The Significance of a, h, and k in My Equation:
Significance of “a”:
In my equation, “a” is represented as -1/2. Since the number is between 0 and -1, the parabola became wider than the parent equation. I know this value is between 0 and -1 because it is a fraction of “negative one half.” When “a” is in between the values of 0 and -1 or 0 and 1, the parabola curve expands wider. In addition, “a” is also a negative (-) number, causing the parabola to open downward on the graph. This is because whenever the “a” is a negative value, the parabola will open downwards and when it is a positive value the parabola will open upwards.
In this case, my given parabola equation consisted of “a” being in between 0 and -1 (also a negative number), causing the parabola to extend wider and open downwards.
Significance of “h”:
In my equation, “h” is + 7. It may be confusing when looking at the equation itself as it reads as +7, however, the “h” determines the x-value in the vertex which is actually -7. The vertex is (-7, -4). The +/- signs change when you plug -7 into the standard equation of y = a(x – h)^2 + k :
Plugging in -7 into standard formula: y = a(x – – 7)^2 + k –> y = a(x + 7)^2 + k
Demonstrated above shows how plugging in a negative number (-) with another negative (-) changes the value into a positive (+). This follows the rule of how two negatives equals a positive.
In addition, the “h” value also represent the horizontal shift of the parabola. In this case, for my given equation, the vertex of the parabola will shift 7 points to the left towards -7 from the origin of the parent vertex (0,0). This is because when the shift in a parabola equation is negative, it will shift to the left, and when it is positive, it will shift to the right.
Significance of “k”:
In my equation, “k” is represented as – 4. In any quadratic equation, the h value determines the y-value of the vertex in the parabola. In this case, my given equation shows that the parabola’s vertex is (-7, -4). The “k” also affects the vertical shift of a parabola. For instance, if it is positive (+) it will shift upwards along the y-axis, and if it is negative (-) it will shift downwards. In this case, my given parabola equation shows that the vertical shift will move down by 4 spaces on the graph.
What my given equation would look like without an “a” value:
y = (x +7)^2 – 4 :
The parabola is now opening upwards rather than downwards as it doesn’t have a negative “a” value.
What my given equation would look like without an “h” value:
y = -1/2(x)^2 – 4 :
Since there is no horizontal shift, the vertex’s x-value stays at 0. The parabola does not shift left or right. The vertex stays at (0, -4).
What my given parabola would look like without a “k” value:
Self-Assessment:
y = -1/2(x + 7)^2
Since there is no “k” value, the parabola will have no vertical shift along the y-axis. “K” resembles the y-value of the vertex so if there is no “k” the vertex will stay at (-4, 0).
- Give an example form this assignment where you represented the same mathematical idea in multiple ways.
While completing this assignment, I represented the same mathematical idea when demonstrating what my given equation would look like without the “a,” “h,” and “k” values separately in order to evaluate the differences and how each of these values affect the parabola in significant ways.
2. Give an example of this assignment where you used mathematical vocabulary to demonstrate your understanding.
During this assignment I was able to use mathematical vocabulary to demonstrate my understanding when explaining the purposes/significance/roles of each “a,” “h,” and “k” values. Specifically, I used mathematical vocabulary to explain the horizontal and vertical shifts of my given equation and they have shifted in that direction. In addition, I also used mathematical vocabulary when explaining why my “h” value/horizontal shift is represented as +7 when the x-value of the vertex (h) is actually -7. In order to explain why the +/- signs changed, I discussed how two negative numbers equal a positive, and I showed it by written formulas and bolding/colouring what numbers were being affected.
3. Give an example from this assignment where you used formatting to share the information in a clear and organized way.
In this assignment, I used formatting to share the information in a clear and organized way by visual images, typed equations, bolding and highlighting/colouring important texts, etc. As a visual leaner myself, I found it helpful and easier to use DESMOS and screenshots to show my thinking and understanding revolving parabolas. I was able to use these images, as well as bolding/highlighting text to explain the significance of a, h, and k in my equation and also what the parabola would look like without/if I changed one of these values.