Assignment # 2

Explorations with Second Degree Equations: Observing the Effects of A, B, and C on y = Ax

^{2 }+ Bx + Cby

Michael Ferra

Proposed InvestigationExamine graphs for the equation of a parabola, y = Ax

^{2 }+ Bx + C for different values of A, B, and C to observe the individual effects of A, B, and C on the equation. To do this we will:

i.Fix B = C = 0. Observe different values for A.

ii. Fix A = 1, C = 0. Observe different values for B.

iii. Fix A = 1, B = 0. Observe different values for C.

iv.Examine the relationship between the two standard forms of the graph of a parabola y = Ax^{2}+ Bx + C and y = A(x - H)^{2}+ K.

i. Fix B = C = 0. Observe different values for A.Let's vary the values of A from -3 to 3 for integer values when B = C = 0. Assume A ≠ 0.

What observations can we make from these graphs?

- Each graph shows a parabola that has a vertex located at the origin.
- The graphs where A > 0, the parabola is concave up.
- The graphs where A < 0, the parabola is concave down.
- The graphs are symmetric to the line of x through the vertex. In this case, the graphs are symmetric to the line x = 0 which is the y-axis.
- The width of the parabolas decreases as |A| increases. For example, the the graph of y = 3x
^{2}has the same width but as y = -3x^{2}except y = 3x^{2}is concave up and y = -3x^{2}is concave down. Thus these two equations are reflections of each other over the line of y through the vertex. In this case, the graphs are reflections to each each other when y = 0 which is the x-axis.For our viewing pleasure let's remove the axes and show our same equations except this time let's also show a graph where A = 0 thus y = 0x

^{2}= 0. Now we can see that y = -Ax^{2}is a reflection of y = Ax^{2}when y = 0.

ii. Fix A = 1, C = 0. Observe different values for B.Let's now vary the values of B from -3 to 3 for integer values when A = 1 and C = 0.

What observations can we make from these graphs?

- From our previous observations of A, we know that when A > 0 that our graph is concave up. Since A = 1, this holds true. Notice that changing the value for B does not alter this previous assumption.
- Notice another previous observation of A, that the width of the parabolas decreases as |A| increases. Since A is remaining constant at A=1, this aspect does not change, even when we change the value for B.
- Each graph shows a parabola that passes through the origin.
- The vertex of each parabola is different. From our observations of different values of A, we saw that each parabola had a vertex at the origin when B = C = 0. Notice the graph of y = x
^{2}that is labeled on this illustration. It has B = C = 0. Compare this graph to the others when A = 1, C = 0 and we vary B. The vertex, and the graph for that matter, appears to be shifted both horizontally and vertically. Finally observe that to find the x-coordinate of the vertex, we take the negative value of B and divide it by two. More simply put, the x-coordinate at the vertex is -B/2. The graphs maintain the same notion of being symmetric to the line of x through the vertex.

iii. Fix A = 1, B = 0. Observe different values for C.Let's now vary the values of C from -3 to 3 for integer values when A = 1 and B = 0.

What observations can we make from these graphs?

- From our previous observations of A, we know that when A > 0 that our graph is concave up. Since A = 1, this holds true. Notice that changing the value for C does not alter this previous assumption.
- Notice another previous observation of A, that the width of the parabolas decreases as |A| increases. Since A is remaining constant at A=1, this aspect does not change, even when we change the value for C.
- The vertex of each parabola is different. From our observations of different values of A, we saw that each parabola had a vertex at the origin when B = C = 0. Notice the graph of y = x
^{2}that is labeled on this illustration. It has B = C = 0. Compare this graph to the others when A = 1, B = 0 and we vary C. The parabolas seem to maintain the x-coordinate value at vertex which is 0, but has different y-values. The vertex, and the graph for that matter, appears to be shifted vertically by a value of C, thus the y-coordinate of the vertex is C. The graphs maintain the same notion of being symmetric to the line of x through the vertex which in this case is x = 0, the y-axis.

iv. Examine the relationship between the two standard forms of the graph of a parabola y = Ax^{2}+ Bx + C and y = A(x - H)^{2}+ K.Let's now show the relationship between the two standard forms of the graph of a parabola. Here we will change the form

toy = Ax^{2}+ Bx + C. This will give us a better understanding of the shifts in the parabolas.y = A(x - H)^{2}+ KIf the vertex is the point (H, K), then H gives the horizontal shift in the graph with respect to the origin and K gives us the vertical shift in the graph with respect to the origin.