SOLUTION: Determine the algebraic rule for the line that: satisfies f(0) = -2 and f(4) = 5

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Question 124962This question is from textbook Intermediate Algebra Review
: Determine the algebraic rule for the line that:
satisfies f(0) = -2 and f(4) = 5 This question is from textbook Intermediate Algebra Review

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Since f(0) = -2, we have the first point (0,-2). Also, because f(4) = 5, we have the second point (4,5)


So let's find the equation of the line that goes through (0,-2) and (4,5)

First lets find the slope through the points (0,-2) and (4,5)

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: is the first point (0,-2) and is the second point (4,5))

m=%285--2%29%2F%284-0%29 Plug in y%5B2%5D=5,y%5B1%5D=-2,x%5B2%5D=4,x%5B1%5D=0 (these are the coordinates of given points)

m=+7%2F4 Subtract the terms in the numerator 5--2 to get 7. Subtract the terms in the denominator 4-0 to get 4

So the slope is
m=7%2F4

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Now let's use the point-slope formula to find the equation of the line:



------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

y--2=%287%2F4%29%28x-0%29 Plug in m=7%2F4, x%5B1%5D=0, and y%5B1%5D=-2 (these values are given)


y%2B2=%287%2F4%29%28x-0%29 Rewrite y--2 as y%2B2


y%2B2=%287%2F4%29x%2B%287%2F4%29%280%29 Distribute 7%2F4

y%2B2=%287%2F4%29x%2B0 Multiply 7%2F4 and 0 to get 0%2F4. Now reduce 0%2F4 to get 0

y=%287%2F4%29x%2B0-2 Subtract 2 from both sides to isolate y

y=%287%2F4%29x-2 Combine like terms 0 and -2 to get -2
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Answer:


So the equation of the line which goes through the points (0,-2) and (4,5) is: y=%287%2F4%29x-2 (note: in function notation the equation is f%28x%29=%287%2F4%29x-2)

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=7%2F4 and the y-intercept is b=-2

Notice if we graph the equation y=%287%2F4%29x-2 and plot the points (0,-2) and (4,5), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=%287%2F4%29x-2 through the points (0,-2) and (4,5)

Notice how the two points lie on the line. This graphically verifies our answer.