Now that Grant's has such great ideas from the glasses expo about how to run his business, and how to make all of his different products better, he's ready to get his glasses wipers, and nozzles onto the market. That means that it's time to talk about profit again. A while ago, Grant picked out some prices for his wipers and his nozzles, but he wants to use the algebra skills we've taught him, to check out if these really are the best choices he can make. So lets revisit the idea of profit for a moment. If profit, which we'll call p, is equal to the money that Grant earns, minus the money that he spends, and if we call money spent m, then can you write an equation for profit, involving just single letter variables? Assume that the money that they earned is only coming from wipers, and that the number of wiper sets sold is equal to n, and the price per wiper set is equal to c.
The answer is p equals nc minus m. This is just a simple case of substitution, like we've done many times before. We do need to pay close attention to the money earned term right here. If Grant is only earning money from wiper set sales, then the money he brings in is going to be equal to the number of wiper sets he sells times the price per wiper set.
This equation that we just developed is a great start. But one thing that Grant learned from a businessman at the expo, is that the price of an item actually effects the number of it that you can sell. So, in this case that would mean that n actually depends on c. If we let the horizontal axis here represent the price of an item, or c, and the vertical axis, represent the number of the item we sell, n. Which of these graphs best represents the relationship between c and n? Think about what each of them says about how change in the price will affect the number of units sold. This is a pretty simplistic view of economics, but just think about which of these is the best choice out of the four.
This one would be correct or at least the most correct of these four. As the price per item goes up, the number of units sold goes down. People are going to be less and less willing to purchase an item, the more and more expensive it is.
Based on data from competitors at the expo, Grant thinks that he will sell will sell 2000 in a month if he lets them cost $80 each. I'd like you to write an equation in slope-intercept form that relates the number of wiper sets sold to the price per wiper set. I'd like you to use this information to write an equation in slope-intercept form that relates the number of wiper sets sold in a month to the price per wiper set.
We're trying to develop a linear equation based off of two points of data that we have. Remembering that our graphs earlier had c on the horizontal axis and n on the vertical axis, we can translate this information into two points. We have the point 20,14,000 and the point 80, 2,000, we can now use these two points to find the slope of a line connecting them and then pick one of these two points to use with the slope in the point-slope form. The slope of the line connecting these two points, we can find is negative 200. I chose to use the point 80, this line, and then I transform that to be in slope-intercept form. So our final answer is, n equals negative 200 c plus 18,000.
Grant just doesn't care about how many wipers he sells though. He really cares about the money he's going to bring in, or his profit, which we came up with this equation for earlier. And I'd like you do now, is substitute in this new equation we have for n, the number of wipers that's sold, as it depends on c, the price per wiper set. Please simplify this equation as much as you can. We should come up with an equation for p, that doesn't actually mention n anywhere, or at least not explicitly.
All we need to do here is substitute this expression into the slot for n and the equation for p. Once we've done that, we just distribute multiplication by c to the two terms in the parentheses here. And that gives us a final answer of p equals negative 200 c squared plus 18,000 c minus m.
Our equation for profit is looking really good. Except, right now, I have no idea what m is equal to. Remember that m is just the money that Grant has spent. Now, we've talked many times before about how the money that Grant spends is going to be a combination of fixed costs, like his rent, for example, and costs that scaled the number of items that he makes or the manufacturing costs. Let's say that each glasses wiper set costs $10 to make, and that Grant makes exactly as many wiper sets as he sells. In addition, all these fixed costs together, equal $20,000 per month. If all of this is the case, then how shall we change the equation for Grant's profit each month? I'd like you to first calculate m, the money that Grant spends each month, and then plug this value for n into the equation that we had for p before. Then, simplify to get your final answer. I know this is a lot to think about in one problem and there are several steps here. So, just make sure you go through things slowly, take everything one step at a time, and write all of your work down on a piece of paper.
Our first step is going to be to calculate m, the amount of money that Grant spends each month. Here are the two important pieces of information that we got from the question. We can replace fixed costs with 20,000. Since we know that each wiper set costs $10 to make, and Grant makes n wiper sets, going to have it in the spot of m in our equation for p. Since we're subtracting m, we need to make sure that we're subtracting this entire quantity, both terms. This looks pretty good right now, but remember you want to replace n here. With the expression that we found in terms of c which was just negative 200c plus 18,000. Now, we can finally go through and simplify our equation. This gives us a final equation for p of, miraculously, negative 200c squared plus 20,000c minus
So, let's graph this new equation for profit. Which of the four graphs over here do you think best represents this equation?
The answer is c, this red curve. To figure this out, we could plug a few values for c into our equation and find the corresponding values for p. Those c and p values will be the coordinates of points on this graph. So, this is interesting. As we could have guessed from our equation, since we have a c squared term, we have a parabola. This is going to give us a great opportunity to use some of the skills that we learned in the last few lessons, and also some new ones.
Here's the graph of our equation for profit in terms of price per wiper blade, by itself. Looking at this, what price do you think that Grant should set per wiper set in order to maximize his profit? Remember that, the horizontal axis here is price per wiper set, and the vertical axis here is profit.
And the answer is $50. We're looking for the peak of this parabola, which is at this point right here. We trace this down, the spot it corresponds to on the horizontal axis, you see it's directly between 40 and 60, so it should be 50, or $50.
We've got this wonderful equation but yet again, Grant has neglected to take everything into account that he needs to think about in terms of his finances. It turns out that his rent actually went up $10,000 per month. And he forgot to tell us that in the information we used to create this equation. So thinking about this new piece of information, how do we modify our equation for profit? Think about this carefully. If Grant's rent increases, then the amount of money that he spends will increase. So think about what that should do for profit overall.
Our new equation for profit is going to look pretty much the same as the old one, except, we need to change this constant term on the end to be 10,000 lower than it was before. If Grant's rent increases, then his overall profit is going to decrease by a set amount.
Now that we have this new equation, we're also going to have a new graph. So what do we need to change about our old graph, or the graph of the old equation, to make it match this new equation? Do we need to shift it up by 10,000, shift it down by 10,000, shift it to the right 10,000 or shift it to the left 10,000?
We need to shift the graph down by 10,000, or in other words, the new graph will be 10,000 lower than the old graph. This is because Grant's profit, the p values, are all the vertical values on the graph will be each 10,000 lower than they were in the old graph. Let's take a look at this for a second. Here are the graphs for our old equation, the one in red. And our new equation, the one in blue. And we can see that the blue graph is shifted just below the red one. It's shifted down by 10,000. So the p coordinate of every single point along the blue curve is 10.000 lower. And the p coordinate of the point with the same c coordinate on the red curve. So it looks like changing the constants as final terms out here can shift the graph in vertical direction and we have a parabola. That's pretty interesting.
To try to compensate for the higher amount that Grant now has to spend on rent, he wants to increase all the possible wiper set prices that he could set by $4. Before, we represented a potential wiper set price by c, so my question is what do we need to replace c with now? Instead of charging c dollars for a given wiper set, how much should he charge now?
If we want the new wiper set prices to be 4 higher than they were before, Grant is going to need to charge c plus 4 dollars instead of c dollars.
Now that we've thought a little bit about how increasing the price of all wiper sets would affect our variables, what's your new equation for profit be? Think about the answer for the last quiz to come up with this result. Also, make sure you simplify the right hand side of our new equation as much as you can.
In the quiz before this one, we said that if we wanted to increase each potential wiper set price by $4, we should replace c with c plus 4. So that's exactly what I've done. In our new equation, everywhere we had the quantity c in the old equation, we now have a c plus 4. Note that we need to insert c plus 4 with parentheses around it so that whatever operator was applied to c applies to both parts of c and of 4. Now, we just need to simplify this expression. After a few steps, we finally get to a final equation of p equals negative 200 c squared plus 18,400 c minus 133,200.
We just saw that if we replaced c in our original equation with c plus 4, we end up with a simplified equation that is what I just read off. Now, I'd like to show you how the graphs of the old equation and the new equation compare. The first equation we had is shown by this blue curve, and the new equation is shown by this orange yellow one. Now, the difference between these two graphs is really that the orange one is shifted four units to the left of the blue one. And we can see that shift written in the equation in the first version of the new equation that we had, where we have these c plus 4s visible. However, this form is really not very pretty, which is why we wanted to simplify it. Although the simplified version looks better, we can't see this shift. The situation for horizontal shifts then, seems kind of different from the vertical shifts situation. We saw earlier that if we wanted to lower a graph, we just needed to change this constant term, in order to change all of the vertical values. I think we're going to need new form for the equation of a parabola, in order to show changes in horizontal direction a little bit more clearly.
If we are going to be talking about shifting parabolas around, we should probably have a sort of point of reference, a standard parabola that we can compare all the parabolas to. So I'd like you to tell me what the simplest equation you can think of for a parabola is. I've already filled in the y equals part for you, so just fill in the rest on the right side.
The equation should be y equals x squared. Remember, that in order to graph a parabola, we need to have a squared term in our equation, and this needs to be the highest power of any term.
If we started with a graph of our super simple equation, y equals x squared, what do we need to do to shift the graph of this down three units? So, in other words, what equation would produce a graph that is shifted down 3 from the graph of y equals x squared. Fill in the rest of the equation in this box.
The equation would need to be y equals x squared minus 3. Let's look at that graph.
Here, the graph of y equals x squared is this red curve and the graph of y equals x squared minus 3 is the greenish-yellow one. You can see that these curves are actually identical. The only difference comes from the presence of this minus 3 in the equation, y equals x squared minus 3. This shifts the graph down 3 units. If we shifted the green curve up 3, then it would exactly overlap with the red one. This is pretty interesting. This is exactly what we saw earlier when we were dealing with Grant's profit equation and graph. When we changed the constant term on the end, it moved the graph up and down.
What if instead we wanted to shift the graph of y equals x squared up 3 units? What equation would we need then?
The equation we need to get a graph that's just shifted up from the graph of y equals x squared by three units, is y equals x squared plus 3. Just as we saw with y equals x squared minus 3, where every point on the lower parabola was just three units below the corresponding points on the red one, this time, everything in the green parabola is three units higher than everything in the red parabola. So if you move the green parabola down three units, it would exactly match the red one. So you see that adding constant terms to the side of the parabola equation that has x on it, moves the entire equation up. And subtracting constant terms over here moves it down.
Right now, when we have the equation y equals x squared minus 3. The part of the equation showing the shift in the graph, compared to y equals x squared, is on the right hand side of the equation, next to the x term. But what if I wanted the shift to be shown on the left side of the equation, where the y is. After all, that's the direction we're moving things in, the vertical one. So what I'd like you to is rewrite this equation so the constant term is on the left hand side, next to the y, instead of on the right hand side.
Through of the constant term from being on the right-hand side to being on the left-hand side, we usually need to add 3 to both sides. This gives us an equation of y plus 3 equals x squared. So, in other words, if you want to shift to the problem y equals x squared down by 3 units, we just need to replace y with y plus 3.
So, in general, if we want to shift a parabola down by k units, what do we need to replace y with?
We need to replace y with y plus k, to shift the parabola y equals x squared, down by k units. This is exactly what we saw with y plus 3 equals x squared. Which remember was a translation of the graph y equals x squared down 3 units.
If instead, we want to shift the parabola up by k units, what then do we need to replace y with?
Shifting up or moving in the positive y-direction, means replacing y with y minus k. This goes along with the same reasoning that we used for the shift downward. We just need an opposite sign here to indicate the opposite direction of motion.
Using the same reasoning, what do you think that we need to replace x with, if we want to shift our parabola to the right 2 units? Remember that right is the positive direction in x.
We need to replace x with x minus 2 if we want to shift this parabola to the right by 2. In the y direction we saw that moving in the negative direction in y, required adding to y. And moving in the positive direction required subtracting from y. The same is true of x.
Now I've got five different curves over here for you, and I'd like you tell me which one of these represents the graph of y minus 3 equals the quatity x minus what we said replacing x with x-2 would do to a curve
The answer is, the curve I'm calling D, which is this purple one up here. Replacing x with x minus 2, meant a shift 2 to the right, from the original equation, y equals x squared, who's graph is this grey one right here. And replacing y with y minus 3, meant shifting that curve up 3, so if we move this curve over 2 and then up 3, we end up with this purple one, right here. Note that just as we saw in the example with Grant, we replace x with something else like x minus 2 in this case, that entire quantity has the same operations applied to it, as x had. So this entire thing is squared, not just the x part.
When we were talking to Grant earlier about his future profit, he was really interested in the maximum point on this parabola. Since this point indicated the vertical value, the p value, of the maximum profit he could make. This maximum point is called the vertex of the parabola, or at least it is for this parabola. Now considering how much it mattered to Grant, and just looking at what a special point this is on the given parabola, the vertex is clearly pretty crucial. So let's focus on this for a second. If you have a parabola shaped like this, opening upward, does the vertex have the maximum or minimum y value for any point on the curve? What about for a parabola shaped like this, opening downward?
Since the parabola is opening upward from this lowest point, when we have this sort of u-shaped curve, the vertex is the minimum of the graph. Or in other words, the vertex is the point that has the smallest y value. The opposite is true if the parabola opens downward. Since every point below this one right here will have a smaller y value, the vertex has the highest y value, or it's the maximum.
If we go back to our super simple equation y equals x squared, where is the vertex of its graph?
The vertex here is at the point 0,0, which we can see is the minimum of the curve since it's opening upward.
This time, I've given you the graph and the equation y minus 3 equals the quantity x minus 2 squared. So, where is the vertex of this parabola?
We can see the vertex is right here, at the point 2 comma 3. Since the graph was shifted over to the right 2 and up 3, from the graph of y equals x squared.
Thinking about what we've just seen in recent graphs, how do you think the graph of the equation y plus 3 equals the quantity x minus 2 squared, will be shifted away from the graph of y equals x squared? How many units will it be shifted in the vertical direction, and how many units in which horizontal direction? Please pick one choice from each of these columns to fill in each blank.
Since we have y plus 3, instead of just y, this graph will be shifted three down, from the graph of y equals x squared, and because we have x minus 2 instead of x, it'll be shifted two, to the right. Remember that when were adding constant terms, whether it's next to y or next to x, we're shifting in the negative direction along the respective axis. So since we're adding 3 vertically, we're moving down and since we're subtracting 2 horizontally, we're moving in the positive direction along the x axis.
Now, that we know how this graph is shifted compared to y equals x squared, where do you think the vertex is?
The vertex of the equation y plus 3 equals the quantity x minus two squared is that two comma negative 3, this point down here. Even though I didn't give you the graph, you could figure this out, by thinking about how this graph is shifted away from the graph of y equals x squared. Here's a quick hand-drawn version, not very spectacular I'm sorry, of y equals x squared. Our point of reference are our parent problem. Since we already knew that the red curve would be just an exact shift of the black curve to the right 2 and down 3. We can pick any point on the black curve and find the corresponding point on the red curve. We can do this for the vertex, which is originally at 0, 0 on the black curve. So then if we move it to the right 2, the x coordinate becomes 2. And if we move it down 3, the y coordinate becomes negative 3.
One more practice finding the vertex of a parabola from its equation. Where is the vertex of y equals the quantity x minus 7 squared plus 5? Think about the form that this equation is in right now, and think about what it needs to be in, in order for us to find a vertex.
The vertex of the parabola y equals x minus 7, squared, plus 5 is at 7,5. The way we originally have this equation written, there is no constant term next to y on the left hand side of the equation. Instead, outside of the portion of the equation, where x is being modified, there's another constant term. It isn't receiving the same kind of operations as what's inside the parentheses here with the x. To get this in the form that we've been working with, we need to subtract the quantity x minus 7 squared. From here, we can see that this graph should be shifted 7 to the right, and up 5 from the parent graph. So that gives us a vertex of 7, 5. For your viewing pleasure, here is that curve. The black one is y equals x squared, and this green one is our new graph. Here I've written its equation in the form that I prefer, so that we can see very easily where the vertex is.
We know that for parabolas that point upward, like the two that I have shown on the graph here. The vertex is the minimum point on the curve, or in other words, the point with the lowest y value. So let's look back at our equation right here for the green problem. Y minus 5 equals the quantity x minus 7 squared. Let's take a second to dissect the different parts of our equation. Looking only at the right side, we have the quantity x minus 7 squared. Now what's the smallest value that this quantity can have? Think about different things that you could plug in for x, and what quantities for this whole expression each of those would yield.
The answer here is 0 since anything squared can't be a negative number. The smallest non-negative number is 0.
What value of x makes this equation true?
We just need to solve this equation for x. When we do that, we get x equals 7.
If we plug in x equals 7 to our equation for the green parabola over here, then what does y equal?
If we insert a value of 7 for x in our equation, we end up with a value of 5 for y. This is perfectly in light of what we would expect, considering we have a means. We just said that the smallest value we could get for the quantity x minus 7 squared was zero. We got that by plugging in 7 for x. We're finding a y value here based on an x value. So the x value that we pick minimizes an expression that's helping to determine the y value. And the y value is going to be as small as it can be. Which, as we can see, is true of the point that we just graphed, since it's the vertex of this upward facing parabola.
We've seen a lot of different parabolas in this lesson so far. So I've written down the equations for a few of them here, and then also the corresponding vertex for each one. Thinking about the relationship between a vertex of each equation, and the equation itself, could you write an equation in the same form as these over here? For a parabola whose vertex is at the general coordinates h comma k? One more note, for simplicity's sake, assume that this parabola is just a shifted version of y equals x squared. The only thing that's different about it from this original graph is that the vertex has moved to h comma k.
Assuming nothing has changed about this parabola, compared to y equals x squared, aside from the entire thing shifting. We know that the whole thing needs to shift by h coordinates in the horizontal direction, and k coordinates in the vertical direction. So the vertex will move to (h,k). If we replace x, with the quantity x minus h, and y with the quantity y minus k, this gives us a final equation of y minus k equals the quantity x minus h squared. So anytime we see an equation written in this form, we can automatically tell where its vertex is. This is going to be super useful as you move forward, doing more and more with parabolas.
Now that we have a general form for the shifted version of the parabola y equals x squared. Let's work with this a little bit. A lot of parabolas we see don't have equations that are already in this form. So to see what other equations we could eventually get back to this general form, I'd like you to multiply out the right side of the equation. Then make sure that you combine like terms.
Multiply it out, the right- hand side becomes x squared minus 2hx plus h squared.
We ended the last quiz with this equation, but if we add k to both sides, we could end up with this equation, u equals x squared minus 2hx plus h squared plus k. So here, y is by itself on one side, and then all of the x terms and constants are on the right side. We've seen a number of equations for parabolas that are in this form, even if we didn't see general forms of them written with constants like these. So let's compare this to another parabola. Maybe y equals x squared, plus 6x plus 5, now because these two equations are in the same form, we should be able to figure out the value of each of the constants over here, based on the value of constants over here. So compared to this general equation, in equation y equals x squared plus 6 x plus 5, what number is h equal to? So find where the constant h appears in this general equation, figure out where that corresponds to in our equation with particular values inserted, and then calculate age based on that. This is a little bit tricky, but just give it a few tries.
In this equation, h equals negative 3. We see h appear in two spots in the general equation. We have a term minus 2hx, and also a term h squared. And the h squared is going to be a little bit harder for us to pick out using this equation. Since we have one constant term of 5 over here, but we have two constant terms each squared and k here. And we can't tell which of these is contributing how much to 5. However, we have one term in the general equation that has an x to the first. And also only one term in the particular equation that has an x to the first. So that means that we have negative 2hx is equal to both sides of the equation by negative 2x. Our xes will cancel out. And 6 divided by negative 2 is just negative 3. So that's our answer.
So now we have a value for h, for this particular equation. But remember what h stood for, in the original equation we started working with. H was the amount that this graph is shifted horizontally, compared to y equals x squared. Now that we found a value for h in this equation, h equals negative 3. Can you find a value for k as well?
k is only involved here in the constant term at the end, and the constant term in our equation with numbers is 5. So, you need to take all of the constant terms from this equation which are h squared and k, and have them equal to the constant here, which is 5. We already know what h is so now, we just solve for k. I find that k equals negative 4.
We can use these values for h and k and insert them into that general expanded form of the equation for a parabola to rewrite this, like this. Y equals x squared minus 2 times negative 3x plus negative 3 squared plus negative 4. Now, we're ready to write this in the form we saw those other equations in. Y minus k equals the quantity x minus h squared. Please do that in this box right here. Once you've done that, I'd also like to, to tell me the coordinates of the vertex of this parabola.
This is really easy to do considering we are reminded of the general formula right here, and we have values for h and k. We just plug those in to get a final equation of y plus 4 equals x plus 3 squared. Here, the vertex is at negative 3, negative 4.
I started to show you over the past few videos, how we can begin with an equation in the standard form for a parabola, and then transform it so that it's written in a form where we can read off the coordinates of the vertex right away. This is a really powerful tool. Since we've learned now how to recognize the vertex of a parabola from it's graph and also the role that, that vertex plays in characterizing the equation of the parabola. As we delve more deeply into how to get equations in this form, we're going to be more and more successful about recognizing and graphing parabolas.