Question

How do you find the minimum of a sine function?

Answer 1

minimum of sin(anything)=-1 so min sin(2x-pi)=-1

Answer 2

so min of g(x)=2*(-1)+4=2

Answer 3

What is the amplitude of the equation?

Answer 4

I'm supposed to find the minimum x and y value and the amplitude is 2

Answer 5

The question says "Using complete sentences, explain how to find the minimum value for each function and determine which function has the smallest minimum y-value." I already found the minimum of f(x) which is (-2,4)

Answer 6

Well, that may be A minimum, not the only one.

Answer 7

I graphed it in GeoGebra and got a sine curve looking like that one and my minimum y value is always 2

Answer 8

Is that right or is it supposed to be written in radians?

Answer 9

Also, the answer isn't correctly stated as (-2,4). That implies that it takes place at x = -2, and at x = -2, \(g(-2) = 2\sin(2(-2)-\pi)+4 \approx 2.4864\)

Answer 10

(-2,4) was for the f(x) not g(x). Sorry for the confusion

Answer 11

Ah, I must have misunderstood. Sorry! Didn't realize there was another function under discussion...

Answer 12

What is the function f(x)?

Answer 13

f(x) = 3x2 + 12x + 16. I figured out f(x) I just need to know how to find the minimum of g(x)

Answer 14

Okay, agreed that the vertex of that parabola is at (-2,4) so the minimum value of f(x) = 4.

Answer 15

Can you help with g(x). Is the minimum y value 2?

Answer 16

The minimum value of g(x) happens whenever \[x=\frac{1}{2}(\frac{\pi}{2}+2\pi n), \,n\in integers\]

Answer 17

To be honest @whpalmer4 , I have no idea what that means

Answer 18

it's just a way of saying that the minimum happens at x = pi/4, x = pi/4 + 2pi, x = pi/4 + 4pi, x = pi/4 + 6 pi, etc...

Answer 19

Unlike the parabola described by f(x) which has its minimum in only one spot (the vertex), the sin function has a minimum over and over and over again

Answer 20

All that formula says is how to find the values of \(x\) where g(x) has a minimum value, which is 2.

Answer 21

Not really important to understand it today, and I apologize for muddying the waters for you.

Answer 22

It's fine @whpalmer4 you actually gave a great explanation. Thank you :)

Answer 23

The \(\sin\) function has a range from -1 to 1. Multiplying it by 2 gives it a range of \(2*-1 \text{ to } 2*1\) or -2 to 2. When we add 4, that gives us a range of -2+4 to 2+4 or 2 to 6.

Answer 24

So the minimum value ever produced by \(g(x) = 2\) and the maximum value is \(g(x) = 6\). Therefore, \(g(x)\) is the function which has the smallest minimum y-value here.

Answer 25

I'm going to screenshot my answer. Will you tell me if everything looks good?

Answer 26

g(x) = 2*sin (2x - Pi) + 4.
g(x) min when sin (2x - Pi) = -1 = sin 3Pi/2
2x - Pi = 3Pi/2 -> 2x = 3Pi/2 + Pi = 5Pi/2 -> x = 5Pi/4.
g (x) min = 2* (-1) + 4 = 2

Answer 27

Does this look ok?

Answer 28

What did you use to graph that?

Answer 29

Yes, though it might be better to use the range approach to find the minimum values for g(x). (you know, the range of sin is -1 to 1, we multiply by 2, so it becomes -2 to 2, we add 4, so it becomes 2 to 6)

Answer 30

That graph was produced by a program called Mathematica. It's sort of a Swiss army knife for math, but unfortunately rather expensive and complicated to learn.

Answer 31

But all I had to do was type "Plot[2 Sin[2 x - \[Pi]] + 4, {x, 0, 8 \[Pi]}, AxesOrigin -> {0, 0},
GridLines -> {Table[1/2 (\[Pi]/2 + 2 \[Pi] n), {n, 0, 8}], {2, 6}},
Ticks -> {Table[1/2 (\[Pi]/2 + 2 \[Pi] n), {n, 0, 8}], {1, 2, 3, 4,
5, 6}}]"

:-)

Answer 32

As graphs are easy to make and fun to view, here's one of f(x):

Answer 33

Whoa lol. I guess I wont be using Mathematica, ever. But do you think my answer is "passable" at least?

Answer 34

Yes, I do.

Answer 35

Awesome, I've already given you best response. Thanks again!

Answer 36

But do try to understand the range approach, as sometimes graphing isn't practical...

Answer 37

I will. When I speak to my teacher again I'll ask her to go in depth and explain it to me