What equation belongs to both families?
"Find an equation for the family of linear functions with slope 4. Find an equation for the family of linear functions such that f(4) = 1." these two questions were asked first and you need them to answer the question above. use standard coordinates x and y
(1) The family of linear equations with slope 4 is `f(x)=4x+k`
** We have `f'(x)=4` , so `int 4dx=4x+C` , replace the constant of integration with parameter `k` **
(2) The family of linear equations with `f(4)=1` is `g(x)=a(x-4)+1`
(3) The equation(s) that are in both families satisfy both conditions.
When x=4 we have:
Thus the equation that lies in both families is `h(x)=4x-15`
Linear functions can be expressed as
So an example of a linear function with slope 4 is
Now let's make up a random linear equation such that f(4) = 1.
How about `f(x)=0x+1` ? That's a horizontal line through (4,1). Or maybe `f(x) = x-3` ? This would also pass through the point (4,1).
But if the slope is 4 and the function passes through (4,1), we have
`f(4) = 4*4+b=1`
so b, the y-intercept, is -15.
`f(x) = 4x - 15`