Assignment 2:
Exploring Quadratics in Vertex Form

by
Jenny Johnson

Quadratic functions are usually recognized in the standard form y = ax² + bx + c. This is a quadratic equation when y = 0, so when 0 = ax² + bx + c. The purpose of this exploration is to examine the graphs of quadratic functions when they are in the form y = (x – d)² + f. The graphs of quadratic functions in both of these forms are in the shape of a parabola.

What happens to the parabola as d varies in the function y = (x – d)² + f ?

If we keep f = 1, we can explore the effects that d has on the parabola. First, what happens when d = 1?

The parabola is concave up with a vertex of (1, 1).

The following illustration includes six different graphs with d values ranging from -1 to 4.

As the parabolas show, when d = -1, the vertex is (-1, 1). When d = 0, the vertex of the parabola is (0, 1). When d = 1, the vertex of the parabola is (1, 1). When d = 2, the vertex of the parabola is (2, 1). When d = 3, the vertex of the parabola is (3, 1). When d = 4, the vertex of the parabola is (4, 1). Clearly, d is always the x-value of the vertex of the parabola. Varying d does not change the shape of the graph, only the position with respect to the x-axis. The following movie shows the parabola as d ranges from -5 to 5.

What happens to the parabola as f varies in the function y = (x – d)² + f ?

If we could d constant at d = 1, we can explore the effects f has on the graph. The following illustration includes the parabolas as f ranges from -1 to 4.

As the parabolas show, when f = -1, the vertex is (1, -1). When f = 0, the vertex of the parabola is (1, 0). When f = 1, the vertex of the parabola is (1, 1). When f = 2, the vertex of the parabola is (1, 2). When f = 3, the vertex of the parabola is (1, 3). When f = 4, the vertex of the parabola is (1, 4). Clearly, f is always the y-value of the vertex of the parabola. Varying f does not change the shape of the graph, only the position with respect to the y-axis. The following movie shows the parabola as f ranges from -5 to 5.

We can conclude from our explorations that when the quadratic is in the form y = (x – d)² + f, the vertex of the parabola will be (d, f). To illustrate this concept, we can plot the points (d, f) on our graphs using a parametric equations for a point, as is shown in the picture below (when d = 1 and f = 1).

The following movie shows the point (d, f) and the parabola as d ranges from -5 to 5.

As we see in the movie, the point (d, f) is always the vertex of the parabola. Similarly, we can view a movie that shows the point (d, f) and the parabola as f ranges from -5 to 5.

We also see in this movie that the point (d, f) is always the vertex of the parabola.

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