Question

which of the following functions opens upward?

\\(f(x) = \frac{x^2}{3} - x - 11\\)

\\(y = -2x^2 + x + 3\\)

Explanation:

Identify the standard form of quadratic functions

We examine the general structure of Quadratic Functions.
A quadratic function is written in standard form as:

$$f(x) = ax^2 + bx + c$$

where \(a\), \(b\), and \(c\) are real numbers, and \(a
eq 0\).

Determine the rule for parabola direction

We apply the rule for Parabola Direction based on the Leading Coefficient.
The leading coefficient is the coefficient \(a\) of the \(x^2\) term:

• If \(a > 0\), the parabola opens upward.
• If \(a < 0\), the parabola opens downward.

Analyze the first function

We analyze the function \(f(x) = \frac{x^2}{3} - x - 11\).
We rewrite the function to clearly identify the leading coefficient:

$$f(x) = \frac{1}{3}x^2 - x - 11$$

Here, the leading coefficient is \(a = \frac{1}{3}\).
Since \(a = \frac{1}{3} > 0\), this function opens upward.

Analyze the second function

We analyze the function \(y = -2x^2 + x + 3\).
The leading coefficient is \(a = -2\).
Since \(a = -2 < 0\), this function opens downward.

Answer:

\(f(x) = \frac{x^2}{3} - x - 11\)