What is b in math?

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.

study.com - B-Value: Definition & Explanation - Video & Lesson Transcript

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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:

b=2

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.

b=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:

b=0

b=-1

b=-2

b=-3

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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Can running give you abs?

“Yes, running can help give you defined abs,” said Todd Buckingham, Ph. D., exercise physiologist. But before you get too excited, it's important to note that running alone isn't enough to improve muscular definition in your midsection.

nike.com - Can Running Really Give You Abs? . Nike.com

“One of the biggest factors contributing to a visible six-pack is the amount of body fat stored around the stomach,” Thompson said. Running can help to create a caloric deficit, which may reduce the padding around your abdomen (but remember, some fat is necessary to cushion vital organs). Sprinting in particular requires your abdominals to contract more forcefully, which may increase muscle size, Buckingham said.

(Related: How To Get Faster at Sprinting, According to Experts)

But there are bigger dial-movers than running that come into play for carving abs: mainly, diet and genetics. Genetics play a big role in determining where you tend to store body fat. “If you’re predisposed to store fat around the abdomen, it might be more difficult for you to have well-defined abdominal muscles, even if you have a low body fat percentage,” Buckingham said. However, if you tend to store more fat around the waist and hips, you could have defined abs with a higher body fat percentage, he added. Admittedly, there’s not much you can do about your genetics. However, if you’re hoping to achieve more muscular definition in your abdomen (or anywhere else), consider working with a doctor, registered dietitian, and personal trainer to come up with a holistic game plan that works for you. In general, following a healthy diet and doing strength exercises that target the rectus abdominis, obliques, and transverse abdominis can help make them more defined, Thompson said. Exercises like sit-ups, crunches, reverse crunches, hollow body holds, and planks all qualify.

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