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We are on problem fourteen. If the function f is defined by f(x) = x2 + bx + c, where b and c are positive constants, which of the following could be the graph of f? So b and c are positive so what does that tell us? a couple of things you immediately know that x2 term whatever is the coefficient of x2, that tells us whether the parabola opens up or opens down. It opens up if the coefficient on the x term is positive which it is positive one. So we know the graph is going to open up. It’s not going to be a downward opening graph, is another graph is going to look like a U as supposed to I don’t know. They’re to say I don’t know what this is. You know that it’s going to be a U and what else do we know about it? What do we know about it is y intercept? Well, this is the y intercept, when x is equal to zero, f (0) is equal to y. these two terms are going to be zero so f (0) = c. so we know that the y intercept is positive. Let’s see if we ca. if that alone allows us to solve the problem. We know that the y intercept id positive so when f is zero, it’s going to intersect some place on the positive y axis and we know that it is upward opening graph so the graph could look like this. It could also look like. This b term, I won’t get too much and to do intuition to b term and so it would actually. We actually know that it just shifted to the left but based on just what I said that it’s opening upwards and that it has positive y intercept. This would also have been a legitimate graph. Anything that is opening upwards like a U because this is a positive x2 here and intersects the y axis in the positive area would be a correct answer. If you look at the choices, A opens down. That’s not the right answer. B opens down not the right answer. C intercepts the y axis at zero so that’s not the right answer. D opens up but it intersects the y axis at the negative y. finally, E is very similar to what I drew in yellow so that is our answer. E.
Next problem. Almost finish with this section. Okay they drew us a cube. Let me draw it big because it looks like it that involves some fancy. Let me draw the cube so this is the front face. All my time has spent for drawing diagrams cube. Drawing a diagram goes back like that. That should always draw the back first if it’s good enough. I think you get the point and then there you know. It draws up the dotted line and maybe I'll draw that later if I have too.
So they tell us that this is point A and that this right here is point B. let’s say the cube shown above has edges of length two so each of these sides are two and A and B are midpoints of the two edges so A and B are midpoints so this side is equal to this side. If the whole side is equal to two, and this A is the midpoint, then we now that this is one. What is the length of AB. So this is what they want us to figure out. They want us to figure out. This is a pure visualization problem and it first like point us two dimensions and its crazy. What am I going to do? What we’re going to have to do is do the Pythagorean Theorem twice so what do we know? Well we know everything we need to solve this problem and let’s see if we can do it. Let me draw that bottom surface, the bottom surface of the cube. So ask your question. Can we figure out - so let me draw a line along the bottom surface. Let’s say that this line on the right is along the bottom surface of the cube. So let me draw this line. That’s along the bottom surface of the cube. Let me label this out like this and actually really fun problem. I'll do this in yellow. Can we figure out what that magenta dotted lines’ length is? Well sure we know that. We know first of all that is a right triangle that we had here right I can redraw it like this where this is yellow, brown, and then dotted line magenta. All I did is I kind of flipped it up so that you could see it. So what is that?
We know that the brown length, we know that’s one. We know the yellow length as what? It’s two right? Because all the sides is a cube or two so we use the Pythagorean Theorem to figure out the magenta line so it’s the square roots. You know 12 + 22 is equal to this side squared so you can just take the square root of it. So that equals the square root of 5. This is magenta line. So the magenta line is √5.
Now, can we figure pout that gray line that I do it first? Well, sure because now we have another right triangle the gray line is just the hypotenuse of this right triangle. That’s kind of you know at an angle. Let me make sure you understand what I'm saying. If I were to flatten it out, I have this green side right here. I have this doted line base and then I have the gray line which is the hypotenuse. So you know what this point right here would be A and then this point right here would be B. do we know the sides? Yes we do. We know this green line here is one so this side here is one. I'm jus redrawing it here. We know the magenta dotted line at the bottom which I had to switch colors of it. We just figured out that that magenta bottom-line √5.
So now, we can just use the Pythagorean Theorem to figure out the length of this line. The length of this line squared is going to be equal to the (√5)2 + 12 so the hypotenuse is crazy equal to this. We can take the square root of that. So that equals the squares root of - what’s the square root of 52? Then that’s just 5 right? Plus 1 so it’s equal to the square root of 6. So this line is equal to the square root of 6 and that is choice D. we luckily have not made a mistake.
Last problem. Problem 16. Let x with this kind of oval looking thing around it be defined as x2 - x for all vales of x. if a, - so they are telling us that a = a - 2, what is the value of a? What does this mean? A kind of this oval. Well, that just means that a2 - a right? That much to that just like that and what is a - 2 map two? And I'm just bringing the equal sign down. A - 2 map two. Everywhere place where I see an x, I'll put an a - 2 in there so that equals a - 22 - a- 2. Now, we just keep solving. Mostly, solving is on the right hand side so you see that equals a2 - 4a + 4 - a + 2 so we get a2 - a is equal to a2 - 5a - 4a - a + 6. I could subtract a from both sides, a2 from both sides. Let me add 5a to both sides so 5a so you get 4a = 6. Divide both sides by 4. We get a is equal to 6/4 which equals 1 1/2 or 3/2. That is choice C.
We are done in this section. I'll see you in the next practice test.