course mth164
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Ok here are my questions for the test.
I am confused about the shifting and stretching of the graph. Can you please explain to me the formula and what each letter does and stands for on the graph. I never quite caught on when we were doing this.
If y = A sin( B x + C) + D:
A is the vertical stretch.
B is the horizontal compression.
D is the vertical shift.
The horizontal shift is -C / B. This is because you can factor the expression Bx + C to get B ( x + C / B), which is the same as B ( x - (-C / B)), which is of the form B ( x - h) for h = -C / B.
The vertical shift changes the amplitude from 1 to A.
The horizontal compression changes the period from 2 pi to 2 pi / B.
The vertical shift moves the horizontal axis of symmetry D units in the vertical direction.
The horizontal shift moves the graph h = -C / B units in the horizontal direction.
And on the identities...
Which exact identies do we need to know to derive the others? Can you give me the algebraic reasoning you owuld use to solve for these?
The algebraic reasoning is outlined on the summary page I handed out. The identities you need to know are in boldface; the rest you can derive from them.
Let me know if you have specific questions about that page.
When we know the cos adj/hyp to find an angle why do we use the inverse of cos to find theta?
You don't use cos = adj/hyp to find an angle, you use cos = adj/hyp to get the cosine of the angle, which is a number. If you want to find the angle whose cosine is equal to a given number, you use the inverse cosine.
Just as we use the inverse of the exponential to solve e^x = c, getting ln(e^x) = ln(c) so that x = ln(c), we generally use the inverse function to solve any equation of the form f(x) = c, where x is the variable and c is a known value.
The symbolic representation of this process is
f^-1 ( f(x) ) = (f^-1) (c) so that
x = (f^-1) (c).
Here f^-1 is understood to be the inverse function. For example if f(x) = e^x, then f^-1(x) = ln(x).
Note that the inverse function has nothing whatsoever to do with the reciprocal of the function.
The cosine function applies to angles and for any angle it gives you a number between -1 and 1.
The inverse cosine function applies to numbers between -1 and 1 and gives you an angle between 0 and pi.
You use the cosine to get numbers from angles, and the inverse cosine to get angles from numbers.