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Current time:0:00Total duration:11:31

Examples: Graphing and interpreting quadratics

Video transcript

We're on problem 36. It says which of the following sentences is true about the graphs of y is equal to 3 times x minus 5 squared plus 1 and y equals 3 times x plus 5 squared plus 1? So let's do something very similar to what we did in the past. If you think about it, both of these equations, y is going to be 1 or greater. Let's just analyze this a little bit. This term right here, since we're squaring, is always going to be positive. Even if what's inside the parentheses becomes negative, if we have x is minus 10, inside the parenthesis becomes negative, but when you square it, it always becomes positive. You're going to multiply 3 times a positive number, so you're going to get a positive number. So the lowest value that this could be is 0. The lowest value that y could be is actually 1. The same thing here, this number can become very negative, but when you square it, it's going to become positive. So this expression with the squared here is going to be positive, and you multiply it by 3, and it's going to be positive. So the lowest value here is always going to be zero when you include this whole term. So similarly, the lowest value y can be is 1. I just want to think about it a little bit just give you a little bit of an intuition. Let's think of this in the context of what we learned last time with the shifting. So let me draw it in a color that you can see. So if that is the y-axis, and I'll just draw mainly the positive area, so if I were to just draw y is equal to x squared plus 1, it would look like this where this is 1. That's y is equal to 1. The graph would look something along this. y is equal to x squared plus 1. That's a horrible drawing. Normally, I wouldn't redo it, but that was just atrocious. y is equal to x squared plus 1 looks something like that. It's symmetric. You get the idea. You've seen these parabolas before. This is y is equal to x squared plus 1. Now, if we were to do x minus 5 squared plus 1, what happens to it? Well, let me think about it. What is 3x squared plus 1? Well, then it just increases a little bit faster. So if I were say y equals 3x squared plus 1, it might look something like this. It'll just increase a little bit faster, three times as fast actually. So that would be 3x squared plus 1. The rate of increase in both directions just goes faster because you have this constant term 3 out there multiplying the numbers. OK, now what happens when you shift it? So let's do x minus 5. So where x equals 0 was the minimum point before, now if we substitute a 5 here, that'll be our minimum point. Because then that whole term becomes zero. So this vertex will now be shifted to the right. Let me do it in another color. So if this is the point 5, now this would be the graph. If you just took this graph and you shifted it over to the right by 5-- I won't draw the whole thing-- that graph right there would be 3 times x minus 5 squared plus 1. Remember, the y shift is always intuitive. If you add 1, you're shifting it up. If you subtract 1, you're shifting it down. The x shift isn't. We subtracted 5, x minus 5. We replaced x with x minus 5, but we shifted to the right. The intuition is there, because now plus 5 makes this expression zero. So that's 3x minus 5 squared. In the same logic, 3 times x plus 5 squared is going to be to here, plus 1. That's going to be shifted to-- let me pick a good color-- to the left. This is going to look something like this. It's going to be this blue graph shifted to the left. This is minus 5. So this is the graph right here of 3 times x plus 5 squared plus 1 Now, hopefully, you have an intuition. So let's read their statements and see which one makes sense. Which of the following is true? Their vertices are maximums. No, that's not true of any of these. Because the vertices is that point right there. It's actually the minimum point. A maximum point would look something like that. We know that, because you just go positive. This term can only be positive. If this was a negative 3, then it would flip it over. So it's not choice A. The graphs have the same shape with different vertices. Yeah, both of these graphs have the shape of 3x squared, but 1 vertices is 10 to the left of the other one. So I think B is our choice. Let's read the other ones. The graphs have different shapes with different vertices. No, they have the same shape. They definitely have the same shape. They both have this 3x squared shape. One graph has a vertex that is a maximum, while the other has-- no, that's not right. They both are upward facing, so they both have minimum points. So it's choice B. Next problem, problem 37. Let me see what it says. What are the x-intercepts? Let me copy and paste that. OK, I'll paste it there. What are the x-intercepts of the graph of that? Well, the x-intercepts, whatever this graph looks like, I don't know exactly what it looks like. This graph is going to look something like this. I actually have no idea what it looks like until I solve it. It's going to look something like this. When they say x-intercepts, they're like, where does it intersect the x-axis? So that's like there and there. I don't know if those are the actual points, right? To do that, we set the function equal to zero, because this is the point y is equal to 0. You're essentially saying when does this function equal zero because that's the x-axis when y is equal to 0. So you set y is equal to 0, and you get 0 is equal to 12x squared minus 5x minus 2. Whenever I have a coefficient larger than 1 in front of the x squared term, I find that very hard to just eyeball and factor, so I use the quadratic equation. So negative B, this is the B. B is minus 5. So negative negative 5 is plus 5. Negative B plus or minus the square root of B squared, negative 5 squared is 25, minus 4 times A, which is 12, times C, which is minus 2. So let's just make that times plus 2 and put the plus out there. A minus times a minus is a plus. All of that over 2A, all of that over 24, 2 times A. So that is equal to 5 plus or minus the square root-- let's see, it was 25 plus 4 times 12 times 2. Because that was a minus 2, but we had a minus there before, so 8 times 12, so 96, all of that over 24. What's 25 plus 96? It's 121, which is 11 squared. So this becomes 5 plus or minus 11 over 24. Remember, these are the x-values where that original function will equal zero. It's always important to remember what we're even doing. So let's see, if x is equal to 5 plus 11 over 24, that is equal to 16/24, which is equal to 2/3. That's one potential intercept. So maybe that's right here. That's x is equal to 2/3 and y is equal to 0. The other value is x is equal to 5 minus 11 over 24. That's minus 6/24, which is equal to minus 1/4, which could be this point. I actually drew the graph not that far off of what it could be. So this would be x is equal to minus 1/4. Those are the x-intercepts of that graph. So 2/3 and minus 1/4 is choice C on the test. We have time for at least one more. Oh boy, they drew us all these this graphs. Let me shrink it. I want to be able to fit all the graphs. So let me copy and paste their graphs. So this is one where the clipboard is definitely going to come in useful. OK that's good enough. I've never done something this graphical. So the graph they say is y is equal to minus 2 times x minus 1 squared plus 1. So that's what we have to find the graph of. So immediately when you look at it, you say, OK, this is like the same thing as y is equal to minus 2x squared plus 1, but they shifted the x. They shifted the x to the right by 1. I know it says a minus 1, but think about it. When x is equal to positive 1, this is equal to 0. So it's going to be shifted to the right by 1, plus 1. We know that. We know that it's going to be shifted up by 1, so up plus 1. Then we have to think is it going to be opening upwards or downwards? Think of it this way: If this was y is equal to 2x squared plus 1, then this term would always be positive. It'll just become more and more positive as you get further and further away from zero, so it would open up. But if you put a negative number there, if you say y is equal to minus 2x squared plus 1, then you're going to open downward. You're just going to get more and more negative as you get away from your vertex. So we're shifted to the right by 1, we're shifted up by 1, and we're going to be opening downwards. So if we look at our choices, only these two are opening downwards. Both of them are shifted up by 1. Their vertex is at y is equal to 1. But this is shifted 1 to the right and this is shifted 1 to the left. Remember, we said it was x minus 1 squared. So the vertex happens when this whole expression is equal to zero. This whole expression is equal to zero when x is equal to positive 1. So that's right here. So it's actually choice C. When your shifting graphs, that can be one of the hardest things to ingrain. But I just really encourage you to explore graphs, practice with graphs with your graphing calculator and really try to plot points and try to get a really good grasp of why when you go from minus 2x squared plus 1 to minus 2 times x minus 1 squared, why when you replace an x with an minus 1, why this shifts the graph to the right by 1. Anyway, I'll see you in the next video.