Black Parabola; y=x²

Red Parabola: y=4(x+6)²+ 4

Blue Parabola: y=-2(x-3)²+ 5 

Green Parabola: y=-¹⁄4 (x+3)²-2

Purple Parabola: y=-10(x+8)²+ 1

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Parabola Explanation

By changing the k, h, and a values [in vertex form y=a(x-h)²+k], I can develop parabolas that vary from the parent function (y=x²) in numerous ways. Regarding my Red Parabola you can see that is shifted left and up, opens upwards, and is more compressed in comparison to the parent function. This is because of it’s k, h, and a values. The a value of this parabola is positive making it open upwards, it also makes it narrower and more compressed because the absolute value is greater then 1. The h value of this parabola is negative making it shift horizontally 6 to the left, h is also the axis of symmetry which is why the axis of symmetry of this parabola is -6. My k value of this equation is 4 and is positive which makes the parabola shift up, it also means that this parabola’s minimum value is 4. My blue parabola opens downwards because it has a negative a value. The maximum value of this parabola is 5 and it has a vertical shift downwards of 5 spaces. The vertex (x,y) is equivalent to (h,k) so the k value will always determine where the parabola starts and what its minimum or maximum value is. My green parabola appears to be wider in comparison to the parent function, this is due to the absolute value of a being less then 1. This parabola has a maximum value of -2 and has been shifted 3 spots to the left. My purple parabola is quite vertically stretched and narrow because my a value is a larger number. It is also shifted horizontally 8 to the right, and opens down with a maximum value of 1.

Self Assessment

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

Throughout the assignment I showed the same mathematical idea of transformations of parabolas in comparison to the parent function in multiple different ways. I showed different possible adjustments that can be made to parabolas using a graph and written equations. I also showed and explained how these adjustments are made and how certain parts of the equation effects your end result. Desmos allowed me create a graph and use different colours for different parabolas to visually show how my parabolas different from the parent function, rather then looking at a bunch of equations and trying to understand how they differentiate from each other.

What are some relevant mathematical vocabulary words you used to demonstrate your understanding?

Some words I used to display my understanding include; vertical and horizontal shift, compressed, vertically stretched, and vertex. I learned of the words vertical/horizontal shift through online research. A vertical shift occurs when a function is moved up or down on the graph and a horizontal shift moves the function left or right. Compressed is another word I go online when looking for a synonym to narrow. It means flattened by pressure; squeezed or pressed together Vertically stretched is a word I got from class notes. It means taller and narrower (in comparison to the parent function.) Vertex is a word I got from my own memory (we learned it when referring to points of a square). It means a point where two or more lines, edges, or faces meet.

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

I used formatting to help share my information in a clear and organized fashion in a few different ways. An example of this is using different colours to differentiate the parabolas. This makes it easy to view the parabolas and made it easier for me to explain them. Another example is I thickened the lines of my parabolas so they stand out and are easy to read. I also made the graph easier to read by having a step of 1 for x and y as well as removing the minor grid lines.