y = x²

y = 6(x+4)² – 5

y = -3/2(x – 3)² + 1

y = 1/6(x – 7)² + 1

y = 1/65x² – 10

In each of these equations, the parameters a, h, and k play crucial roles in determining the transformations of the parabola y = x². The value of “a” controls the vertical stretch or compression and the direction of the parabola. When “a” is greater than 1, as in y = 6(x+4)² − 5, the parabola is vertically stretched, making it narrower than y=x². In contrast, when “a” is between 0 and 1, as seen in y = 1/6(x−7)² + 1, the parabolas are vertically compressed, appearing wider; this transformation is made more dramatic in y = 1/65x² – 10. If “a” is negative, as in y = −3/2(x−3)² + 1, the parabola opens downward, indicating a reflection across the x-axis. The values of “h” and “k” independently control the horizontal and vertical translations. The “h” value moves the parabola left or right; for example, in y = 6(x+4)²− 5, shifting the parabola 4 units left. The “k” value, like in y = −3/2(x−3)² + 1, where k = 1, moves the parabola vertically, in this case, 1 unit up. By examining the graphs of each equation, I could see how each transformation affected the width, direction, and position of each specific parabola, with the differences in stretch, compression, and translations visible between the graphs.

Self-Assessment

How did you represent the same mathematical idea in multiple ways in this assignment?
I represented the transformations of parabolas by using equations and corresponding graphs. The equations showed how changes in “a”, “h”, and “k” values transformed the shape and position of the parabolas, and the graphs provided a visual representation of these transformations.

State some of the relevant mathematical vocabulary words you used to demonstrate your understanding and their definitions. State where you found the definitions (your own memory, class notes, online?).
I used terms like “vertical stretch,” “compression,” “reflection,” “horizontal translation,” and “vertical translation.” These terms came from my memory and class notes, where we discussed how each part of the parabola equation affects the graph.

How did you use formatting to share your information in a clear and organized way?
I used a paragraph format with logical progression, introducing each parameter (a, h, and k) and explaining its effect on the graph. In addition, bullet points for the self-assessment kept my responses organized and easy to follow.