BQ #7: Unit V: Derivation of the Difference Quotient

The difference quotient is derived through the slope of a secant line. We can use this image as a guide to figure out how:

The distance from the origin to Point A on the x-axis is "x". The distance from point A to point B on the x-axis is "h", so therefore the distance from the origin to Point B is "x+h". The distance on the y-axis from the origin to Point A is labeled as "f", and because the x-coordinate of Point A is "x", the y-coordinate of Point A is "f(x)", therefore the coordinate of Point A is: "(x, f(x))". likewise, the distance on the y-axis from Point A to Point B is "h", so the y-coordinate for point B is: "f(x+h)". Therefore, the coordinate for Point B is: "(x+h, f(x+h))".

The distance (slope) from Point A to Point B is also known as the secant line. We plug in the x and y coordinates of both points into the slope formula. Once we plug it in, we can simplify by canceling out x in the denominator. This gives us the difference quotient.

BQ #6: Unit U: Continuities, Discontinuities, Limits, & Values

1. A continuity goes on forever, is predictable, can be drawn with a single long pencil stroke (without having to lift the pencil) and has no breaks, jumps, or holes. Because it is called continuous, that means that the graph is one straight line, like in slope-intercept form, which continues on in the same pattern.
A discontinuity is just the opposite of continuous in that it can have holes, jumps, and breaks and can't be drawn with one long pencil stroke. They're unpredictable and all over the place. There are two families of discontinuities: removable and non-removable.

In Removable, there is only one type: a point discontinuity which has a hole. Under Non-Removable, there are three types: jump, oscillating behavior, and infinite. A jump discontinuity is different from both the left and right side and can sometimes be broken. Oscillating behavior is really wiggly and resembles a seismograph. Because it gets so crazy in the middle, it's difficult to determine. Infinite occurs when there is a vertical asymptote which is caused by unbounded behavior, which means that it increases or decreases without bound towards (negative/positive) infinity.

2. A limit is the intended height of a function whereas value is the actual height reached. A graph can have infinite limits. However, a limit is reached when both the right side and left side of the graph reach the same destination, which, in that case, is also the value. To determine whether a limit is reached, we trace the graph until we know that the limit is or isn't reached. This is also how we graphically evaluate a function's limit. A limit is also different from a value in that it can't exist is when the left and right are different (jump discontinuity), when there's unbounded behavior (infinite discontinuity) and during oscillating behavior.

3. Limits can be evaluated numerically, graphically, and algebraically. When evaluating limits numerically, we use a table to help us see how close we get to the asymptote without ever actually touching it. As already mentioned, when evaluating graphically, we trace the left and right sides of the graph to see where we end, whether the left and right sides of the graph actually meet their destinations. There are several ways to evaluate a limit algebraically. It's important that the first thing we should do is apply the direct substitution method, which is basically plugging in the number to see if its works. If we get a numerical answer, or 0, or DNE, then we know that it works. If we happen to get 0/0 (indeterminate), then we have to try something else. Other methods to try if we get indeterminate are the dividing out/factoring method, rationalizing/conjugate method, and limits at infinity (long method).

Unit T Intro + Concepts 1-3

BQ #2:
Trig graphs relate to the Unit Circle because, if we unwrap the Unit Circle, we get our x-axis on which to plot the points and label the segments. The amount of space that the first quadrant takes up ion the Unit Circle is the amount of space it takes up on a straight line as the x-axis. The quadrant angles (90, 180, 270, and 360) are our distances and measurements. When graphing, though, we use radians instead of degrees because it is simpler on a graphing calculator, but the idea is still the same.

a) Period?: The period for sine and cosine is 2pi whereas the period for tangent and cotangent is pi because that's how long it takes to repeat the pattern. On a Unit Circle, sine is positive in the first and second quadrants and negative in the third and fourth quadrants. When the circle is stretched out in a line, that means that, when graphed, the sine would be positive for the first two segments (first two quadrants) and negative in the last two segments )last two quadrants). The pattern for sin is (+)(+)(-)(-), which extends to the whole Unit Circle to complete a pattern, which measures to 2pi.

The same goes for cosine, which is positive in the first and fourth quadrants. Therefore, on the x-axis, cosine would be positive on the first and fourth segments and be negative on the second and third segments. The pattern for cosine is (+)(-)(-)(+) which lasts a whole Unit Circle measurement and amounts to 2pi as well.

The pattern for tangent and cotangent is (+)(-)(+)(-) because it is positive in the first and third quadrants. Therefore, the pattern is shown to repeat itself already just within one revolution of the Unit Circle: (+)(-) and then (+)(-) again. One pattern starts in the first quadrant and repeats itself starting on the third quadrant. Because the pattern starts and completes itself halfway through the Unit Circle, the period for tangent and cotangent is just pi.

b) Amplitude?: Sine and cosine are the only trig functions with amplitudes because they can't have asymptotes. They will always have amplitudes of one because of the trig ratios for each. Sine as a trig ratio is y/r and cosine as a trig ratio is x/r. "r" is always 1. The restrictions for each can't go higher than +1 and can't go lower than -1.


BQ #5:
Trig graphs only get asymptotes when it is undefined, and a trig function is undefined only when divided by 0. Sine and cosine don't have asymptotes because they can't be divided by 0 and therefore can't be undefined. This is because the trig ratios for sin and cosine have "r" as the denominator: x/r (cos) and y/r (sin). "r" always equals 1 so "x" or "y" for sine and cosine can't be divided by 0 because they're always divided by 1 whereas the rest of the trig functions can be divided by 0 when either "x" or "y" equals 0.
Because the ratios for tangent and secant have "x" as the denominator, they will have asymptotes in the same general area: r/x (sec) and y/x (tan). Likewise for cosecant and cotangent: r/y (cos) and x/y (cot).


BQ #3:
Tangent?: The trig ratio for tangent is sin/cos. When graphed, sin and cosine are both positive in the first quadrant, so when applied to the trig ratio for tangent, sin/cos will actually translate into (+)/(+), which makes tangent in the first quadrant segment of the graph positive. We then use this method for the rest of the quadrants to figure out where tangent is positive or negative. We can find the asymptotes wherever cosine is 0, which is when it intersects with the x-axis.

Cotangent?: Because the trig ratios for cotangent are the opposite of tangent's trig ratio, the concept is basically the same. We use the same method of figuring out whether sine and cosine are positive or negative in a quadrant segment in order to determine whether cotangent is positive or negative as well. Because cotangent and tangent are reciprocals of each other, their trig graphs will be opposites of each other: where tangent goes up, cotangent goes down, and vice versa. The asymptotes for a cotangent graph will occur wherever sin(x)=0.

Secant?: Secant and cosine are reciprocals of each other also, so whenever cosine is positive, secant will also be positive, and whenever cosine is negative, secant will be negative as well. Wherever cosine equals 0, there will be an asymptote. In just one period, there is a half parabola facing upwards where there is a "mountain" and a downwards facing parabola where there is a "valley".

Cosecant?: Cosecant is the reciprocal if sine, so we use similar methods as the previous ones. Because sine is positive in the first two quadrants, cosecant will form a full parabola facing upwards in the first two quadrants. Because sine is negative in the third and fourth quadrants, cosecant will form a full parabola facing downwards in the last two quadrants of the period. Therefore, a whole period will actually include two parabolas: one going up and the other going down.

BQ #4:
Because tangent and cotangent are reciprocals of each other, they will be positive and negative in the same quadrants, yet their asymptotes will be different, allowing for them to be different when graphed. Tangent has asymptotes when x=0 whereas cotangent has asymptotes when y=0. Because tangent and cotangent don't have the same variable as 0, when graphed, they will be different, even though the same segments on the graph will be positive and negative.

Reflection #1: Unit Q: Verifying Trig Functions

1. What does it actually mean to verify a trig identity?
To verify a trig identity is to be given a an equation and solve the problem so that the left side of the problem eventually equals to the right side of the equal sign. This means that the right side of the equal sign will not be messed with, or touched, at all. The point of verifying a trig identity is to "verify" that the problem actually amounts to the given answer.

2. What tips and tricks have you found helpful?
I found knowing/memorizing the identities extremely helpful because they're necessary for substituting into trig functions in order to isolate what I want. Another important tip is to convert everything to sin and cosine because it just makes everything easier to work with.

3. 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.
When verifying a trig function, I make sure I never touch the right side of the equal sign. I try to see if I can convert anything to fit into the trig function(s) on the other side. The identities are extremely important for doing this. Complex problems with various trig identities within them have to be taken apart piece by piece in order to reduce the work and make it simpler.

SP 7# :Unit Q Concept 2: Finding ALL Trig Functions when given one trig function and quadrant (using identities)

Please see my SP7, made in collaboration with Gisela Barroso, by visiting their blog here. Also be sure to check out the other awesome posts on their blog. :D

I/D #3: Unit Q Concept 1: Pythagorean Identities

Inquiry Activity Summary:
We can refer back o our knowledge of the Unit Circle and the Pythagorean Theorem in order to find out the derivation of sin^2x+cos^2x=1. Identities, in math, are proven facts or formulas that are always true. For example, the Pythagorean Theorem is an identity because it is always true. The following image shows and explains how the Unit Circle and the Pythagorean Theorem go together to derive the ratio identity.

We can reverse the process and figure out how to derive Pythagorean identities by using a ratio identity. The following image shows that process using Cosecant and Cotangent:

Inquiry Activity Reflection:
The connections that I see between Units N, O, P, and Q so far are that there are constant references back to the Unit Circle and everything ties in together.
If I had to describe trigonometry in THREE words, they would be challenging, time-consuming, and connected.

WPP #13 &14: Unit P Concepts 6 & 7: Application with the Law of Sines & Applications with the Law of Cosines

Please see my WPP13-14, made in collaboration with Gisela Barroso, by visiting their blog here. Also be sure to check out the other awesome posts on their blog :D