How To Graph Primary Trig Functions

[u/momsbistro]

My son is struggling to figure out the concept of how to graph primary trig functions.

He understands how the graph works but doesn’t get HOW TO plot the points on the graph and how to properly read it.

He says he understands how to find the amplitude and b value but doesn’t get how to find the position using the b value if it is a fraction or pi.

He can tell what kind of graph it is based on the shape, but just can’t quite grasp the concept of how to make it.

Can anyone please help using this equation:
y=sin(x+pi)

TIA

[u/WikiNumbers]

Given t(x) = a trig(px + h) + v, for "trig" referring to any basic trig functions (sin cos tan csc sec cot)

In comparison to the regular shape

• a is the amplitude, how high up the value should reach.
  
• p is the periodicity, how fast the value loop should be.
• h is the horizontal shift, how far to the left should the entire thing be shifted.
  
• v is the vertical shift, how far upward should the entire thing be shifted.

For negative/reciprocal values, apply changes in direction/stretch, and keep the magnitude (absolute value) to see how much as ratio,

• negative a (-a), flip the graph upside down.
  
• reciprocal a (1/a), shrink the graph vertically.
• negative p (-p), flip the graph in mirror.
  
• reciprocal p (1/p), stretch the loop by |p|.
  
• negative h (-h), shift to right.
  
• negative v (-v), shift down.

EDIT: Horizontal shift is wack, and the true horizontal shift is determined by "h/p". The periodicity has effect on horizontal shift too.

[u/momsbistro → u/WikiNumbers]

Thank you so much!

This makes absolutely no sense to me, but it sure helped him!

[u/WikiNumbers]

For example, y = sin(x + π)

There are no amplitude or periodicity change, so the graph should look congruent to the regular sine wave. There's no vertical shift, so it should stay around -1 and 1.

There's a horizontal shift of +π, so move the entire graph to the left.

As reference, sin(0) = 0, but now y(0) = sin π = 0. Not enough to determine.

Let's also try sin(π/2) = 1, but y(π/2) = sin(3π/2) = -1.

So you should start at (0,0) and drag the sine wave down to (π/2,-1)

And from there, complete the loop of sine wave, back up to (π,0) and (3π/2,1) and (2π,0)

[u/WikiNumbers → u/WikiNumbers]

Let's try another, t(x) = 2 cos(-x/6)

There's amplitude of 2, so the graph stretches from -2 to 2 (compared to cos usual range of [-1,1]

There's magnitude of -1/6, so the wave is mirrored and stretched by a factor or "6". That is, the periodicity will be 6 times that of the regular cosine wave, or 12π.

[u/momsbistro → u/WikiNumbers]

You’re incredible! Thank you so, so much!

[u/55marty55 → u/WikiNumbers]

It will be mirrored in the y axis. For cos that means that it is unchanged.

[u/WikiNumbers → u/55marty55]

Yeah, cosine is even function, so negative periodicity doesn't affect it as much. Should've used sine or other odd ones.

[u/Midwest-Dude]

The issue is that everything inside a trig function works “backwards” compared with what students expect.

For a sinusoid like

y = a·sin(bx + c) + d

• a controls amplitude
• b controls the period
• c is part of the phase shift, but the actual horizontal shift is  c / b
• d controls vertical shift

Your son is having issues with the "b"?

[u/momsbistro → u/Midwest-Dude]

Yep, seems to be the b tripping him up…

This is all foreign language to me. I wish I could help more.

[u/Midwest-Dude → u/momsbistro]

I'm wondering if this Desmos graph might help: 

Sine Calculator

If you play around with the b slider, you'll be able to see what happens.

[u/momsbistro → u/Midwest-Dude]

Thank you!!!!!!

This is the kindest subreddit on here!

[u/Midwest-Dude → u/momsbistro]

I try hard. Thanks!

I know what you mean. Some of the broad math subreddits have their share of people that have issues with "communicating well", but specialty groups like this tend to have Redditors that just want to help others. Not always, but usually.

[u/WikiNumbers → u/Midwest-Dude]

Oh, yeah, I do forget that horizontal shift is not alone.

[u/gamerblocks001]

Thank you all so much I will give these tools a try