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Let's say we are given a quadratic polynomial $ax^2+bx+c$. Without the use of derivatives, what is the meaning of the coefficients when it comes to graphing the polynomial?

The coefficient $c$ determines the intersection with the y-axis by setting $x=0$.

$a$ determines if the polynomial opens upwards or downwards and how "large" the polynomial is, however I have a hard time seeing why this the case? It's easy to see if we use derivatives, however how can one see this intuitively without the use of derivates?

$a$ and $b$ determines the placement of the vertex. If $sign(a)=sign(b)$ then the vertex is placed below the x-axis, otherwise it's placed above the x-axis. This is clear, since the x-coordinate for the vertex is given by $\frac{-b}{2a}$. But how about the placement of the vertex in regards to the y-axis? How and why does $a$ and $b$ determine this placement?

TJ123
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By “completing the square” method, one can change the given expression into its corresponding “vertex form”, namely $y = f(x) = a(x – h)^2 + k$.

Then, the vertex ($V$) is located at $(h, k)$ where $h = -b/(2a)$ and $k$ equals to something like $-\triangle/(4a^2)$, --- the residue coming from the completing square process.

I think “the placement of the vertex in regards to the y-axis” is referring to the value of k.

In practice, we don’t try to remember the above clumsy formula for $k$. Since $V = (h, k)$ is a point on $y = f(x)$, this means $V = (h, f[h])$. That is, $k$ can easily be found by evaluating $f(h)$ instead, once $h$ is found.

Mick
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