Is this quadratic pointing up or down? How do I know?

Question

The equation is $-2x^2 + 4x + 30 = 0$. I simplified it to $-2(x^2 - 2x - 15)$.

To know if it points up, I need to look at $ax^2$, and if $a > 0$ it is up and if $a$ is $< 0$ it is down.

However, which version do I look at? In the original version, it would be pointing down. In the simplified version, it would be pointing up

Answer 1

You must analyze the function $f(x)= -2x^2+4x+30$. You cannot set the function equal to $0$ unless you are trying to solve for the $x$ intercepts.

To determine the concavity of the function (whether it is up or down), you must look at the coefficient of the $x^2$ term. In either case, $-2x^2+4x+30$ or $-2(x^2 - 2x - 15)$, the coefficient is $-2$, indicating that the function is facing downwards, since $-2<0$.

Answer 2

It might help to state the result you're trying to apply more clearly:

The function $f(x)=ax^2+bx+c$ is pointing up when $a>0$ and is pointing down when $a<0$.

I write this out to emphasize that this requires $f$ to be written in standard form as $ax^2+bx+c$ - so you should look at that equation. That said, if you have something like $-2(x^2-2x-15)$, it's pretty easy to see that when you distribute out to get this in standard form, the coefficient of $x$ will be $-2$, so it points down, but it's still good to think about the intermediate step of distributing everything, even if you don't carry it out in full.

Answer 3

You started your post by talking about the equation The equation is $$-2x^2 + 4x + 30 = 0.$$ And you correctly noticed that $-2x^2 + 4x + 30 = -2(x^2 - 2x - 15)$, which means that this equation is equivalent to $$x^2 - 2x - 15=0.$$

If you are only interested in the roots, then it does not matter which of the two equations you take. The roost are the $x$-intercepts of the graph of $$f(x)=-2x^2 + 4x + 30.$$ But the $x$-intercepts of the graph of $$g(x)=x^2 - 2x - 15$$ are exactly the same.

The function $f(x)$ is a quadratic function which is concave, the function $g(x)$ is convex. One of the graphs is directed downwards, the other on upwards. But the difference between the graphs of these two functions is only important if you also are interested in other properties of these functions. Not if you only need to find points for which $f(x)=0$. (Which is equivalent to $g(x)=0$.)