Determining Negative Regions of the Quadratic: y = -2x² - 8x - 10

To solve this problem, we'll follow these steps:

• Step 1: Analyze the quadratic function's graph from given coefficients.
• Step 2: Calculate discriminant to find root characteristics.
• Step 3: Discuss the implications on positivity or negativity of $f(x)$.

Now, let's work through each step:

Step 1: The quadratic function $y = -2x^2 - 8x - 10$ has a leading coefficient $a = -2$, which is negative. This indicates that the parabola opens downwards, potentially sitting below the $x$-axis.

Step 2: Calculate the discriminant $\Delta = b^2 - 4ac$.

Here, $b = -8$, $a = -2$, and $c = -10$. Plug these into the formula:

$\Delta = (-8)^2 - 4(-2)(-10) = 64 - 80 = -16$

Since the discriminant $\Delta$ is negative, there are no real roots. The parabola does not intersect the $x$-axis.

Step 3: Discuss implications.

Because the parabola opens downward and has no real roots, it lies entirely below the $x$-axis. This means for all real values of $x$, $f(x) < 0$.

Therefore, the function $y = -2x^2 - 8x - 10$ is negative for all $x$.

The function is negative for all $x$.