Domain Analysis: Finding Valid Inputs for y = -x² + 6¼

To find the positive and negative domains of $y = -x^2 + 6\frac{1}{4}$, we first rewrite it as $y = -x^2 + 6.25$.

Step 1: Solve for the roots of the function:
Set $y = 0$:

$0 = -x^2 + 6.25$

Rearrange to find $x^2 = 6.25$.

Take the square root: $x = \pm \sqrt{6.25}$.

This simplifies to $x = \pm 2.5$.

Step 2: The function $y = -x^2 + 6.25$ is a downward-opening parabola with vertex at zero leading term, meaning its maximum occurs at $x = 0$.

Step 3: Identify intervals:

• Interval $x < -2.5$: $y$ is negative (as the parabola is below the x-axis).
• Interval $-2.5 < x < 2.5$: $y$ is positive (as the parabola is above the x-axis).
• Interval $x > 2.5$: $y$ is negative again (as the parabola curves back down).

The positive domain is $-2.5 < x < 2.5$. In conventional mathematical form, this can be noted as:

$x > 0 : -2\frac{1}{2} < x < 2\frac{1}{2}$

The negative domain occurs when $x < -2.5$ and $x > 2.5$:

$x > 2\frac{1}{2}$ or $x < 0 : x < -2\frac{1}{2}$

Therefore, the correct answer is Choice 3.

Thus, the positive and negative domains of the quadratic function are:

Positive domain: $x > 0 : -2\frac{1}{2} < x < 2\frac{1}{2}$

Negative domain: $x > 2\frac{1}{2}$ or $x < 0 : x < -2\frac{1}{2}$