Question

using complete sentences explain how to find the minimum value for each for each function and and determine which function has the smallest y value
f(x) = 3x^2 + 12x + 16
And
g(x) = 2sin (2x-pi) +4

Answer 1

@jim_thompson5910 I'm pretty sure the minimum point for f(x) is (-2, 4). Is that correct?

Answer 2

yep the min for f is y = 4 when x = -2

ie the min f(x) = 4 occurs at the point (-2,4)

Answer 3

Okay how to we find g(x)?

Answer 4

when does sin(x) have a minimum?

Answer 5

I don't know

Answer 6

look at the unit circle

what is the lowest point on it?

Answer 7

(0, -1)?

Answer 8

what is the corresponding angle

Answer 9

I'll be right back in a few minutes

Answer 10

(0, 1)? and okay

Answer 11

I'm back

no look at where it shows the angle theta

Answer 12

what angle theta is where (0,-1) is located?

Answer 13

I don't know what it is. I'm confused

Answer 14

look at this

http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/1024px-Unit_circle_angles_color.svg.png

and tell me what the angle is

Answer 15

I don't know I'm lost..

Answer 16

do you see the 270 degrees?

Answer 17

Yes

Answer 18

so that's why sin(270 degrees) = -1

or sin(3pi/2 radians) = -1

Answer 19

the min occurs when sin(x) = -1, ie when x = 3pi/2

so you have to determine when 2x-pi is equal to 3pi/2

Answer 20

2x-pi = 3pi/2

what is x equal to?

Answer 21

0?

Answer 22

you should get x = 5pi/4

Answer 23

I see a shortcut though

Answer 24

they just want the min y value

sin( anything ) has a min of -1

so 2sin (2x-pi) +4 turns into 2*(-1) +4 when you replace all of "sin..." with the smallest it can get, which is -1

Answer 25

This is confusing to me.. I'm trying to understand it but it's a lot of info.. I'm not good with trigonometry

Answer 26

2*(-1) +4 turns into 2, so this is the smallest that g(x) can get

Answer 27

notice on the wavy graph, the lowest points have a y coordinate of y = 2

Answer 28

Yea I noticed that. So y=2 but what is x? Do we need to know x?

Answer 29

they just want to know the smallest output of the function

Answer 30

Okay. So g(x) has the smallest minimum?

Answer 31

yes

Answer 32

and you can see that on your graph you posted

Answer 33

Yeah I see that now haha. Do you mind helping with one more?

Answer 34

alright

Answer 35

Prove: \[\sin \theta-\sin \theta \times \cos^2 \theta = \sin^3 \theta\]

Answer 36

hint: factor out sin(theta)

Answer 37

only alter the left side

do not change the right side

Answer 38

So it would be \[\cos^2 \theta = \sin^3 \theta?\]

Answer 39

you should have

\[\Large \sin(\theta)\left(1-\cos^2(\theta)\right) = \sin^3(\theta)\]

Answer 40

then try to do something with the 1-sin^2

Answer 41

How would you get 1-sin^2?

Answer 42

oops

Answer 43

I meant 1-cos^2

Answer 44

Would you multiply

Answer 45

nope

Answer 46

there's an identity you use for 1-cos^2

Answer 47

hint: sin^2 + cos^2 = 1