In this lesson, learn about how the b-value in a quadratic equation affects the location of the parabola. Also learn how the other letters affect the parabola in conjunction with the b-value.

## Definition of the *B*-Value

Before we get into the meat of the lesson, let’s go over just a couple of definitions.The **quadratic function** is *f(x) = a * x^2 + b * x + c*. The **b-value** is the middle number, the number next to the *x*.

The other letters, *a* and *c*, are also numbers like *b*. Each of these can be any number. In combination, they tell you what the quadratic function will look like when graphed.

## The Quadratic Parabola

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 + 3*x* + 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.

## How B Affects the Parabola

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:

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:

I’ve drawn the apparent parabola the graphs make as the *b* changes (the bright pink colored curve).

Another interesting thing can be seen here as well. Do you see the one point that all these graphs have in common? Yes, they all share the same point where they cross the *y*-axis. We can say that when *a* is positive, as *b* increases, the graph moves from left to right, following a downward-opening parabola whose tip occurs at (0, *c*). Our *y*-intercept is given by our *c*-value.

Now, what about when *a* is negative? What happens to the graph then when our *b* changes? Let’s see again by changing our *b* values. We will use the same numbers again:

Stacked together, all these graphs look like this:

Well, it looks like our *b* is still shifting our graph to the right as it decreases, but now instead of following a parabola that opens down, it’s following a parabola that’s opening up. I also see that our *y*-intercept is still the same for all of my graphs.

What can we say? We can say that when *a* is negative, as *b* decreases, our parabola moves to the right, following an upwards-opening parabola whose tip occurs at (0, *c*).

## Lesson Summary

We’ve learned that in a **quadratic function**, *f(x) = a * x^2 + b * x + c*, the **b-value** is the middle value, the one multiplied by the *x*. The general graph of a quadratic is a parabola. The only exception is when *a* is 0, in which case we get a straight line.

When we keep *a* and *c* constant, we can summarize how changes in *b* affect the parabola in this way:

- With
*a*positive, as*b*decreases, the parabola shifts to the right, following a downwards-facing parabola whose tip is at (0,*c*). - With
*a*negative, as*b*decreases, the parabola shifts to the right, following an upwards-facing parabola whose tip is at (0,*c*).

## Vocabulary ; Definitions

**Quadratic function**: The quadratic function is *f(x) = a * x^2 + b * x + c*, which tells you what the function will look like graphed.**B-value**: The b-value is the middle number, the number next to the *x* and summarizes how changes in *b* affect the graph.

## Learning Outcome

After viewing this lesson, you should be able to summarize how changes in the b-value affect the graph of the quadratic function.