Question

Compare the functions shown below:
f(x) = 4 sin (2x − π) − 1
h(x) = (x − 2)2 + 4

Answer 1

g(x)
x y
−1 6
0 1
1 −2
2 −3
3 −2
4 1
5 6

Answer 2

Compare ^^ those 3 functions
And hey @mathmate long time no see, lol

Answer 3

Oh and the question is:
Which function has the smallest minimum y-value?

f(x)

g(x)

h(x)

Both f(x) and g(x) have the same minimum y-value.

Answer 4

^^ That's the question that goes with those functions, I forgot to put it at the beginning, lol

Answer 5

That's ok.
Do you know how to find the minimum of a function?

Answer 6

Ummm, not really :/

Answer 7

Have you done calculus yet?

Answer 8

I'm in algebra 2, but the last module of this class is like and advance to what Calculus is gonna be like, and that's what I'm on now, but it's hard and confusing, to me at least

Answer 9

*an

Answer 10

For g(x) is the smallest minimum y-value -3?

Answer 11

No, that's ok.
There are many ways to solve the problem. Telling me what you're doing helps to find the appropriate way.

Answer 12

As a matter of fact, you don't need calculus to find the minimum of each of the functions. Algebra II is all you need to find the minimum of each.
You already gave me correctly the minimum of g(x).
Which one do you want to start, f(x) or h(x)?

Answer 13

Hmm, f(x)

Answer 14

Would I just plug in a 0 where the x is in the function?

Answer 15

Not really, for sin(x).
ok, f(x)=4sin(2x-\(\pi\))-1
Do you know what is the range of sin(x)?

Answer 16

hint:
https://www.google.ca/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=graph%20of%20sin(x)

Answer 17

From -1 to 1 right?

Answer 18

Exactly.
So what can be the minimum of f(x)?

Answer 19

Oh, so the minimum is -1 then

Answer 20

If the minimum of sin(x) (or sin(whatever)) is -1, what is the minimum of
f(x)=4sin(2x-pi)-1 ?

Answer 21

taking into account the minimum of sin(whatever) is -1! :)

Answer 22

Oh, ok, give me a sec

Answer 23

I got -1.4, but I feel like it might be wrong.

Answer 24

not quite 1.4, read on.
Or, you can apply the transformation, of the parent function sin(x) to
f(x)=4sin(2x-pi)-1
which implies vertical stretch factor of 4, a horizontal shrink of factor 2 (does not affect the minimum) and a vertical translation of -1.

Answer 25

So the answer for function f(x) is 1.4?

Answer 26

Well, if you put the min. value of sin(2x-pi)=-1, which is its minimum value, and put it into f(x), tell me what you get.

Answer 27

So I'd basically do 4 times -1 minus 1?
Because that gives me -5

Answer 28

exactly! So you got the minimum of f(x).
Are you ready to move on to h(x)?
(...mentally review second degree curves, vertex,....) :)

Answer 29

Oh yayyy, now I get it.
But now what do I do for h(x)?

Answer 30

h(x) = (x − 2)^2 + 4
It is already in the vertex form, it tells you off the cuff where the vertex is.
Can you tell me where the vertex is?

Answer 31

4

Answer 32

Actually, the vertex is (2,4). I'll take your answer as correct, since we are working with max/min.
Now is the vertex a max or a min? and why?

Answer 33

Sorry, lol I confused the vertex with the intercept

Answer 34

Umm I think the vertex is minimum...

Answer 35

Yes, because when you expand (x-2)^2, the coefficient of x^2 is +1.
When the leading coefficient (the coefficient of the highest degree) is positive, the vertex is at minimum.
So what is the minimum of h(x)?

Answer 36

So it's 4, because the 4 is the y, not the 2

Answer 37

exactly. (2,4) means x=2, y=4. So four is the y-coordinate of the vertex, the minimum.
Have you made up your mind about the minimum of h(x)? lol

Answer 38

Yes, now I understand :)
And the answer to the question is f(x) because the minimum of f(x) is -5, which is the lowest

Answer 39

Thanks!!

Answer 40

Exactly! you're welcome! :)