6.5.1 Trigonometric Graphs

Remember 6.3 transformations y = ±a sin(bx - h) + k a is a dilation h is the horizontal shift k is the vertical shift a is now going to be called the amplitude h is now called the phase shift b is a new variable we didn’t look at before, it will be the period, or frequency (physics term)

The book goes by: y = a sin(bx + c) a in our book will be amplitude (common to both math and physics) b will be period (math term), frequency (physics term) c will be the phase shift (math term) horizontal translation (math theory term) They also discuss Vertical Translation which we have used before but they leave the theory out of this section, for more information refer to 6.3

Examples: The book does a really good job of explaining the ideas of this section. It includes full picture color graphs please refer to page 448 for the graphs. y = sin (x) y = 2 sin (x) Notice the difference of amplitude, it has doubled thus the range of y – values have doubled refer to these tables:

Table set: Tables: the range of values for y For y = sin(x) R = { y| -1 ≤ y ≤ 1} For y = 2sin(x) R = { y| -2 ≤ y ≤ 2}

Period: For period it is a little tougher to imagine On the unit circle, it would be like increasing the rotation of a wheel For y = sin (x) the b value is 1 and the period is 2  The period is the point when the graph repeats y values 22 1 period

The period is found by a linear equation Since, b = 1 and b period = 1 period = 2  Then the period for a translation is

Finding the period for b values -1 < b < 1 we see an expansion of period ie 1 Period (4  ) for b values 1 < b and -1 < b we see a compression of period ie 1 period (  )

Phase Shift Notice that we have already talked about horizontal translations This is simply new terminology Horizontal Translation right  Horizontal Translation left 

To put it all together: y = a sin (bx + c) The amplitude is: |a| The period is: And the phase shift is:

Why ? Notice amplitiude relates to y values since bx + c is the argument they both relate to x Thus we are doing a two part translation for x values Up till now we have only looked at b = 1 so the Horizontal Translation was always = -c But now that b may or may not be 1 we must use the new formula

Homework p 458 1, 2, 5-31 odd, 41, 61, 62