SOLUTION: I really need help on this problem: If f is a linear function and f(0)= 10 and f(10)= 14, find f(5) and f(-5). PLEASE HELP ME!!!!!!!!

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: I really need help on this problem: If f is a linear function and f(0)= 10 and f(10)= 14, find f(5) and f(-5). PLEASE HELP ME!!!!!!!!      Log On


   

Question 87678: I really need help on this problem: If f is a linear function and f(0)= 10 and
f(10)= 14, find f(5) and f(-5).
PLEASE HELP ME!!!!!!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If f%280%29=10 this means we have a point (0,10). If f%2810%29=14 this means we have a point (10,14). So we can find the equation of the line through the two points


Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (0,10) and (10,14)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (0,10) and (x%5B2%5D,y%5B2%5D) is the second point (10,14))


m=%2814-10%29%2F%2810-0%29 Plug in y%5B2%5D=14,y%5B1%5D=10,x%5B2%5D=10,x%5B1%5D=0 (these are the coordinates of given points)


m=+4%2F10 Subtract the terms in the numerator 14-10 to get 4. Subtract the terms in the denominator 10-0 to get 10




m=2%2F5 Reduce



So the slope is

m=2%2F5





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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 (x%5B1%5D,y%5B1%5D) is one of the given points


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


y-10=%282%2F5%29%28x-0%29 Plug in m=2%2F5, x%5B1%5D=0, and y%5B1%5D=10 (these values are given)



y-10=%282%2F5%29x%2B%282%2F5%29%280%29 Distribute 2%2F5


y-10=%282%2F5%29x%2B0 Multiply 2%2F5 and 0 to get 0%2F5. Now reduce 0%2F5 to get 0

y=%282%2F5%29x%2B0%2B10 Add 10 to both sides to isolate y


y=%282%2F5%29x%2B10 Combine like terms 0 and 10 to get 10

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Answer:



So the equation of the line which goes through the points (0,10) and (10,14) is:y=%282%2F5%29x%2B10


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=2%2F5 and the y-intercept is b=10


Notice if we graph the equation y=%282%2F5%29x%2B10 and plot the points (0,10) and (10,14), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%282%2F5%29x%2B10 through the points (0,10) and (10,14)


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





So now we have the equation

y=%282%2F5%29x%2B10

which means f%28x%29=%282%2F5%29x%2B10

Lets now find f%285%29

f%285%29=%282%2F5%29%285%29%2B10 Plug in x=5

f%285%29=%2810%2F5%29%2B10 Multiply

f%285%29=2%2B10 Reduce

f%285%29=12 Add

So f%285%29=12


Lets now find f%28-5%29

f%28-5%29=%282%2F5%29%28-5%29%2B10 Plug in x=-5

f%28-5%29=%28-10%2F5%29%2B10 Multiply

f%28-5%29=-2%2B10 Reduce

f%28-5%29=8 Add

So f%28-5%29=8

Notice if we graph y=%282%2F5%29x%2B10 we get:


graph of y=%282%2F5%29x%2B10 which goes through the points (0,10),(10,14),(5,12),(-5,8)

Notice how the line goes through all of the points. This verifies our answer.