Math 4 Unit 4: Rational Functions

Graphing Rational Functions

Identify the holes, intercepts, HA, VA, and SA, then graph.

Multiplying Expressions

Multiply and simplify.

Dividing Expressions

Divide and simplify.

Adding Expressions

Add and simplify.

Subtracting Expressions

Subtract and simplify.

Solving Equations

Solving Inequalities

Solve the inequality.

Math 4 Unit 3: Polynomials

Factoring Quadratics

Completely factor x^2+6x+8

= (x+4)(x+2)

Factoring Quadratics II

Completely factor 3x^2+4x-7

= (3x+7)(x-1)

Factoring Quadratics III

Completely factor

= x^2(x-7) – 3(x-7)

= (x^2-3)(x-7)

Factoring Quartics

Completely factor 2x^4-28x^2+90

= 2(x^4-14x^2+45)

= 2(x^2-9)(x^2-5)

= 2(x+3)(x-3)(x^2-5)

Factoring/Solving Difference of Cubes

Solving Quartics

Completely factor.

= 2x^4-28x^2+90

= 2(x^4-14x^2+45)

= 2(x^2-9)(x^2-5)

= 2(x+3)(x-3)(x^2-5)

x=-3, 3, +-sqrt 5

Vertex Form

Rewrite in vertex form.

Graphing Polynomials

  • Determine the End Behavior based on the Leading Coefficient Test and write the end behavior of this polynomial in Limit Notation.
  • Find the zeros of the polynomial function
  • Find the y-intercept of the polynomial function
  • Graph the polynomial function.

Write a Polynomial with Roots

Find a polynomial function with the given zeros: 0, 2 mult 2

Long Division

Divide with Long Division

Synthetic Division

Divide with Synthetic Division

Rational Roots Theorem

Use the Rational Roots Theorem to find all possible roots and then solve.

Math 4 Unit 2: Functions and Their Graphs


Domain – the span of all the x-values of a function

Range – the span of all the y-values of a function

x-intercept(s) – the point(s) where a function touches the x-axis

y-intercept – the point where a function touches the y-axis

Horizontal asymptote(s) – the imaginary line(s) where the domain of a rational function approaches on both sides but never touches.

  • There are three rules that define where the asymptote will be and how it will look like:
    • If the degree of the numerator is less than the degree of the denominator, the asymptote will, by default, be y = 0.
      • The asymptote will be higher or lower if there is a k value in the function.
    • If the degree of the numerator is equal to the degree of the denominator, the asymptote will, by default, be at y = (degree of the numerator/degree of the denominator)
      • The asymptote will be higher or lower if there is a k value in the function.
    • If the degree of the numerator is greater than the degree of the denominator, there will not be a horizontal asymptote.

Vertical asymptote – the imaginary line(s) where the range of a rational function approaches above and below but never touches.

Local minimum – the point where a positive even polynomial’s vertex is, where the function does not go any lower.

Local maximum – the point where a negative even polynomial’s vertex is, where the function does not go any higher.

End behavior – the direction, attributes, and/or limit of a function as it goes left or right.



Rational Function

Radical Function

Rational Function

Radical Function


The height of a passenger on an escalator is modeled as

where t is the time in seconds after the passenger has boarded the escalator on the second floor and h is the height in feet above the first floor.

This means it takes the passenger 16 seconds to reach the first floor (0 feet).

This means the passenger is 12 feet above the first floor when he/she first gets on the escalator. Likewise, this also means that the second floor is 12 feet above the first floor.

Inverse Functions

Find the inverse of the following function:

Composition of Functions

Find f(g(x)) and its domain if:


Decomposition of Functions

Find what h(x) decomposes into if:


Math 4 Unit 1: Sequences and Series

Recursive Sequence

A sequence expressed in the form of an operation that is performed on the previous term of the sequence to find the next. To use a recursive sequence, the previous term must be given to calculate any following terms.

Example: Find the first 4 terms of the recursive sequence below.

Explicit Sequence

A sequence expressed in the form of an equation in terms of n, where n is the term’s position in the sequence. This method of expressing a sequence does not require any previous terms to find following terms; it only requires you have the position of the term (n) you are looking for and the first term of the sequence.

Example: Find the first 4 terms of the explicit sequence below.

Arithmetic Sequence

Example: Find the 4th term of an Arithmetic sequence with a first term of 7 and a common difference of -4.

Arithmetic Series

Example: Find the sum of the first 10 terms of an Arithmetic sequence with a first term of 7 and a common difference of -4.

Geometric Sequence

Example: Find the 6th term of a Geometric sequence with a first term of 2 (a1) and a common ratio of 3 (r).

Geometric Series

Example: Find the 6th term of a Geometric series with a first term of 2 (a1) and a common ratio of 3 (r).

Application Problem

A cybersecurity technician has a starting salary $106,000. Per his contract with the company, he is given a 5% raise each year. If he works for 20 years, how much will he have earned total by the end?

a1 = 106,000 because the salary starts at $106,000

r = 105% or 1.05 because the salary increases by 5% each year

n = 20 because the technician is working 20 years

Infinite Sequences

Find the sum of the following sequence, if there is one:

1/20, 1/40, 1/80…

Convergent & Divergent Series

A convergent series has an r value with an absolute value less than 1, meaning the sequence will get closer and closer to 0 as it progresses. This means you can calculate the sum of the series to the infinite term, as there will be a finite sum.

For example, the geometric series in the previous question has an r value of 1/2 (|1/2|<1), because its terms are getting closer and closer to 0 (in this case, smaller and smaller)

On the other hand, a divergent series has an r value with an absolute value greater than 1, meaning the sequence will get farther and farther away from 0 as it progresses. Because of this, you cannot find the sum to an infinite term as the sequence has no limit to how large its terms it can get.

For example, this geometric series has an r value of 2 (|2|>1), which means the terms will get larger and larger, farther and farther away from 0, meaning there is no real way to find an infinite total of the series since there is no limit to how high it will go.

Summation Notation

Find the sum using the correct sequence formula.

Binomial Theorem (all terms)

Expand completely.

Binomial Theorem (single term)

Find the 3rd term in expansion of

Reality Series Video Project

Math 3 Unit 9 – Percentiles

Part 1

I think the most important thing i learned was what percentiles were and how to find/use them. I always heard about these, like the weight percentiles or height percentiles the doctor gives you at checkups, but never really understood what they meant. Now, I can understand what these stats mean in terms of the data.

Since percentiles are used so often to help explain data points and such, they will be important moving on. One thing that helped me understand were the notes we captured on what they were. They explained to me how a percentile was a measurement or indication of how many data points in a given set are lower than the specific data point you are looking at. This gives a sign of how extreme that data point is, and can be used to calculate whether that point is statistically significant, a very important skill that we will use a lot now and later.

Part 2 – Designing an Experiment Project

I specifically learned what statistically significant meant. Before, I would just look that the means of data and if one was higher, or higher enough, then I would conclude there was a significant difference. I understand now that you need to run a large amount of randomization tests and compare if the original mean is within the most extreme 5%.

For this project, we first had to come up with what we wanted to test. This was honestly the hardest part, because we really had to idea what to do. After quite a bit of consideration, and some inspiration from some classmates, we decided on the color influence experiment. Next, we decided how we wanted to test it. We choose to use a Google Form so it would be easy to distribute them to participants and collect data in one place.

One part of the project that did turn out very well was our data collection design. The idea of the Google form was very easy to use to get participants. We didn’t find people at lunch and use their time to do it, so we got a fairly large amount of respondents. We could post announcements on our messaging platforms and find quite a few people there.

One thing that didn’t go well was the presentation. My partner and I did not have communications over the weekend as she did not have internet, and she didn’t know what we had to present on Monday, so we barely had time to prepare. We had a bit of miscommunication during the presentation, but it wasn’t too bad. It just wasn’t very smooth. Next time, I would make sure we had reliable communication to avoid something like this from happening again.

Math 3 Unit 8 – Unit Circle

Part 1

The most important thing I learned in this unit was the unit circle. It will be important in the future as it provides a way to find sin, cos, and tan values without a calculator, along with their degrees and side lengths. When we go to NC State, where we cannot use calculators, we will need to rely on this to do our calculations. Even when we do have access to calculators, however, this tool will still be a very convenient tool to evaluate trigonometry.

To learn about the unit circle, we constructed one ourselves. Using our new knowledge about radians and trig rations, we put down all of the measurements, angles, and numbers. This showed us exactly how the unit circle worked. Now, we can construct one ourselves using the methods we learned without a calculator or notes.

Part 2 – Cycles in Nature Project

I learned that the hours of daylight throughout the year actually follow an exact cycle. Growing up, hours of daylight always affected my family, since my dad had vision issues and could not drive in the dark. Learning exactly when and how the time of day changed was a very educational experience. I never knew what the seasonal equinoxes were before.

To model the data for our project, we had to manually find each day of the year that were the 1/8th milestones of 365 days. Then, for those 8 dates plus the 4 equinoxes, we had to, again, manually search how many daylight hours each day had. It was a very tedious process and my partner and I very nearly messed some of our data. Then, we had to figure out how to write our model equation, which was difficult because we were confusing our radians and degrees. This whole process was the hardest part of the project. In the end, however, it mostly worked out; we just had to focus our effort, time, and concentration onto this part of the project. One thing that helped was having one teammate read off numbers and have the other put them in, so the latter would not have to keep switching around and risk entering incorrect data. It was also much easier and faster than having only one person do it at a time. Everything else, like the poster, was not really difficult compared to that.

One good thing about this project was our poster. This was the first time I ever developed a poster theme from scratch (though it was inspired by a infographic design and has similar elements). It has a very clean look, with effective use of background images. My grade level poster also used the same theme, and I worked on both posters alongside one another.

One thing I would do differently next time is to put all of our information on the poster. When we made the poster, we assumed that some of the rubric’s requirements could be left on our project documents, since it did not explicitly state that everything had to be on our poster. As a result, we missed many points.

Math 3 Unit 7 – Triangles

Part 1

The most important thing I learned this unit was the centers of triangles. There are 3 types that can easily be mistaken for each other; circumcenter, incenter, and centroid. The additional page that was included in our task helped visualize (I am a visual learner) just where each point was and how they were positioned in relation to the sides and vertices. The circumcenter is the point where all the perpendicular bisectors of the sides meet. The incenter is where all the angle bisectors of the vertices meet. The centroid is where all the lines connected between a vertex and the midpoint of the opposite side meet. This will be important in the future as we will need to find points that need to be equidistant from different sides or angles, and/or divide areas.

Part 2 – Castlerigg Stone Circle Project

I realized during this project that volume of non-standard 3D shapes is often very difficult to manage. Given a 10 oz. requirement, we first had to convert that into a cubic volume requirement, then guess and check several dimensions to finally find a size that was functional. All the heights, lengths, bases, slant lines, etc. that we had to manage all at the same time was very difficult.

One specific detail I learned was how to calculate the volume of a milk carton. The bottom part was easy; a simple rectangular prism, but the top triangle part was confusing. This led to a miscalculation in volume. At first, I calculated a volume with over 40 cubic centimeters over what we needed, which left us plenty of space. Then I realized that the triangle was not a triangular prism and could not be counted as one; after a bit of thinking, I then realized that it was actually a pyramid! the slanted faces of the carton and the slanted triangle on the sides formed a square-based pyramid with the height of the triangular prism. Even after that, I realized that the diagonal folds were reducing the height, and accounting for that I found the true volume to be exactly our initial goal, with 0.1 ounces to spare. I was very lucky that I made the design larger than I thought I needed, because in the end, it was exactly what we were required to make.

That was the greatest challenge of the project: handling all the numbers. I am prone to random mistakes, and in calculating the numerous volumes and areas I made several errors. In the end, I had to go back and fix them all last minute. Next time, I will have another team member double check all my numbers before I move forward, as to prevent another total do-over again.

But in the end, our project was a success in what it outputted. Our milk carton was very solid, functional, and good looking. Our presentation was effective in presenting all our information in a concise, non-overwhelming manner. I was able to use my design not just in a slideshow, but on a physical product.

Math 3 Unit 6 – Equations of Circles

Part 1

I believe the most important thing I learned in this unit is how to write the equation of a circle. This will be useful as I have never had a method to plot a circle onto a coordinate plane, allowing comparisons to lines and other equations on a physical space.

The activity that definitely explained how the equation of a circle worked is the circle we constructed using the right triangles. The moment I visualized how the paper would look like with all the triangles on it, I understood how the concept worked. This was a very creative way to show this idea.

Part 2 – Castlerigg Stone Circle Project

An interesting fact I learned about Castlerigg (and one of the reasons I choose this circle) is that this stone circle predates infamous Stonehenge by 2 centuries. As for mathematics, I learned how to convert degrees to physical distance. Once Mrs. Parker explained how the minutes and seconds worked, I finally understood why they were called minutes and seconds; they were on a 60 scale.

But the hardest part of this project was the video; it required dozens of videos and images to compile into a 6 minute video. Then after hours of work on putting the video together, I needed to cite all 38 links. But with planning and time management we were able to complete all we needed on schedule, having eliminated other tasks to allow time to finish it all. But next time I would start grabbing plenty of images and videos from the very beginning and make sure we had all the content first; otherwise, we will have to re-render the video again and again with all the changes we had to make.

One thing that did turn out very well though was our poster; with experience from both my partner and I we were able to create a very attractive and clean poster design that accurately captured the theme of the stone circle.

Math 3 Unit 5 – Irrational Functions

Part 1

I think the most important thing I learned this unit was how to sketch irrational graphs without a graphing calculator or Desmos. This is important because being able to do math without technology is an important skill. On some tests, sometimes calculators are not allowed and being able to still make a rough graph of a complicated irrational equation.

The table in the beginning of Task 4.3 was a very helpful way to help me understand how to find the features of an irrational graph could be found. Given an equation, I could look at the characteristics, compare it to the chart, and get instructions to easily find what features could help me graph it.

Part 2 – Technology Manufacturing Project

After researching the Tech Manufacturing industry, I found it interesting how there were so many companies developing different things, with different strategies. Some always did the same thing over and over again, like Dell which always sells PCs. Others were always trying new things in an attempt to always keep up with the latest developments, like IBM. Others do a mix of both, like my company, Intel, which both continuously does computer chips, improving them by doing the same thing over and over again in new ways, as well as joining the fray in new fields like drones.

About financial investments, I learned that math isn’t always a great way to predict things that have so many factors, including a lot of human influence. Using Desmos to create a graph using a couple of data points isn’t going to produce a result that will mathematically, and more importantly, accurately, predict the success of a company which is run by human CEOs and makes money from the work of human employees selling to human consumers. Math cannot predict the stock market. That’s why humans are the ones who are looking at data as well as numerous other factors in the real world that math cannot take in account for. For them, math is a tool, not the answer.

The hardest part of this project was spending so much time looking and manipulating data. We had to navigate databases and then manually copy numbers to our documents, then hand type them into Desmos, just to get ridiculously inaccurate graphs. I understand why accountants and people in the financial industry get paid so much now.

One part of this project that turned out well was we were able to make a solid conclusion on which company we should invest in. We didn’t have to struggle to decide, but that might be because we only had a few data sources.

Next time, I would allot more time to do the data work. It took an extremely long time, and if I did this again I would plan to spend longer on that part.