Linear function example: spending money
Jill just received $40. The number of dollars she has left, y, after x days, is approximated by the formula y is equal to 40 minus 2.5x. Graph the equation and use the graph to estimate how much money Jill will have 8 days later. So let's just make a table of x and y values. Then we can use that table to actually plot the graph. And then ask us to do everything else they want us. How much money she'll have after 8 days. And we should actually just put that right in the equation, we might as well do that. So we're doing numbers of days. We're not going to go back in time. She starts with $40, so we can start with 0 days. So 0 days. So she just received the $40, you don't even have to look at the equation here. What's 0 days after that, how much money will she have? Well she hasn't had a chance to spend it yet. So you could just think about it. She'll have $40. Or you could look at the equation and see that the equation verifies this. When x is 0, so the y value is going to be 40 minus 2.5 times 0, which is just 40. Because that part right over there is 0. So at time, at 0 number of days, she will have $40. Now we could do one day later, but then we're going to have these decimal points in here. So that this part of the equation always ends up with clean numbers, let's multiply it by multiples of 2. So then at 2 days, how much money will she have? Well, it's going to be 40 minus 2 times-- I'll do the same order-- minus 2.5 times 2. 2.5 times 2 is 5. So 40 minus 5 is $35. After 4 days, it's going to be 40-- let me do this in a different color, so when I plot the points, you'll see where I got my information from. After 4 days she's going to have 40 minus 2.5 times 4. 2.5 times 4 is 10. So 40 minus 10 is 30. You see, every 2 days that goes by, she is spending $5. $5 every 2 days, or $2.50 every day. And you actually see that right over here. She is spending. That's a negative sign. $2.50 dollars every day. Every time you increment x by 1, $2.50 goes away. So let's keep going. So then after-- I'll look for another color here-- 6 days, it's going to be 40 minus 2.5 times 6. 2.5 times 6 is 15. 40 minus 15 is equal to 25. Then finally we could do after 8 days. After 8 days, she'll have 40 minus 2.5 times 8. 2.5 times 8 is 20. So 40 minus 20 is $20. So we actually answered our question. Our estimate for how much money Jill will have 8 days later is actually $20. But let's do this first part. Let's actually graph the equation, see it visually. So let me draw some axes here. This will be a hand-drawn graph, but I think it'll get the job done. So let's make that our y-axis, or in this case it's the number of dollars she has. And let's make this my x-axis. This is our x-axis. And we only need to focus on the first quadrant. Because at least in this context, we're assuming she won't have a negative number of dollars. So the y values will be positive. And we assume that the days are only going to be positive. We're not to deal with negative time. So the x values are always going to be positive. So we're only going to be operating in the first quadrant. So that's all I have to draw. And so she starts off at $40. Let me mark off the y-axis in increments of 10 first. So this would be $10, this would be $20, this would be $30, this would be $40. And then I could do the $35, the $25, the $15, and then the $5. And then let me mark off the days. So this is-- let me do it, I'll do that same yellow color-- so this is after 2 days, this is 4 days, this is 6 days, this is 8 days. We could keep going if we like. So after 0 days, so this right over here, after 0 days, she has $40. So that's this point right over here. That's that right over there. Then after 2 days, she has $35. 2 in the x direction, then we go up 35. So that's that point right over there. Then after 4 days, she has $30. So you go x 4, remember the days are in x, or x are the days, actually you should mark it, these are days. And the y-axis is the dollar axis. So after 4 days, she has $30. Then after 6 days-- I'm going to do the same color-- after 6 days she has $25. So x-coordinate is 6, y-coordinate is 25. And then finally, after 8 days, she has $20. And so we've plotted those points, and we could connect them. We could actually just, if we had a nice ruler, we could just connect two of those, and we would have the line. But our line looks something-- let me do this in a new color-- our line would look something like that. That shows how much she has after every day. And we're done. We've graphed the equation, and we know she'll have $20 left after 8 days.