Parabola above $x$-axis

Question

Why is the quadratic(or maybe other degrees) polynomial $ax^2+bx+c$ with $a$ positive has a parabola having both its ends always above the $x$-axis?

I am not getting the logic behind it.

Answer 1

Think about what happens as $x$ gets really really big in either the positive or negative direction.

Let's start with $x = 1$. Then, you have $y = a + b + c$. So, $b$ and $c$ play just as important a role in determining the sign of $y$ as $a$ does.

Then, let's see what happens at $x = 10$: $y = 100a + 10b + c$. If $b$ and $c$ are large enough, they're still important in determining the sign of $y$.

Then at $x = 100$, we have $y = 10000a + 100b + c$.

At $x = 1000$, $y = 1000000a + 1000b + c$.

Notice, as $x$ increases, the effect of $a$ on $y$ becomes much larger than the effects of $b$ and $c$. Since $a$ is positive, eventually, regardless of what's going on with $b$ and $c$, $ax^2$ will be a very large positive number with absolute value bigger than $|bx+c|$.

The exact same logic works with the negative side, eventually $ax^2$ will dominate $bx+c$ in terms of the absolute value. Since $ax^2$ is always positive (except at $x = 0$), you'll end up with $y$ being positive as $x$ gets really big in either direction.

Answer 2

$$ax^2+bx+c=a\left(x-\frac{b}{2a}\right)^2-\frac{b^2}{4a^2}+c$$ The number in brackets grows arbitrarily when $x$ goes far from $b/(2a)$. Since $a$ is positive, this yields, arbitrarily great positive numbers.

It can be explained much better using limits, do you know about them?