B-Value: Definition & Explanation

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  • 0:00 Definition of the B-Value
  • 0:29 The Quadratic Parabola
  • 1:55 How B Affects the Parabola
  • 4:12 Lesson Summary
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Lesson Transcript
Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

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 + 3x + 1, which is below. The b-value in this equation is 3.

The parabola

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:


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:

Stacked graphs.

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Additional Activities

Finding the Equation of a Parabola from a Graph

We learned from the video lesson that the b value in the quadratic equation y = ax2 + bx + c affects the location of the parabola. Let's use the information we learned in the lesson to find the equation of two parabolas from their graphs.

Consider the two parabolas on the graph below. Each parabola has the same a value.


Using your knowledge from the video, decide if the a values of these parabolas are positive or negative. What are the c values of the parabolas? Which parabola has a smaller b value?

Answers to Discussion

The a values should be positive since the parabolas open up. Both parabolas have c = -1, since the y intercept of each parabola is -1. The blue parabola will have a smaller b value than the red parabola, since the graph is further to the right.


Given that a = 2 for these parabolas, find the equations of the parabolas.


We know from the video lesson that the x coordinate of the vertex is given by x = -(b)/(2a). So, for the blue parabola, we have

-1 = -(b)/(2)(2))

4 = b

and so the equation of the blue parabola is

y = 2x2 + 4x - 1.

For the red parabola we have

-2 = -(b)/2(2)

8 = b

and so the equation of the red parabola is

y = 2x2 + 8x - 1.

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