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A cell phone plan has a basic charge of 35 dollars a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost $ C $ as a function of the number x of minutes used and graph $ C $ as a function of $ x $ for $ 0 \le x \le 600 $.

$C(t)=\left\{\begin{array}{ll}{35} & {\text { for } 0 \leq t \leq 400} \\ {35+.1(t-400)} & {\text { for } 400<t \leq 600}\end{array}\right.$

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So here we have this cell phone plan situation and we want to find a function, and we want to graph the function. So let's start by listing some things we know. We know that if you talk zero minutes, um, you still have to pay $35 for that cell phone. And we know if you talk 400 minutes, you still pay $35 because you get the 1st 400 minutes included. Now, if you talk 600 minutes, you're going to have to pay 10 cents a minute for those additional 200 minutes, and that's going to cost you an extra $20. So the cost is now going to be $55. Okay, we can start by plotting these points, and then we can go back and find the function. So we have a cost for a number of minutes. Zero cost $35. Number of minutes. 400 cost $35. We have a horizontal line segment. Number of minutes 600 cost $55. Now we have a line segment that's increasing. All right, let's figure out the equation for each of those parts. So basically we have a piece vice function here, so it's going to have to equations involved in it. And each one has a domain. So the first piece, that was the horizontal line. Why equals 35? So in the function, I just write f of X equals 35 the F of X is already there. All we have to do is include the 35 and the domain for that would be from 0 to 400 and I'm gonna go back and call this see instead of F because it's the cost function. We were given C in the book. All right, Now let's find the equation of the other piece. So we have the point here, which was 400 comma 35 we have the point here, which was 600 comma 55. And we could use those two points to find the equation of the line. 400 35 is one point 600 55 is one point. Let's find the slope. Why? To minus y 1 55 minus 35 over x two minus x 1 600 minus 400. And that gives us 20 over 200 that gives us 2000.1. And then at this point, we should go. Oh, yes, of course. That's the slope. That was a charge per minute. That makes sense. That's the rate of change. Slope is rate of change. Okay, now, what about the y intercept for that line? So let's use why, minus y one equals M times X minus X one. And for my point, I'm going to plug in just the 435. You could use either one. So let's make a little room here. That's going to be why minus 35 equals 0.1 times X minus 400 will distribute the 0.1 0.1 x minus 40 and then we'll add 35 to both sites. Why equals 0.1 X minus five? All right, so that's what we're graphing in the other piece. So that can go in here 0.1 time. And if you would prefer to write 0.1 as 1/10 that's okay, 0.1 X minus five. And the domain from that piece was greater than 400 minutes and less than or equal to well for what we're drawing just less than or equal to 600. But who knows? Maybe it goes further than that in real life. Okay,