SOLUTION: Please write the linear function for which f (3) =6 and f (-4) =2 Thank you

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Question 146541: Please write the linear function for which f (3) =6 and f (-4) =2
Thank you
Found 2 solutions by jim_thompson5910, nerdybill:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
f (3) =6 tells us that when x=3, then y=6. So we know that the equation goes through the point (3,6)


f (-4) =2 tells us that when x=-4, then y=2. So we know that the equation goes through the point (-4,2)


So let's find the equation of the line that goes through the points (3,6) and (-4,2)




First let's find the slope through the points and


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula.


m=%282-6%29%2F%28-4-3%29 Plug in y%5B2%5D=2, y%5B1%5D=6, x%5B2%5D=-4, x%5B1%5D=3, ,


m=%28-4%29%2F%28-4-3%29 Subtract 6 from 2 to get -4


m=%28-4%29%2F%28-7%29 Subtract 3 from -4 to get -7


m=4%2F7 Reduce


So the slope of the line that goes through the points and is m=4%2F7


Now let's use the point slope formula:


y-y%5B1%5D=m%28x-x%5B1%5D%29 Start with the point slope formula


y-6=%284%2F7%29%28x-3%29 Plug in m=4%2F7, x%5B1%5D=3, and y%5B1%5D=6


y-6=%284%2F7%29x%2B%284%2F7%29%28-3%29 Distribute


y-6=%284%2F7%29x-12%2F7 Multiply


y=%284%2F7%29x-12%2F7%2B6 Add 6 to both sides.


y=%284%2F7%29x%2B30%2F7 Combine like terms. note: If you need help with fractions, check out this solver.


y=%284%2F7%29x%2B30%2F7 Simplify


So the equation that goes through the points and is y=%284%2F7%29x%2B30%2F7


Notice how the graph of y=%284%2F7%29x%2B30%2F7 goes through the points and . So this visually verifies our answer.
Graph of y=%284%2F7%29x%2B30%2F7 through the points and

Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
When they give you:
f(3)=6 and f(-4)=2
.
'y' is defined as 'f(x)', therefore they gave you two points:
(x,y) = (3,6)
(x,y) = (-4,2)
.
Calculate the slope of the line from:
slope = m = (y2-y1)/(x2-x1)
.
slope = m = (y2-y1)/(x2-x1)
slope = m = (6-2)/(3-(-4))
slope = m = (4)/(3+4) = 4/7
.
Remember, the slope-intercept of a line is:
y = mx + b
plug in one (let's use (3,6)) of the two given points (doesn't matter which one) and the calculated slope and solve for b:
y = mx + b
6 = (4/7)(3) + b
42 = 12 + b
30 = b
.
Recapping, we found:
m = 4/7
b = 30
.
Plug it all back into:
y = mx + b
y = (4/7)x + 30
.
That's it, you're done.