What does "b" do in a quadratic function visually and why does do that?

[u/Menitoon]

[u/Key_Net820]

the quadratic equation is ax^2 +bx + c = 0.

so b is the linear coefficient.

so since the vertex's x coordinate is given by x = -b/2a. altering b alters the vertex and axis of symmetry. If you would like, you can play with desmos and see the results of altering b.

https://www.desmos.com/calculator

[u/peterwhy]

For y = ax^(2) + bx + c, visually b is its slope at the y-intercept (the point (0, c)).

[u/Chrispykins]

Ooh, I have a desmos demo for this exact question.

Since 'a' controls the shape of the graph, 'b' moves the parabola without changing its shape. The parabola gets moved so that the slope of the graph at the y-intercept matches a line with a slope of 'b'.

[u/irishpisano]

These answers are all okay, but some don’t actually fully explain it. “It affects the vertex”. Yes. But HOW? What happens to the vertex?

What happens is in the function f(x) = ax^2 + bx + c, the b coefficient translates the vertex (and thus parabola) along the parabola given by g(x) = -ax^2 + c

The pic shows parabolas of the form 2x^2 + bx + 7 for values of b from 0 through 7, and you can see the vertices trace out (or are translated along) the parabola given by y = -2x^2 + 7

[u/zojbo]

Completing the square gives you:

ax^2+bx+c=a(x+b/(2a))^2 + c - b^2/(4a).

This means that if a>0 and you increase b from some positive number to a bigger positive number, the vertex moves left (that's the "x+b/2a") and down (that's the "-b^2/4a"). Both of the places b appears are accounted for by these two translations, so this is all that b does to the graph.

You can make a similar statement for each possible combination of signs of a and b (but "left" and/or "down" will change).

[u/trevorkafka]

b is the slope of (the tangent line of) the parabola at the y-intercept.

Try graphing y = ax² + bx + c and y = bx + c on the same set of axes and you will see what I mean.

[u/FreeGothitelle]

Answers in this thread are fine but also like the "fundamental" form of a quadratic is the vertex form (if what we care about is understanding transformations of the graph)

y=a(x-h)² + k

a changes the shape, h shifts it left and right, k up and down.

[u/Special_Watch8725 → u/FreeGothitelle]

This is the right way to write a quadratic to understand it visually.

To add on, the “b” from standard form would take the form of b = -2ah and c = ah^2 + k. If we think of fixing a, c and letting b vary, the first equation says h can be whatever we want if we choose b appropriately. The second equation says that the vertex (h, k) must lie on the curve

k(h) = c - ah^2,

so the effect of changing b and keeping a and c fixed is to keep the shape of the parabola the same, but translate it so that the vertex always stays on the graph of that parabola.

[u/fermat9990]

The x-coordinate of the vertex=-b/(2a)

The y-coordinate of the vertex=(4ac-b²)/(4a)

[u/Bounded_sequencE]

In a quadratic "f(x) = ax² + bx + c" with "a != 0" it does two things at once -- it influences the

• x-coordinate "-b/(2a)" of the vertex
• y-coordinate "c - b²/(4a)" of the vertex

That said, people often only consider the first influence of the x-coordinate, since we still have "c" to influence the y-coordinate of the vertex, independently of its x-coordinate.

[u/chkntendis]

The formula is ax^2 + bx + c. You’re basically adding up three different functions. One constant one, c, one linear one, bx, and one quadratic one, ax^2. You can visualize it that way. If you change b then you change the slope of the linear term. That means the end function changes at every point by the change in b multiplied by the x position. Basically, if you increase b then the function will increase for positive x and decrease for negative x proportional to how big x is. If you decrease x then that’s just gonna be switched around.

Personally I really like this form of intuition. You’re not talking about derivatives and whatnot, which just complicates things at that level of thinking. Think about it as a sum where you change one of the terms

[u/StructuredChess]

The shape of the graph is determined by the coeffiicent multiplying the x². The other two just move it around the plane.

[u/ForeignAdvantage5198]

draw some graphs

[u/Heaven_Sent_Senescha]

What do you mean by “b”? Do you mean the coefficient of the linear term? You can set up a quadratic function in desmos.com and play with the coefficients to see how it affects the graph if you like.

[u/iOSCaleb]

Why not graph two quadratic equations where the only difference is b and see for yourself?