Sketch the graph of the function $f\left(x\right)=\frac{1}{x^2}$. Identify the asymptotes on the graph.

• A. Vertical Asymptote: $x=0$, Horizontal Asymptote: None
• B. Vertical Asymptote: $x=0$, Horizontal Asymptote: $y=0$
• C. Vertical Asymptote: $x=0$, Horizontal Asymptote: $y=0$
• D. Vertical Asymptote: $x=1$ , Horizontal Asymptote: $y=0$

Answer: (C)

Verified step by step guidance

1. The function given is \( f(x) = \frac{1}{x^2} \). This is a rational function where the numerator is a constant and the denominator is a quadratic expression.
2. Identify the vertical asymptote by setting the denominator equal to zero. Since the denominator is \( x^2 \), set \( x^2 = 0 \). Solving this gives \( x = 0 \). Therefore, there is a vertical asymptote at \( x = 0 \).
3. To find the horizontal asymptote, consider the behavior of \( f(x) \) as \( x \) approaches infinity. As \( x \to \infty \) or \( x \to -\infty \), \( \frac{1}{x^2} \to 0 \). Thus, the horizontal asymptote is \( y = 0 \).
4. The graph of \( f(x) = \frac{1}{x^2} \) is symmetric about the y-axis because it is an even function. This means the graph will look the same on both sides of the y-axis.
5. In the graph, as \( x \) approaches 0 from the positive or negative side, \( f(x) \) approaches infinity, confirming the vertical asymptote at \( x = 0 \). As \( x \) moves away from zero in either direction, \( f(x) \) approaches zero, confirming the horizontal asymptote at \( y = 0 \).