Why does the width of the graph of a parabola depend only on $a$, not $b$?

Question

Lets assume a quadratic function $y = ax^2 + bx +c$. My book says how wide or narrow the graph is depends on the size of $|a|$

My question is why doesn't it depend on $b$ also? If you, say, increase the value of $b$ then it must increase the value of $y$, mustn't it? Even if you reformat it as $y = a(x-h)^2 +k$ you still get $b$ there since $h = -(\frac{b}{2a})$

So if I am right till now, then for the same value of $x$, we would get a bigger value of $y$ if $b$ is increased which would make the graph narrower. By the same logic the direction of the opening of a parabaloa also should depend on $b$, but it doesn't. Why?

Answer 1

You almost answered your question yourself.

Looking at $$y=a\left(x + \frac{b}{2a}\right)^2 + k,$$

and setting $\tilde x=x+ \frac{b}{2a}, $ we get a parabola $y=a\tilde x ^2 +k$, where the width of the parabola is influenced by $a$ and only $a$ (with respect to the new variable $\tilde x$ which has its minimum at $\tilde x = 0$. The parameter $k$ is still there, but it influences the offset in $y-$direction.

What's the effect change of variables from $x$ to $\tilde x$? Well,it's shifting $x$ by $b/(2a)$ to the right. This offset in $x-$direction does not change the shape of the parabola.

Answer 2

If you reformat it as $y = a(x-h)^2 + k$, well, $h$ only tells you the horizontal displacement of the graph, and $a$ tells you how wide or narrow it is. But $a$ only depends on $a$, not $b$!

For an intuition for why this is, notice that changing $b$ is the same as adding a linear equation to the quadratic equation. If you stack a parabola on a line, the shape of the parabola doesn't change, only the position of its vertex.

Answer 3

As you note, in the version $y=a(x-h)^2+k$ the $h$ is affected by $b$ since $h=-b/(2a).$ [not $-b/2$ as you wrote...] But the value of $h$ only affects the right or left shift of the graph, not the steepness which has to do with $|a|.$