The Correct Answer and Explanation is:
Here is the completed analysis of the Rational Function Family: Reciprocal Squared, based on the image:
Parent Function:
y=1x2y = \frac{1}{x^2}y=x21
Shape:
U-shaped on both sides of the y-axis, with a vertical asymptote at x = 0 and a horizontal asymptote at y = 0
Domain of y=1x2y = \frac{1}{x^2}y=x21:
All real numbers except 0, or written as: x∈(−∞,0)∪(0,∞)x \in (-\infty, 0) \cup (0, \infty)x∈(−∞,0)∪(0,∞)
Range of y=1x2y = \frac{1}{x^2}y=x21:
All positive real numbers, or written as: y∈(0,∞)y \in (0, \infty)y∈(0,∞)
Table of Values:
xy=1×2−214−110undefined11214\begin{array}{|c|c|} \hline x & y = \frac{1}{x^2} \\ \hline -2 & \frac{1}{4} \\ -1 & 1 \\ 0 & \text{undefined} \\ 1 & 1 \\ 2 & \frac{1}{4} \\ \hline \end{array}x−2−1012y=x21411undefined141
Explanation
The function y=1x2y = \frac{1}{x^2}y=x21 belongs to the rational family and is classified as a reciprocal squared function. The defining feature of this function is that the variable xxx appears in the denominator, raised to the second power. This creates a situation where the function becomes undefined at x=0x = 0x=0, resulting in a vertical asymptote. As xxx approaches zero from either direction, the values of yyy grow infinitely large, producing the sharp rise on both sides of the y-axis seen in the graph.
Since squaring any real number always gives a positive result, the output of this function is always positive as well. Therefore, the range is all positive real numbers. The function also has a horizontal asymptote at y=0y = 0y=0, meaning that as xxx becomes very large or very small, the value of yyy approaches zero but never reaches it.
The graph is symmetric with respect to the y-axis, indicating even function behavior. This symmetry stems from the fact that squaring negative and positive values of xxx gives the same result. For example, x=−1x = -1x=−1 and x=1x = 1x=1 both result in y=1y = 1y=1.
The bottom section of the image includes transformed examples of the parent function. These include vertical shifts, horizontal shifts, and reflections, which alter the graph’s position but maintain the fundamental shape and asymptotic behavior. This demonstrates how transformations can affect rational functions while preserving their core characteristics.
