# 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...

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

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(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.

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`