Question

Plot the curve for \(f(x)=\frac{1}{x^2}\) and label any asymptotes.

Short answer

Vertical asymptote: x = 0. Horizontal asymptote: y = 0.

Step-by-step solution

Understand the Function

The given function is a rational function: \[ f(x) = \frac{1}{x^2} \]This function has a denominator that can never be zero, so it has specific behaviors to identify such as asymptotes.

Identify Vertical Asymptote

A vertical asymptote occurs where the denominator is zero. For the function \( f(x) = \frac{1}{x^2} \), the denominator is zero when \( x = 0 \). Therefore, the vertical asymptote is at \( x = 0 \). Label this on the graph.

Identify Horizontal Asymptote

As \( x \) approaches infinity or negative infinity, the value of \( f(x) = \frac{1}{x^2} \) approaches 0. Thus, the horizontal asymptote is at \( y = 0 \). Label this on the graph.

Plot Key Points

Calculate and plot some key points to understand the behavior of the function. For example, when \( x = 1 \), \( f(1) = 1 \); when \( x = 2 \), \( f(2) = \frac{1}{4} \); when \( x = -1 \), \( f(-1) = 1 \); and when \( x = -2 \), \( f(-2) = \frac{1}{4} \). Plot these points on the graph.

Sketch the Curve

Using the asymptotes and key points, sketch the curve of the function. The curve will approach but never touch the asymptotes at \( x = 0 \) and \( y = 0 \), and it will be in the first and second quadrants because \( f(x) > 0 \) for all real values of \( x \) except at the vertical asymptote.

Key concepts

Vertical Asymptotes

Understanding vertical asymptotes is crucial when plotting rational functions. A vertical asymptote represents a value of x where the function tends toward infinity. For the function, \( f(x) = \frac{1}{x^2} \), look at the denominator, \( x^2 \). The denominator becomes zero at \( x = 0 \). Therefore, we have a vertical asymptote at \( x = 0 \). This means the graph will approach this line but never touch or cross it. It's essential to recognize this behavior to understand how the function behaves near this point.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches very large positive or negative values. For \( f(x) = \frac{1}{x^2} \), as \( x \to \infty \) or \( x\to -\infty \), the value of \( f(x) \) tends to zero. Therefore, the horizontal asymptote is at \( y = 0 \). This tells us that no matter how large or small x gets, the function's value will approach 0, but never actually touch or reach zero. This is another characteristic you should mark on your graph to help sketch the curve accurately.

Key Points Plotting

Plotting key points helps to form a clearer picture of the function's behavior. With the function \( f(x) = \frac{1}{x^2} \), let's find some simple values:
- When \( x = 1 \), \( f(1) = 1 \).
- When \( x = 2 \), \( f(2) = \frac{1}{4} \).
- When \( x = -1 \), \( f(-1) = 1 \).
- When \( x = -2 \), \( f(-2) = \frac{1}{4} \).
Plotting these points gives us visuals on both sides of the asymptote at \( x = 0 \). These points highlight how quickly the function values drop as x moves away from zero, yet remain positive.

Curve Sketching

Curve sketching takes into account all information compiled from asymptotes and key points plotting. Begin by drawing your asymptotes: a vertical line at \( x = 0 \) and a horizontal line at \( y = 0 \).
Then, plot the key points you calculated earlier. Keep in mind:
- The curve will approach the x-axis as it extends to infinity in either direction.
- The function will never touch or cross the vertical asymptote at \( x = 0 \).
- The graph exists only in the first and second quadrants because all y-values are positive.
The careful plotting of points and understanding of asymptotes helps in drawing an accurate and informative graph.