Transforming Parabolas

Graph A (Red) : Quadratic parent function 𝑓(𝑥)= x^2

Graph B (Blue) : Opens up and is wider 𝑓(𝑥)= 1/2x^2-2

Graph C (Green) : Opens down and is more narrow 𝑓(𝑥)=-111(x+4)^2+2

Graph D (Purple) : Shifted to the right 𝑓(𝑥)= (x-8)^2+1

Graph E (Black): Shifted to the left 𝑓(𝑥)=15(x-5)^2-2

Significance of “h”, “a” and “k”

The “a”, “h” and “k” parts of my graphs are fundamentally important for the outcome of the parabola and are the defining features that determine a function. Each of these functions follow the equation 𝑓(𝑥)=a(x-h)^2+k. So, following these rules, you can see that “a” represents 1/2 in Graph B (Blue), and -111 in Graph C (Green). The coefficient of the brackets in the function, “a”, determines the compression and direction of the parabola. This means that the higher the number that “a” represents, the thinner the parabola, and when “a” is negative, it opens down. Graph A (Red) follows the quadratic parent function of 𝑓(𝑥)= x^2. This still follows the “𝑓(𝑥)=a(x-h)^2+k” formula, but implies that “a” represents 1, while “h” and “k” represent zero. Therefore, if “a” represents a number that is a fraction of one, the wider, or less compressed, the parabola will be. This is shown in Graph B when a wider parabola was needed. Other clear examples of “a”‘s purpose can be seen in Graph C (Green) or Graph E (Black). Graph C and E are compressed, and therefore “a” had to be a rational whole number that is greater than one. The purpose of the Graph C was to open downwards, which means that “a” has to be a negative number. The “k” is significant because it dictates what the value of “y” will be for the vertex of the parabola. In Graph D, the “k” was +1, and that is the value of “y” for its vertex, making it the minimum value of “y”. However, if the parabola were to open downwards, “k” would be the maximum value. In some of the graphs, “h” had a very important purpose. Subtracting a value from “x” before it is squared, shifts the parabola to the right, but if the value of “h” is negative, it shifts to the left. An example is how in Graph E, the value of “h” was -5, and made the value of “x” a +5 in the vertex.

Self Assessment

How did you represent the same mathematical idea in multiple ways in this assignment?

I represented the same mathematical idea in multiple ways by bringing up mathematical standpoint with function equations and what it means for the vertex. The vertex helps you visualize what the parabola may look like on the graph, and gives a more visual perspective. I provided a variety of different parabolas with a written explanation between the differences between them. This shows that parabolas can be narrow, wide, open up, open down, shift up, or shift down, all depending on the different ways the function is written. I used mathematical language to explain why I had put in the exact numbers in my function equations. In order for this to work, it was important to make sure that my graphs were well formatted and accurate. This makes the graphs easy to read and informative to the viewer.

State some of the relevant mathematical vocabulary words you used to demonstrate your understanding.

In order to demonstrate my understanding, I used of relevant mathematical vocabulary. For example, in Graph B, I referred to the value of “a” as a fraction, because it represented the number 1/2. When talking about “k” I mentioned how it could tell you the maximum or minimum value of “y”, which is mathematical language. I also mentioned how “a” needed to be a rational whole number, greater than one, to make the parabola more narrow. In my paragraph, I elaborated on how “k” determined the value of y in the vertex as well.

How did you use formatting to share your information in a clear and organized way?

I used formatting to share organized information in a myriad of ways. The definition of clear and organized is for something to be legible, and easy to understand. Based on this, the main goal of mine was to make sure that there is limited overlap between graphs, and that each label was accurate, and easy to read. Graph A was the only graph I could not leave to my own recreation, so I put that one in first. Graph B was supposed to be wider, and so I experimented with putting it on each side of the parent function. However, due to it’s width, I decided to put it underneath Graph A to save space, and make things look more clean. Since Graph C was thin, I knew that I could put it on either side, and so I put it on the left. I decided to put graph D farther away, on the right, so that it wouldn’t overlap to much with the other graphs. This also gave the perfect amount of space for Graph E, that I chose to make thinner for convenience.