases of 5 units in the 1 unit interval of x:
    enter image source here

    As you may guess the slope is a measure of the "inclination" of your line!!!

    Try to guess what a slope of zero means!!!!!!

  • Use the slope formula (#m = (y_2 - y_1)/(x_2 - x_1)#) to calculate the slope given two points #(x_1, y_1)# and #(x_2, y_2)#.

    Here is an example of finding the slope, given two points (-2,3) and (4,-5).

    #(-2, 3) = (x_1, y_1)#
    #(4, -5) = (x_2, y_2)#

    #m = (y_2 - y_1)/(x_2 - x_1)#

    #m = (-5 - 3)/(4 - (-2))#

    #m = (-5 - 3)/(4 + 2)#

    #m = (-8)/(6)#

    #m =-4/3#

    The slope of of (-2,3) and (-4,5) is #-4/3#

  • Nuzhat has already discussed how you can find the slope of a line from two points that lie on the line. I'll discuss two other methods of finding the slope from a graph.

    1. From the angle made with the x-axis

    Since the slope of a line is basically the ratio of the y-component of the line to its x-component,

    The slope of a line can be found out by taking tangent of the angle between the given line and the x-axis.

    Consider the following figure:

    enter image source here

    In this case, the angle between the x-axis and the line is #theta#.

    Therefore,
    Slope of the given line = #tantheta#

    Note: Angles in the counterclockwise direction are taken as positive, and those in the clockwise direction are taken as negative.

    For example, if the angle between the x-axis and the given line is #30^o#,

    Slope of the given line = #tan30=1/sqrt3#

    2. From the equation of the line

    The slope of a line can also be determined from its equation. The standard form of the equation of a line is:

    #Ax^2+By+C=0#

    where #A,B and C# are some constants.

    First, the equation of the line must be written in the standard form.

    Then, the slope of the line = #-A/B#

    For example, let the equation of the given line be #x^2+3=2y#.

    Rewriting in the standard form, we get: #x^2-2y+3=0#
    and we can see that:
    #A=1#
    #B=-2#
    #C=3#

    Therefore, the slope of the line #=-A/B=-(1)/(-2)=1/2#

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