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CAHSEE practice: Problems 28-31

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Problem 28. Which equation best represents the part of the graph shown below? So we have a bunch of equations here. Let's see. This one is just a linear equation, so this is just some number times x. So that graph in general would be a straight line. This one and this one are straight lines. This one would be a straight line with a positive slope, maybe look something like that. This choice C right here would be a straight line with a negative slope, like that. They would both go through the origin, because they don't have any y-intercept right there. So neither of these are the answer. So let's look at these two-- I guess we could call them quadratics. So here we have 1.75x squared. So the graph of x squared-- let me do it in a better color-- the graph of x squared looks like this. It's a parabola. You've seen that before. And they're drawing it in the first quadrant right there, and that looks about right. And you're like, hey, but what about the 1.75? Well, that's just a scaling factor. If you throw a larger number there, it'll curve up a little bit faster. If you throw a smaller number there, it'll curve up a little bit slower. But the general idea is that they all have the general shape of a parabola. So this one looks pretty good. It's an upward opening parabola. It's kind of an upright U. So this looks like a pretty good solution. If we were to draw the whole graph, it would continue on this side like that. Then choice D, it's just like choice B, except they have a minus here. And the reason why I know that's not going to work is that the graph of minus x squared, or minus anything times x squared is going to look like this. It's going to be downward opening. So this is the graph of minus x squared, the graph of minus 1.75x squared, might open up or go down a little bit faster. So it's going to have the same general shape. So we know that this is also not the right answer. So our choice is B. Next question. 29. Lisa typed a 1,000-word essay at the average rate of 20 words per minute. If she started typing at 6:20 PM and did not take any breaks, at what time did Lisa finish typing the essay? So she had a 1,000-word essay and she was typing at 20 words per minute. So let me write that down. So she had 1,000 words and she's typing at 20 words per minute. So if you're writing 1,000 words and you're doing 20 per minute, how many minutes is it going to take? Well, what's 1,000 divided by 20? You could do this in your head. You say 100 divided by 20 is 5. 1,000 is 10 hundreds. So it's going to be 50. But if you don't want to do it in your head, you can just say 20 goes into 1,000-- 20 goes into 10 zero times. 20 goes into 100 five times. 5 times 20 is 100, and then you bring down the 0. 20 goes into 0 zero times, and then 0 times 20 is 0, and then you are done. 20 goes into 1,000 fifty times. Or you could say 2 goes into 100 five times. All the same thing. Just 20-- and then-- sorry, 50. Want to be careful. So it'll take her 50 minutes, and actually the units work out. You have words divided by words per minute, or that's the same thing as words times minutes per word. They cancel out, and you end up with 50 minutes. If that confuses you, don't worry about it too much. The intuition is what's important. You had a 1,000-word essay, you did 20 words per minute. So it was going to take you 50 minutes. 50 minutes times 20 words per minute is equal to 1,000 words. So you're going to have 50 minutes. So she starts at 6:20 PM. So at 7:00 PM, she's worked for 40 minutes. Because there's 60 minutes in an hour. But she works for 50 minutes, so she does another 10 minutes. So that gets her to 7:10. So the choice is D. Problem 30. What does-- let me do it in red. What does x to the fifth equal when x is equal to minus 2? So it's minus 2 times minus 2 times minus 2 times minus 2 times minus 2. Now we can just multiply it all out. Let's just do that, just for fun. Minus 2 times minus 2 is positive 4. Then positive 4 times minus 2 is minus 8. Minus 8 times minus 2 is positive 16. Positive 16 times minus 2 is minus 32. So our answer is A. Now, if you wanted to do that really fast, you might recognize, well, 2 to the fifth is 32. So it's either going to be that choice or that choice. And then we have a odd exponent. So we're going to be multiplying a negative times itself an even time, and then one more negative. So you're going to get a negative number. When you have an odd exponent, you're going to end up with a negative. You have an even exponent, you'll end up with a positive. So that would have been the fast way to do it, but even if you didn't do that, you could have just multiplied it straight out like we just did. And just remember a negative times a negative is a positive, and then a positive times the negative is a negative. It's minus 32. 31. The graph below compares the weight of an object on Earth to its weight on the Moon. Interesting. An object's weight on the Moon, OK, so this is the Moon weight, this is the Earth weight. So something that-- let's say-- well, I actually do weigh 150 pounds. So I weigh 150 pounds on Earth, and according to this chart, I would weigh 25 pounds on the Moon. Or I would weigh essentially 1/6 my weight that I am on Earth. What is it asking me? What is the approximate weight on the Moon of an astronaut who weighs 120 pounds on Earth? Well, I just said, essentially this thing is just taking 1/6. You can look at any point. When you weigh 10 pounds on the Moon, you're weighing roughly 60 pounds on Earth, or vice versa. So you just take 1/6, or we could actually just look at the chart. 120 pounds-- so if we go halfway, that's about 125, between 100 and 150. 120 pounds would be right there. You go up the graph, just like that. And you go there, and it looks like 18 or 19 pounds. And they say, what is the approximate weight? So the closest thing we are here, let's see, the choice of 15 and 20. We're definitely closer to 20 than we are to 15. So I will go with 20 right here. And you know, if we really wanted to look at it, this is actually the point that we should have looked at, this point right here would have been 120 pounds. But you just approximate where 120 is, you read the graph, you say, oh, it's closer to 20 than any of these other choices. So that'll be our answer.