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Question 117503: Find the slope of the graph of the linear function "f".
f(2) = -3, f(-2) = 5
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Since you know that the function produces a linear graph, you can represent this function in
slope intercept form as:
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f(x) = mx + b
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where m is the slope of the graph and b is the point where the graph crosses the y-axis.
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The problem tells you that f(2) = -3. And this tells you that when x = 2, then f(x) = -3.
Substituting these values into the slope intercept equation results in:
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-3 = m(2) + b
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Which, when you rearrange the order of multiplication in the first term on the right side becomes:
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-3 = 2m + b <=== call this equation 1
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Similarly, the problem also tells you that f(-2) = 5. And this tells you that when x = -2, then
f(x) = 5. Substituting these values into the slope intercept equation results in:
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5 = m(-2) + b
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And again, rearranging the order of multiplication in the first term on the right side makes
this equation become:
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5 = -2m + b <=== call this equation 2
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So now we have the set of equation 1 and equation 2 as being:
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-3 = 2m + b and
5 = -2m + b
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You can eliminate the variable m by adding the two equations together vertically. Note
that the 2m and the -2m cancel each other when added. Therefore, the sum of the two equations
results in:
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+2 = 2b
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Solve for b by dividing both sides of this equation by 2 to get 1 = b.
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Now that you know what b is, you can return to either equation, substitute that value for b
and solve for m. Let's return to equation 1:
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-3 = 2m + b
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Substituting +1 for b results in this equation becoming:
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-3 = 2m + 1
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Subtract 1 from both sides to get rid of the 1 on the right side and the equation reduces to:
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-4 = 2m
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Divide both sides of this equation by 2 to solve for m and you get:
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m = -4/2 = -2
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But in the slope intercept form, m (which is the multiplier of x) equals the slope of
the line. So the answer to this problem is that the slope is -2.
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Check ... now that you know the values of m and b you can write the slope intercept form
for this equation as:
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f(x) = mx + b = -2x + 1
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So you have just:
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f(x) = -2x + 1
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To find f(2) substitute 2 for every x and you get:
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f(2) = -2(2) + 1 = -4 + 1 = -3 .... that checks.
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and to find f(-2) substitute -2 for every x and you get:
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f(-2) = -2(-2) + 1 = +4 + 1 = +5 ... and that also checks.
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We found the correct equation f(x) = -2x + 1 because everything checks out as specified
in the problem. Therefore, the slope of -2 (which is the multiplier of x) is correct.
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Hope this helps you to understand the problem.
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