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Math Help - Slope and Gradient Proof

  1. #1
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    Slope and Gradient Proof

    Hello,

    I have to show that for some constant m:
    \frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}=m

    For all x_{1} x_{2}, show that f(x)=mx+b, where b is some constant. The hint given is to fix x_{1} and take x=x x_{2}; then solve for f(x).

    I figured that I should change it to point slope form like this (referring to the hint given):

    But I'm not sure if this step is correct or not and how I can turn it into the slope-intercept form? Please help.
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  2. #2
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    Quote Originally Posted by Kataangel View Post
    Hello,

    I have to show that for some constant m:
    \frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}=m

    For all x_{1} x_{2}, show that f(x)=mx+b, where b is some constant. The hint given is to fix x_{1} and take x=x x_{2}; then solve for f(x).

    I figured that I should change it to point slope form like this (referring to the hint given):

    But I'm not sure if this step is correct or not and how I can turn it into the slope-intercept form? Please help.
    Yes, that step is perfectly good (it uses the "distributive" law) and it is in "slope-intercept form". Just think of it as y= mx+ (f(x_1)- mx_1). m is the slope and f(x_1)- mx_1 is the intercept.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Yes, that step is perfectly good (it uses the "distributive" law) and it is in "slope-intercept form". Just think of it as y= mx+ (f(x_1)- mx_1). m is the slope and f(x_1)- mx_1 is the intercept.
    How would f(x_1)- mx_1 be a constant though since it consists of 2 variables?
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