Unit V BQ #7

Where does the difference quotient come from?

PC: Kelsea; graph

In the quadratic graph above, there is a specific point labeled (x, f(x)). This specific point creates a tangent line. A Tangent line is a line that only touches the graph once. The second point, (x+h, f(x+h)) creates a secant line. A secant line is a line that touches two points on the graph. Now, you're probably wondering where we go the two points. Well, it's kind of simple even though it looks complicated on the picture right now.

So, let's begin by observing the point (x, f(x)). This point is the original one given to us. Since on the graph, the x value is stated, all we need to do to find the y value is plug it into the function, that is why it is f(x). Now, to the solid point on the x-axis, we call it h since we do not know the exact value of it. Now, how do we find the point on the graph above h? Well, for the x value of it, we know that if we add x+h, it will give us the x-value. Now, to find the y-value, all we do is plug in the x to the function, making it f(x+h). Yay! Now we have the two exact points for our secant line! Now, how do we find the equation for the tangent line of (x+h),f(x+h))? Well, in the picture below, we use the slope equation to help with that. It is worked out below. Now, we can see that this is where the difference quotient comes from! Yay! So, in short, the difference quotient comes from slope equation, that is solved but just didn't use actual numbers.

PC: Kelsea: equation

References:

Mrs. Kirch's videos

Photo's: Kelsea

Unit U BQ#6

1. What is continuity? What is discontinuity?

Continuity is when we are able to draw the graph without lifting the pencil from the paper. So that means the graph cannot have any jumps, holes, or break. It is also predictable. This means that we now it is approaching infinity on both sides, either positive or negative. Now what is discontinuity? Look above, and picture the exact opposite of what I just explained. Discontinuities can be draw by lifting the pencil. So there are jumps, holes, and breaks. It is also unpredictable, meaning we do not know where it is headed from the left and from the right at any given point. Now, there are four discontinuities. There are two families of discontinuity: removable and no-removable. Removable contains point discontinuity. Non-removable contains point discontinuity, oscillating behavior, and infinite discontinuity. 

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 the graph. That means the limit can be a hole on the graph, because it was in fact intended to go to that point on the graph. The limit exists only at removable discontinuities, which is specifically point discontinuity. It exists here only because the intended height is specific from both the left and the right of the graph. However, the limit does not exist at non-removable discontinuities, which are point discontinuities, oscillating behavior, and infinite discontinuities. It doesn't exist at point discontinuities because there are two separate intended heights coming from both the left and the right of the graph. It doesn't exist at oscillating behavior because the graph is so wiggly, that there is no specific point on the graph and in general, cannot make up it's mind on where it wants to go. Lastly, it doesn't exist at infinite discontinuity because it occurs when there is a vertical asymptote, which leads to unbounded behavior which means two separate directions from both the right and the left.  The limit is the intended height however the value is the actual height, which only includes shaded in holes and lines on the graph, not open circles. 

3. How do we evaluate limits numerically, graphically, and algebraically. 

We evaluate limits numerically by using tables. We put numbers that are approaching a certain number from the left and the right. That way we can see what is the intended height. For graphically we observe the graph and use onr finger on each side and see if they approach at the same point. If its a hole but they both meet then its a limit. If there are two seperate points, then there is no limit. For algebraically, there are three methods. The first method you should alwats try first is substitution. Thats when we directly substitute x and solce to find the answer. If the answer is 0/0 we use factoring to see if we can cross out any equation which means there will also be a hole. After factoring and crossing out, we then substitute the x and solve. If we cant factor though, we use conjugate if there are any Radicals to find anything that can cancel. And that's how we solve (:

BQ#3 – Unit T Concepts 1-3

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

Before we begin, it is important to note that the equation for sine is (y/r) and cosine is (x/r), which means they can never be undefined (on the unit circle) and therefore not be an asymptote.

Now, let's begin with tangent. Tangent, in terms of identities, has the equation of (sin/cos). In terms of graphing, we know that if cos is 0, it will be undefined and give the equation asymptotes, restrictions. The asymptotes help restrict the graph as well as how it will look. We know that tangent has a pattern, in terms of unit circle, of +-+-. So asymptotes help determine where the graph will lie. It will be easier to show you so look below for reference.

©Mrs. Kirch's Desmos

As we can see, the unit circle is divided into four and the tangent (orange line) pattern in the unit circle helps determine the pattern that the tangent has as well as what it will look like. Now, this relates to sine and cosine because In the first quadrant, all are positive and are above the x-axis. In the second quadrant, only sine is positive, so cosine and tangent are negative. In the third quadrant, only tangent is positive and is the only one above the x-axis. Lastly, tangent and sine are negative so sine is the only one above the x-axis. There are similarities between the two every from quadrant to quadrant.

Now as for cotangent? Same rules apply like tangent! Look below and you can see that the only difference is the direction because cotangent goes left to right and tangent goes right to left.

©Mrs. Kirck's Demos

Now as for secant? Well, lets look at the picture below. (Secant is the blue line.)

©Mrs. Kirch's Demos

Now we can see that there are asymptotes restricting the secant graph and making multiple ones. However, when we look, we can see a similarity between secant and another function. Can you see it? Whenever cosine is positive, the secant graph goes up. When the cosine graph is negative, the graph goes down. I know, I know, you've just been mind blown right now. But hold on, there's one more.

Look at cosecant below. (Cosecant is the blue line.)

©Mrs. Kirch's Desmos

Do you see a similarity despite the asymptotes restricting the cosecant graph? BY JOLLY YOU'VE GOT IT! Whenever the sine graph is negative, the cosecant graph is negative and whenever the sine graph is positive, the cosecant graph is positive.

BQ#5: Unit T Concepts 1-3

Why do sine and cosine NOT have asymptopes, but the other four trig graphs do? Use Unit Circle ratios to explain. 

First, let's find out how to get an asymptote in the first place. We get an asymptote when the equation's answer becomes undefined. Now, let's look at the equations for the trig functions. Now, below are the equations. We know that r will always be 1 because we are referring to the unit circle. Sine and Cosine have a denominator of r so it will be impossible to obtain an undefined answer when the denominator is 1. Cosecant and Cotangent have a denominator of y so if y is 0, then the answer would be undefined and would result in an asymptote. Actually, this would only work for degrees of 90 and 270.  For Secant and Tangent, the denominator is x so if x is 0, that will lead to an answer of undefined and an asymptote. This would only work for degrees of 0 and 180. 

©Kelsea DC

©Kelsea DC

References:

Mrs. Kirch's SSS Packet

Desmos Packet

BQ#2: Unit T Intro

How do the trig graphs relate to the unit circle?

A trig graph is actually a unit circle. Well, not the whole thing since the graph goes on forever, but from 0 to 2π, which is essentially the length around a whole unit circle.
To test this, we can use our trig functions to help. For example, lets use sine. For sine, it is positive in the first two quadrants and negative in the last two. Now, how is this related at all? Well, when we uncoil the unit circle and make a trig graph for sign, essentially what is quadrant one and quadrant two will be above the x-axis and will be positive. What would be the third and fourth quadrant would be below the x-axis since it is negative.
Make sense yet? Somewhat? Let's try another one! Let's use cosine.Cosine is positive in quadrant one and quadrant four and negative in quadrant 2 and 3. Now, let us once again imagine the unit circle unwinding and placing it on a graph What would have been quadrant one and quadrant four would be above the x-axis because it is positive for cosine in those quadrant. Now, the second and third quadrant will be below the x-axis since cosine is negative in those quadrants.
Making more sense? Need just one more example? Let's use tangent! Tangent, is positive in the first and third graph and negative in the second and fourth graph. Now, for the last time, uncoil that unit circle and think, how would our graph look like? ... Quadrant one and three would be above the x-axis because it's positive and quadrant 2 and 4 would be below the x-axis since it is negative.

©Kelsea DC

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

Lets refer back to what we reviewed above to help us. Also, note that a period is when a part of the graph reaches both above the x-axis and below the x-axis. As we know, sine, cosine, and tangent are positive in certain area. Sin: ++--; Cos: +--+; Tan:+-+-. These are the patterns for these. Now a period is how much it takes for something to repeat. When we look at sine, how long does it take to obtain a period of ++--? Well, it takes 2π and also because there are no repeating factors in the format and that how long it would take for it to repeat again. . For cosine, what kind of period would it take for +--+? Well, 2π again since there are no repeating factors and that's how long it takes in order for it to repeat again.  Now, tangent is +-+-. What's it's period? 2π? WRONG! It's π! Why? because in one period, the graph reaches both above the x-axis and below the x-axis. Need a visual? NO WORRY! Look Below!

©Kelsea DC
Green=Quadrant 1 Orange=Quadrant 2 Blue =Quadrant 4 Yellow=Quadrant 4

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

Sine and Cosine can only have amplitudes of 1/-1 because on the unit circle, they are restricted to it. Their denominator in their equation is 1. If we try to use a number larger or smaller than it and plug it into the calculator, it becomes an error. However, tangent has an equation of y/x so it isn't restricted because y and x aren't really a specified number like r is. Same applies to cotangent, only the equation is flipped. 

Reference:

Mrs. Kirch's SSS Packet 

Hand Drawn Photos By Moi 

Reflection#1: Unit Q: Verifying Trig Functions

1. To verify a trig function means to make sure that the two equations are equal to each other. You are unable to touch both sides at the same time so you have to use one side and from that, use identities to get the exact answer as it is on the other side of the equal sign.
2. I have found that remembering and at least knowing the unit circle is a big help because it can be used not only to check answers on one of the concepts, but it is very useful when finding degrees because there are many right triangles referenced in unit Q. Also, to remember which equation have sin and cos first because those are the most vital identities to being successful in this unit.
3. When looking at an identity, whether I need to verify or simplify, I look to see if all the trig functions have seomthing in common. Then I try to use reciprocal and ratio identities if there is nothing squared in the equation. However, if there is something squared in the equation, I reference the Pythagorean Identities to check if anything can be substituted. I often look for the least common denominator, taking out GCF, and factoring. Also, reciprocal and conjugates are my best friend in this unit!

Lastly, as a side note, I would like to mention that the easiest and best way to pass this unit is to simply become familiar with the identities. How you ask? CONSTANT PRACTICING! Becoming familiar with the identities with equations is one of the best ways to master it. And do what feels right. Do not think too hard about the problem. If you do think to hard and find it too difficult, you need to relax and take a breather. This unit is easy. It just takes some time to process all the information.

SP 7: Unit Q: Concept 2: Finding all trig functions when given one trig function and quadrant

Please see my SP7 made in collaboration with Eriq, by visiting their blog post here.  Also, be sure to check out our other collaborative blog post as well as his awesome individual ones! (: