Graph blue: parent function y=x^2

Graph red: opens upwards and wider y=-1/8(x-2) ^2-5

Graph green: opens down and is narrower y=-8(x+2) ^2+10

Graph black: shifted to the right of the parent function y=1/2(x-8) ^2+1

Graph purple: shifted down lower to the left y=(x+8) ^2-4

Significance of A, h, k

There are various parts of a vertex form that decide the characteristics of the parabola.

In a parabola equation, “a” dictates the direction and width of the parabola. In the graphs, altering “a” changes the curve and the shape. For instance, in my red graph, where “a” = -1/8 the parabola widens opening upwards , for the parabola to open downwards the “a” value would be negative.

Other important parts of the vertex form equation are the h value and k value.

The “h” value determines the location of the start, shifting it horizontally on the x axis, this is called a horizontal shift. The “x” value of a vertex can also be defined as the axis of symmetry, so “h” value = axis of symmetry. If “h” were negative it would shift to the left and if it were positive, it would move the parabola to the right. This is because we change the sign of the number when we plug it into the equation y=a(x-h)^2+k. Like in the green parabola, the “h” value is a +2 so the vertex moves 2 units to the right, because the 2 becomes negative. An easier way to figure it out is to make the expression equal zero, like x+2=0 that results in x=-2. The “k” value determines where on the y axis the parabola starts opening. If “k” is positive the parabola shifts up, if negative, it’s shifted down. This is called the vertical shift. The y value of the start of the parabola means the y value of the vertex. It is the minimum/maximum value of the parabola that is found at a vertical shift. It represents the minimum and maximum value of the parabola depending on whether if the graph opens up or down. In my case the black parabola, the “h” value of 8 shows a rightward placement of the parabola, placing its vertex at (8,1) Meanwhile, the purple graph shows a downward shift, which to the “k” value of -4, moving the vertex to (-8, -4). In comparison, the parent function’s k value is 0, which is why it starts opening up at 0 on a higher point.

Self-assessment

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

I illustrated the mathematical concept by showing how changing the values of a, k, and h affects the position and shape of the parabola on the graph. I explained the importance of each variable in my equation and using a graphing tool (DESMOS) to visualize my function.

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

When discussing quadratic functions, I used several important mathematical terms to explain the concepts clearly.

“Horizontal shift” describes how the parabola moves left or right on the graph, based on the “h” value in the vertex form of the equation.

The “vertex” is the highest or lowest point of the parabola, which is key for deciding its maximum or minimum value.

 Lastly, a “parabola” is the U-shaped curve that is the quadratic equation, known for its symmetrical shape around the vertex.

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

I used a clear and structured format by adding titles to each section I discussed in this Edublog. I also added color into each heading to make them stand out and appear more organized.