Lars Kitaen

The general shape of the graph of all quadratic functions is a parabola. The only exception is when the a is 0. Then the graph is a straight line, since we no longer have a quadratic whose highest power is 2, but a linear function whose highest power is 1.

Let's look at a random quadratic function to see what the graph looks like; then we will see how the b-value affects this graph. While changes in the a and c value also affect the graph, in this lesson we're focusing on how changes in the b-value alone affect the graph.

Let's look at the graph of f(x) = x^2 + 3x + 1, which is below. The b-value in this equation is 3.

We see that our graph is indeed a parabola. Our parabola is curving up. The x-value of the vertex, the tip of the parabola, is -3 / 2 or -1.5. We can actually calculate this x-value by evaluating the expression -b / 2a, where a and b are the values from the quadratic function. Our function has an a of 1 and a b of 3, so plugging these into the expression -b / 2a gives us -3 / 2 * 1 = -3 / 2 or -1.5, as expected. The point where the graph crosses the y-axis is given by our c-value. Our c is 1, and our graph crosses the y-axis at 1, as expected.

Now, what happens when we start changing the value of b? Let's see. We're going to keep our other values, a and c, constant while we play around with b to see what changes. Right now our a is positive, so let's see what happens to b when our a is positive.

Changing our b to 2, we get this kind of graph:

What has changed? It looks like our graph has shifted up and to the right. The x-value of our vertex is now at -1.

Okay, so our graph is shifting with the change in b; but what kind of overall shifting is occurring? Let's continue to play.

Let's change our b to 1.

Our vertex has moved to where x equals -1/2 or -0.5.

What about when b equals 0, -1, -2, and -3? Let's see:

Pretty interesting, isn't it? Our parabola continues to shift to the right as our b gets smaller and smaller. The vertex of our parabola also seems to be moving along a parabola of its own, with the tip happening when b is 0. Let's see how all the graphs look stacked on top of each other:

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