(a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family.
(b) Find an equation for the family of linear functions such that $ f(2) = 1 $ and sketch several members of the family.
(c) Which function belongs to both families?
a. $2 x+b$
c. $2 x-3$
All right, let's look at equations for the family of linear functions that have a slope of to. So we know that all equations of lines take on the form y equals MX plus B if we want to put them in slope Intercept form and M is the slope. So we're saying that we have lines that have a nem value of to a slope of to so we could write equations of this form. Y equals two x plus B B stands for the Y intercept, and that's going to change depending on the line. We don't have to use a be there. We could use any letter. We want to stand in for that. Why intercept now? Let's take a look at the graphs of several of these. So suppose we had y equals two x plus zero. Just the line Y equals two X. It would have a Y intercept of zero a slope of to so it would look something like this. Let's say we had y equals two x plus one. It's going to be parallel to the line. I just drew, and it has a Y intercept of one same slope. Let's look at y equals two X minus one. It has a Y intercept of negative one same slope. So it's going to be parallel that didn't live to parallel. We'll try that again one more time. Okay, There we go. Now for part B. We want an equation for the family of functions that all go through the point to one. Basically, when you see f of two equals one, that means the point to one. So if we take our point slope form, which is why minus y one equals m times X minus X one. And if we substitute the point to one in for X one and why one we'll get an equation that looks like this. Why minus one equals m times a quantity X minus two. So this represents the family of functions that all go through the point to one. Now, if we want to change that into slope intercept form, we could go ahead and distribute the M, and then we could add one to both sides. But I think it's more informational if we leave it the other way. All right, let's sketch several of these graphs, so we want point. We went lines that go through the point to one. So we locate the point to one and then we just draw a whole bunch of lines that go through that point. We could have this line. We could have this line. We could have this line. We could have this line. There's infinitely many possibilities. Lastly, we're going to combine both parts, so we want the function that has a slope of two and goes through the point to one. So let's go back to that point slope form again. Why? Minus y one equals M times X minus X one. And let's substitute the point to one. And they're So why minus one equals? We're also going to substitute the slope of two in there. So why minus one equals two times the quantity X minus two. Now, this is the equation of the line. We could leave it like that, or we could change it to slope intercept form first by distributing the to and then by adding one to both sides. 12 y equals two x minus three