(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.
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(3) The equation(s) that are in both families satisfy both conditions.
Then:
`4x+k=a(x-4)+1`
`4x+k=ax-4a+1`
`(4-a)x+(k+4a-1)=0`
When x=4 we have:
`(4-a)4+k+4a-1=0`
`16-4a+k+4a-1=0`
`k=-15`
Thus the equation that lies in both families is `h(x)=4x-15`
Linear functions can be expressed as
`f(x)=mx+b`
So an example of a linear function with slope 4 is
`f(x)=4x+7`
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`