Unit U BIG QUESTION - BQ#6

1. What is continuity? What is discontinuity?
          Continuity is when a function is continuous, meaning it can be drawn without lifting your pencil off the page.
          Discontinuity is when a function is discontinuous, meaning it has either a hole, a break in the graph, a jump, or if it oscillates.


2. What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and a value?
           A limit is the intended height of a function that exists anywhere on a graph as long as you reach the same height from both the left and the right.
           A limit does not exist if the left hand limit and the right hand limit are not equal.
           A limit is the intended height of a function while a value is the actual height of the function.

3. How do we evaluate limits numerically, graphically, and algebraically?
           We evaluate limits numerically by making a table that gets really close to the limit but the x-values never actually touch it; graphically by looking at the graph either drawn out or on a calculator and writing out the limits; and algebraically by directly substituting into the function.

SP 11: Unit R Concept 3 Type 2- Trig of an inverse trig function (use sum and difference to prove)

BQ #4 Unit T Concept 3:

• Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill? Use unit circle ratios to explain.
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     A "normal" tangent graph is uphill while a normal cotangent graph is downhill, not because it is the inverse of the other, but because of the positive/negative sign that corresponds to tangent in each quadrant. For example, tangent (and cotangent) are positive in the first and third quadrants, and negative in the second and fourth quadrants. They are arranged this way because of the unit circle ratios for these trig functions tan=y/x and cot=x/y. So for tangent, the graph will go--in one period--from negative (below the x-axis) to positive (above the x-axis). And for cotangent, the inverse of that rule applies.


Tangent

 Cotangent

BQ #3: Unit T Concepts 1-3 graphing tangent, cotangent, secant, and cosecant

How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.

These pictures show the relationship between sin and cos with all the other trig graphs. Sin and cos are alike in how they continuously swing. Unlike the other graphs with an asymptote that don't continue the way sine and cosine do. The other graphs run along the asymptotes without ever actually touching them. 

BQ#5-Unit T Concepts 1-3

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.
         Sine and cosine will never have asymptotes because asymptotes only happen when the trig ratio is undefined, in neither sine nor cosine will the ratio be divided by 0 because r will always equal 1.

BQ #2: Unit T intro

How do trig graphs relate to the Unit Circle?

              a. Period?- Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

The period for sine and cosine is 2pi because in relation to the unit circle, it takes a full rotation (2pi) for the positive/negative pattern to repeat, whereas it only takes half of a full rotation (pi) for the sign pattern to repeat.

              b. Amplitude?- How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?

Sine and cosine are the two trig functions whose inverse trig function can only be between -1 and 1, if it were more than the absolute value of 1, the function would bust and it wouldn't be possible.

Reflection #1 Unit Q: Verifying Trig Identities

"What does it means to verify a trig function?"
When we are given a problem that is set equal to something, we must prove that it equals the other side.

"What tips and tricks have you found helpful?"
Making sure that it is all a single trig function before trying to do anything else.

"Explain your thought process and steps you take in verifying a trig identity. Do not use a specific example, but speak in general terms of what you would do no matter what they give you."
1. Check for anything that is squared, if a function is squared, try to change it using a pythagorean identity. 2. Try to change it to all being one trig function. 3. Verify to make one side equal to the other.