anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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anonymous
  • anonymous
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anonymous
  • anonymous
@jim_thompson5910 I'm pretty sure the minimum point for f(x) is (-2, 4). Is that correct?
jim_thompson5910
  • jim_thompson5910
yep the min for f is y = 4 when x = -2 ie the min f(x) = 4 occurs at the point (-2,4)

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anonymous
  • anonymous
Okay how to we find g(x)?
jim_thompson5910
  • jim_thompson5910
when does sin(x) have a minimum?
anonymous
  • anonymous
I don't know
jim_thompson5910
  • jim_thompson5910
look at the unit circle what is the lowest point on it?
anonymous
  • anonymous
(0, -1)?
jim_thompson5910
  • jim_thompson5910
what is the corresponding angle
jim_thompson5910
  • jim_thompson5910
I'll be right back in a few minutes
anonymous
  • anonymous
(0, 1)? and okay
jim_thompson5910
  • jim_thompson5910
I'm back no look at where it shows the angle theta
jim_thompson5910
  • jim_thompson5910
what angle theta is where (0,-1) is located?
anonymous
  • anonymous
I don't know what it is. I'm confused
jim_thompson5910
  • jim_thompson5910
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
anonymous
  • anonymous
I don't know I'm lost..
jim_thompson5910
  • jim_thompson5910
do you see the 270 degrees?
anonymous
  • anonymous
Yes
jim_thompson5910
  • jim_thompson5910
so that's why sin(270 degrees) = -1 or sin(3pi/2 radians) = -1
jim_thompson5910
  • jim_thompson5910
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
jim_thompson5910
  • jim_thompson5910
2x-pi = 3pi/2 what is x equal to?
anonymous
  • anonymous
0?
jim_thompson5910
  • jim_thompson5910
you should get x = 5pi/4
jim_thompson5910
  • jim_thompson5910
I see a shortcut though
jim_thompson5910
  • jim_thompson5910
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
anonymous
  • anonymous
This is confusing to me.. I'm trying to understand it but it's a lot of info.. I'm not good with trigonometry
jim_thompson5910
  • jim_thompson5910
2*(-1) +4 turns into 2, so this is the smallest that g(x) can get
jim_thompson5910
  • jim_thompson5910
notice on the wavy graph, the lowest points have a y coordinate of y = 2
anonymous
  • anonymous
Yea I noticed that. So y=2 but what is x? Do we need to know x?
jim_thompson5910
  • jim_thompson5910
they just want to know the smallest output of the function
anonymous
  • anonymous
Okay. So g(x) has the smallest minimum?
jim_thompson5910
  • jim_thompson5910
yes
jim_thompson5910
  • jim_thompson5910
and you can see that on your graph you posted
anonymous
  • anonymous
Yeah I see that now haha. Do you mind helping with one more?
jim_thompson5910
  • jim_thompson5910
alright
anonymous
  • anonymous
Prove: \[\sin \theta-\sin \theta \times \cos^2 \theta = \sin^3 \theta\]
jim_thompson5910
  • jim_thompson5910
hint: factor out sin(theta)
jim_thompson5910
  • jim_thompson5910
only alter the left side do not change the right side
anonymous
  • anonymous
So it would be \[\cos^2 \theta = \sin^3 \theta?\]
jim_thompson5910
  • jim_thompson5910
you should have \[\Large \sin(\theta)\left(1-\cos^2(\theta)\right) = \sin^3(\theta)\]
jim_thompson5910
  • jim_thompson5910
then try to do something with the 1-sin^2
anonymous
  • anonymous
How would you get 1-sin^2?
jim_thompson5910
  • jim_thompson5910
oops
jim_thompson5910
  • jim_thompson5910
I meant 1-cos^2
anonymous
  • anonymous
Would you multiply |dw:1433476582158:dw|
jim_thompson5910
  • jim_thompson5910
nope
jim_thompson5910
  • jim_thompson5910
there's an identity you use for 1-cos^2
jim_thompson5910
  • jim_thompson5910
hint: sin^2 + cos^2 = 1
anonymous
  • anonymous
So 1-cos^2 would become sin^2
jim_thompson5910
  • jim_thompson5910
yep
anonymous
  • anonymous
Thank you so much
jim_thompson5910
  • jim_thompson5910
you're welcome
anonymous
  • anonymous
Wait how did you get the 1 again?

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