This task requires finding the points of intersection of the given function $y = \frac{x^2 - 1}{4x^2}$ with the $OX$-axis (the x-axis).

Step-by-Step Solution:

1. Intersection with the x-axis: For the function to intersect the x-axis, the $y$-value must be $0$. Hence, we solve: $\frac{x^2 - 1}{4x^2} = 0$ A fraction equals $0$ if the numerator is $0$, provided the denominator is not $0$.

2. Set numerator to $0$: $x^2 - 1 = 0$ Solving this gives: $x^2 = 1 \quad \implies \quad x = \pm 1$

3. Check denominator: The denominator is $4x^2$. It is $0$ when $x = 0$, but $x = 0$ does not satisfy the numerator being $0$. Therefore, $x = 0$ is not a solution.

4. Intersection points: The function intersects the x-axis at $x = 1$ and $x = -1$. The corresponding points are: $(1, 0) \quad \text{and} \quad (-1, 0)$

Final Answer:

The points of intersection are: $1,0 \quad \text{and} \quad -1,0$

Would you like additional details or further explanations?

Related Questions:

1. How do you find intersections of a function with the y-axis?
2. What are the conditions for a rational function to intersect the x-axis?
3. Can a function have no intersections with the x-axis? Provide examples.
4. What role does the denominator of a rational function play in finding its zeros?
5. How do vertical asymptotes relate to points of intersection?

Tip:

Always verify if the denominator of a rational function introduces restrictions (e.g., vertical asymptotes or undefined points).