Assignment # 2

Explorations with Second Degree Equations: Observing the Effects of A, B, and C on y = Ax2 + Bx + C

by

Michael Ferra

Proposed Investigation

Examine graphs for the equation of a parabola, y = Ax2 + 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 = Ax2 + 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 = 3x2 has the same width but as y = -3x2 except y = 3x2 is concave up and y = -3x2 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 = 0x2 = 0. Now we can see that y = -Ax2 is a reflection of y = Ax2 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 = x2 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 = x2 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 = Ax2 + 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 y = Ax2 + Bx + C to y = A(x - H)2 + K. This will give us a better understanding of the shifts in the parabolas.

If 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.

If you have the Geometer's Sketchpad program then click here to make your own explorations with different values of A, B and C!