Student Problems

STUDENT PROBLEM#3 NOT ONE 

• What is the problem about?

This photo covers a real life problem of Unit I Concept  1 which deals with the graphing of exponential graphs. In this concept we deal with the equation y=a(b)^x-h+k and this equation is the basis of solving to find the rest of the elements in the problem. Once again, this concept deals with previous concepts that have come back to actually play another role in solving this type of problem such as knowing how to find the x and y intercepts involving logs and ln, how to find the domain and range of the equation, and how to label the numbers correctly with the right letter by remembering the equation used for this section. In this problem we also deal with figuring out the horizontal asymptote since it is an exponential graph and it has specific rules for the domain and range. 

• What must the reader pay close attention to in order not to make a mistake? 

In order for the reader not to make a mistake when trying to graph an exponential graph is that you need to make sure that you memorize the equation y=a(b)^x-h +k. This is the basis of this problem because you need to know which numbers plug into the variables or else you may get the incorrect answer. You also need to make sure you know how to solve for the x intercept by setting y=0 and how to find the y intercept by setting x=0. Since this problem is an exponential graph you also must remember that these graphs have no restriction on domain but the range is restricted. The range depends on the asymptote and also on the number that a ends up being in your equation. The range is also depended on the variable a because if it is positive you have to make sure the range is (the #, inf.), and if (A) is negative it will be (-inf., #). 

Student Problem #4 Unit I Concept 2

• What is this problem about?

This problem goes over a real life example of a problem from Unit I Concept 2. Concept 2 deals with an example that is a logarithmic graph and by identifying it as that, we know that x=k and there are no restrictions on the range. Because it is a logarithmic graph we have a new equation in which we use: log base b (x-h)+k. Also, for this type of graph it is only important that we understand that the domain is dependent upon the asymptote. 

• What does the viewer need to pay special attention to in order to understand the concept?

The viewer needs to make sure to pay attention on how to find the asymptote, instead of it being y=k it is now x=h for the asymptote so you need to make sure to know what h equals by looking back at the formula given. Yo must also play close attention to how you solve for both the x and y intercept because since we are now dealing with logs and natural log then we need to be aware that we know how to get rid of them, so that we can simplify further. Another important fact that we must remember and pay attention to is how to write the domain and range. For this type of logarithmic graph we should remmeber that there are no restrictions on range and for the domain will include the asymptote and an infinity sign. 

Student Problem #5 Unit J Concept 5 

• What is this problem about?

This problem goes over a problem from Unit J Concept 5. Unit J Concept 5 covers a lesson on Partial Fraction decomposition with distinct factors. This concept is much easier than it seems. There are 7 steps in which you must take to complete this problem and they are as follows: Factor out the denominator, separate each of the factors into their own fractions and put the letters A,B,C, etc. on top, get a least common denominator for the factors by multiplying it by what its missing, combine like terms, set the coefficients of the numerator equal to like term letters on the right, set up the system and solve, and finally put A, B,C, ect. back to where there variable is at in the spit up fractions in the beginning. 

• What does the viewer need to pay special attention to in order to understand the concept?

The viewer needs to pay special attention not to forget to put zeroes in the numerator if there is just an x, or if something is missing. This is important because when setting up your system of equations you may forget and get confused of what number goes on the other side of the equal sign. Also, you must pay attention to how you take out the x's once writing your system. You need to be sure to take the x's out so that you can combine terms an solve for A or B, it is not necessary to keep those x's in the equations at this point. 

Student Problem #6 Unit J Concept 6

• What is this problem about?

This problem covers an example from Unit J Concept 6. This concept is very similar to Concept 5 except now we are dealing with Partial Fraction Decomposition with repeated factors. In this type of problem when you try in factor the denominator out you will find that one of the factors is repeated. Meaning that now when you separate the factor into separate equations for the repeated factor you must include the factor as a denominator as many times as the exponent. For example if the denominator as shown is (x-1)^3 you must have one fraction be A/(x-1), the nest one be B/(x-1)^2, and the last one be C/(x-1)^3. This is the only difference, then you would just solve as you did in concept 5. 

• What does the viewer need to pay special attention to in order to understand the concept?

In this concept the viewer must pay special attention to the common denominator so that when multiplying the numerator by what a part of the fraction is missing then your system comes out to being correct. Also, you must pay attention to how to right the denominators when separating the fractions. You must also pay close attention making sure your answers are plugged in correctly in the end. 

• What is this problem about?

This picture covers a problem from Unit K Concept 10. This concept deals with writing repeating decimals as a rational number using geometric series. In this type of problem what you are basically trying to do is break down the decimal to discover the rational number or the sum of the decimal. After breaking down the decimal into repeating units you then are able to figure out the first term of the series and then the common ratio of the series which allows us to then plug it into the equation that has the sigma with the infinity sign. Then, you plug in a sub 1 and r into the summation notation formula in order to simplify and figure out the rational number. 

• What does the viewer need to pay special attention to in order to understand the concept?

In order to understand the concept the viewer needs to pay special attention to how the decimal is broken down. What we do is basically take the repeating decimal and set it as its own. Then we out an addition sign which then allows us to break down the decimal by putting zeroes in front of the repeating decimal. The point of this is to make sure that you get the correct value for a sub 1. Also, when you have wrote out the series it is important that you write the broken down decimal correctly so that when we divide one term to the other, we and up with the correct common ratio (r).

Unit R Concept 1

• What is this problem about?

This picture goes over a problem from Unit R concept 1, dealing with finding the exact values of sums or differences. This is also an example of finding exact values for angles that aren't on the unit circle, but rather now we are able to use our knowledge from the angles given on the unit circle to solve for other angles that are not put on the circle. By using the sum and difference identities we are able to solve for these non apparent unit circle angles and use the other angles given on the unit circle to help figure out the exact values. We basically are trying to figure out which angles on the unit circle can add or subtract to the angle in which we are solving for, and in this concept we can see that there are various ways in which we can go about solving these problems. 

• What does the viewer need to pay special attention to in order to do the problem correctly?

In order to solve the problem correctly the viewer needs to make sure that he/she is very familiar with the unit circle. Due to the knowledge we take away from the unit circle it is imperative that we know the ordered pairs for the magic five and are familiar with identifying sin, cos, and tangent using the ordered pair that corresponds with the angles used. It is also important that we are all familiar with SOHCAHTOA and what that stands for in order to identify the trig functions for each of the angles used. Also, very important is that we need to focus on memorizing the sum and difference formulas so that we don't have to keep referring back to the notes. The last crucial thing one must do to solve correctly is checking your answers and also making sure that it is all plugged in correctly. 

What is this problem about? 
This problem goes over one from Unit R Concept 2 that is dealing with using sum and difference formulas when given values of right triangles. We have basically put together all of our knowledge from the unit circle, right triangles, SOHCAHTOA, and sum/difference formulas in order to solve a problem for this concept. This problem is mainly dealing with recognizing/identifying the trig functions with the limited information given of the triangle so that later we are able to use this information to solve for the exact values using the new sum and difference formulas we have. We then use substitution  to plug in the values into the equations, which leads us to finally simplify the fraction and get our final answer. 

What does the viewer need to pay special attention to in order to do the problem correctly?
When solving a problem like this from unit R the viewer needs to pay special attention to how we are able to identify the values of each trig. function from the two separate triangles. It is important that we do so becuase if you identify one trig function incorrectly that can result in your whole answer being wrong. The viewer also needs to pay special attention to the equations used to solve for the exact values of the trig. functions. Also, it is imperative that one is very careful when solving and simplifying, making it even more necessary to check your answer afterwards. It is also importnat that the viewer realize the difference in the quadrants in which these trig. functions fall in due to their signs (+/-).

What is this problem about?
This problem goes over an example from Unit R Concept 3 which deals with trigonometric function of an inverse trig. function. When solving this we also have to use the sum and difference formulas to help prove that these values for the functions are true. This problem also makes use of previous units such as the unit circle to figure out the actual ordered pair for the trig functions used. Also in this concept our knowledge of the trig functions is able to be put to use such as knowing that sin is equal to the y value of the point and cosine is the x-value of the ordered pair used. 

What does the viewer need to pay close attention to in order to do the problem correctly?
In order to solve this problem correctly the viewer must pay close attention to how we are able to derive u and v from the original equation give; canceling out the inverse to just get the value on one side. This is the most important part because we then need to be careful when identifying our ordered pair that is used. Another spot where kids make mistakes is making sure that they use the correct equation to plug the values into. It is important that the viewer is familiar with the sum and difference formulas to pick the correct one when substituting and simplifying.