Algebra II
-
California Standards Test: Algebra II
-
California Standards Test: Algebra II (Graphing Inequalities
-
CA Standards: Algebra II (Algebraic Division/Multiplication)
-
CA Standards: Algebra II
-
Algebra II: Simplifying Polynomials
-
Algebra II: Imaginary and Complex Numbers
-
Algebra II: Complex numbers and conjugates
-
Algebra II: Quadratics and Shifts
-
Examples: Graphing and interpreting quadratics
-
Algebra ||: Conic Sections
-
Algebra II: Circles and Logarithms
-
Algebra II: Logarithms Exponential Growth
-
Algebra II: Logarithms and more
-
Algebra II: Functions, Combinatorics
-
Algebra II: binomial Expansion and Combinatorics
-
Algebra II: Binomial Expansions, Geometric Series Sum
-
Algebra II: Functions and Probability
-
Algebra II: Probability and Statistics
-
Algebra II: Mean and Standard Deviation
Algebra II: Functions and Probability 70-75, functions and probability
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- We're on problem 70.
- And they still want us to do some sequences and series.
- What is the n'th term in arithmetic series below.
- Which means that it just increases by a constant amount
- every term.
- So let's think about it a little bit.
- The first term is 3.
- And then we increment it by 4 each time.
- Let me write all of this.
- 3, then we go to 7, then we go to 11, and then we go to 15,
- then we go to 19.
- So we go to 3 and we're adding 4 every time.
- So you get the sense that each term in the sequence is going
- to be 4 times the n'th term.
- But that doesn't quite work out.
- Because I've had 4 times-- if this is the first term, 4
- times the first term is 1.
- And 3 is less than 1.
- And then 4 times 2 is 8.
- And 7 is less than 8.
- And 4 times 3 is 12.
- But this is 1 less than 12.
- So it seems like all these are 1 less than a multiple of 4.
- So the n'th term is going to be 4n minus 1.
- And that should work out.
- The third term is 4 times 3, 12, minus 1 is 11.
- And that's the third term.
- So it's 4n minus 1, which is choice D.
- Next problem.
- Problem 71.
- Let me copy and paste it.
- Actually, I'll copy and paste all of them.
- All right.
- Which expression represents f of g of x if f of x is equal
- to x squared minus 1 and g of x is equal to x plus 3?
- So a lot of times, these seem daunting.
- But when you get used to it, these become kind of fun
- problems. So f of g of x means, in f of x , everywhere
- you see an x you have to replace it with g of x.
- So f of g of x is equal to g of x squared minus 1.
- Now, all I did is if I said f of dog, wherever I see an x, I
- would replace it.
- So f of dog would be dog squared minus 1.
- Whenever I see an x I just replace it with whatever I
- input in right there.
- So they want to know f of g of x , I stick a g of x there.
- So it's going to be g of x squared minus 1.
- But what's g of x?
- Well g of x is x plus 3.
- So this is going to be equal to x plus 3 squared minus 1.
- Which is equal to x squared plus 6x plus 9 minus 1.
- Which is equal to x squared plus 6x plus 8.
- Which is choice B.
- Next question.
- Given that f of x is equal to 3x squared minus 4 and g of x
- is equal to 2x minus 6, what is g of f of 2?
- Well here, since they gave us an actual number, so they
- didn't leave it abstract, we could figure out what f of 2
- is and then pop it into g.
- So what's f of 2?
- Yeah, you should be able to see that color. f of 2.
- Wherever you see an x, you replace it with 2.
- So that equals 3 times 2 squared minus 4.
- Which is equal to-- 3 times 2 squared, 2 squared is
- 4 times 3 is 12.
- Minus 4 is equal to 8.
- So f of 2 is equal to 8.
- So g of f of 2 is equal to g of 8.
- And what's g of 8?
- g of 8, everywhere you see an x, just pop in an 8.
- Is equal to 2 times 8 minus 6.
- Is equal to 16 minus 6.
- So this is equal to 10.
- Choice D.
- Problem 73.
- If f of x is equal to x squared plus 2x plus 1 and g
- of x is equal to 3 times x plus 1 squared, which is the
- equivalent form of f of x plus g of x?
- So essentially they just want us to add these two functions.
- So if we say f of x plus g of x-- let me do that. f of x--
- and I'll do it in another color-- plus g of x.
- Well now we just add them.
- What's f of x? f of x is x squared-- I'll do it down
- here-- x squared plus 2x plus 1.
- That's f of x.
- Then we're going to add g of x to that.
- Oh, I see why it's a little-- they want us-- g of x you kind
- of have to expand this out.
- So g of x is 3 times x plus 1 squared.
- Well that's 3 times x squared plus 2x plus 1.
- But then from here we just have to simplify this all out.
- And I'll do that in a third neutral color.
- Brown color.
- x squared plus 2x plus 1 plus-- distribute the 3-- 3x
- squared plus 6x plus 3.
- And now let's see, you have an x squared and a 3x squared.
- You add those together, you get 4x squared.
- And I have a 2x and I have a 6x.
- Add those together, I get plus 8x.
- And then I have a 1 and I have a 3.
- So plus 4.
- 4x squared plus 8x plus 4.
- And that's choice C.
- Next problem.
- Next page.
- All right, let me see.
- What do they want us to do here?
- A math teacher is randomly distributing 15 rulers with
- centimeter labels and 10 rulers
- without centimeter labels.
- All right.
- What is the probably the first ruler she hands out will have
- centimeter labels and the second ruler will not have
- centimeter labels?
- So, OK, let's think about the first one.
- I'll color code it.
- So what is the probability that the first ruler she hands
- out will have centimeter labels?
- So there's a total of 25 total rulers.
- 15 with centimeters and 10 without.
- At least initially, there's a total of 25 rulers.
- And they want to know the probability that the first
- ruler she hands out will have centimeter labels.
- So 15 have centimeters.
- So the probability is 15 over 25.
- That's the probability of the first one
- having centimeter labels.
- That's that one.
- But they want to know, the first one having centimeter
- labels and the second ruler not having-- and you can't see
- that highlight color-- and the second ruler will not have it.
- So this is-- so we're going to have to multiply the
- probability of the first ruler times the probability that the
- second, without labels, and now this is the key-- given
- that the first ruler had labels.
- Now this might all seem really complicated with the notation
- but when you think of it intuitively, it should
- hopefully make some sense.
- I've given away-- so the first ruler there's a 15 out of 25
- chance that I've given a ruler with labels.
- Now on the second ruler, since both of these things have to
- happen, I can assume that I've already given away the first
- ruler having labels.
- So how many rulers do I have now?
- Well I definitely have 24 rulers left.
- And now, how many of these do not have labels?
- Well the first ruler I gave away had a label.
- So I still have 10 rulers that do not have a label.
- If you wanted to ask a similar question, you could say what
- is the probability that the first two ones that I hand out
- have labels?
- That would be 15 over 25 times 14 over 24.
- Because the labels would've been decremented
- after the first one.
- Because we're assuming.
- We have to say both of these things happen.
- Now we're saying, OK, we're going to have one less ruler
- but I haven't reduced the number of
- rulers without labels.
- But anyway, let's just simplify this out.
- This becomes 15 times 10 divided by 25 times 24.
- You can do a lot of canceling so we don't have multiply--
- let's see, divide that by 3, get 3.
- Divide that by 3, get 5.
- Divide that by 5, you get 1.
- Divide that by 5 you get 2.
- 3 times 2 is 6.
- So this is equal to 6 over 24.
- Which is equal to 1/4.
- Which is choice A.
- Next problem.
- Let's see.
- Problem 75.
- On a certain day the chance of rain is 80% in San Francisco
- and 30% in Sydney.
- Fair enough.
- Assume that the chance of rain in the two cities is
- independent.
- What is the probably that it will not rain in either city?
- So it's going to be the probability of not rain, not
- in San Francisco, times the probability of not in Sydney.
- Because we have to have both of these things happening and
- they're independent.
- So the probabilities will be multiplying.
- So what's the probability of not
- having rain in San Francisco?
- They gave us the chance of having rain as 80%.
- So the probability of not having rain is 20%.
- Or, I like to stick to the fractions.
- 20% is 1/5.
- There's a 1/5 probability of not having rain.
- What's the probability of not having rain in Sydney?
- It's 30%.
- Actually, I probably should've stayed in decimal world.
- So let's say it's 20% in San Francisco of not having rain.
- The probability of Sydney of not having rain is 1 minus
- 30%, or 0.7, which is a 70% chance of not having rain.
- And that is equal to-- 2 times 7 is 14.
- We have two numbers behind the decimal point.
- So there's a 14% chance that it will not
- rain in either city.
- And that is choice B.
- Next problem.
- Well actually I'm out of time.
- See you in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.