Trace the curve y is equal to x square upon 1+x square

To trace the curve of the function \( y = \frac{x^2}{1 + x^2} \), we need to analyze its features, including the shape, intercepts, asymptotes, and overall behavior. Let’s break this down step by step.

Understanding the Function

This function is a rational function where the numerator is \( x^2 \) and the denominator is \( 1 + x^2 \). To better understand its behavior, we can consider a few key elements.

Domain of the Function

The domain consists of all the values that \( x \) can take. Since the denominator \( 1 + x^2 \) is always positive for all real \( x \) (as \( x^2 \) is non-negative), the function is defined for all real numbers:

• Domain: \( (-\infty, \infty) \)

Finding Intercepts

Next, let’s find the intercepts of the curve:

• x-intercept: Set \( y = 0 \).
  • \( \frac{x^2}{1 + x^2} = 0 \) implies \( x^2 = 0 \) which gives \( x = 0 \). So, the x-intercept is (0, 0).
• y-intercept: Set \( x = 0 \).
  • \( y = \frac{0^2}{1 + 0^2} = 0 \). The y-intercept is also (0, 0).

Behavior as x Approaches Infinity

To understand the end behavior, we analyze what happens as \( x \) approaches positive or negative infinity:

• As \( x \to \infty \):
  • \( y = \frac{x^2}{1 + x^2} \approx \frac{x^2}{x^2} = 1 \) (dominant terms). Thus, \( y \to 1 \).
• As \( x \to -\infty \):
  • Again, \( y \to 1 \). The behavior is symmetric as the function is even.

Identifying Asymptotes

Since the function approaches 1 as \( x \) goes to positive or negative infinity, we find a horizontal asymptote:

• Horizontal asymptote: \( y = 1 \).

Analyzing Critical Points

To find the local maximum or minimum points, we can take the derivative of the function:

• Using the quotient rule, we differentiate:
• Let \( u = x^2 \) and \( v = 1 + x^2 \). Then:
• \( y' = \frac{u'v - uv'}{v^2} = \frac{2x(1 + x^2) - x^2(2x)}{(1 + x^2)^2} = \frac{2x}{(1 + x^2)^2} \).

Setting \( y' = 0 \) gives \( x = 0 \) as the only critical point. Examining the second derivative or using the first derivative test will show that this point is a local maximum.

Sketching the Curve

Now that we have the key points and features, we can sketch the curve:

• Intercepts: (0, 0)
• Horizontal asymptote: \( y = 1 \)
• Behavior: Approaches the line \( y = 1 \) as \( x \) goes to positive or negative infinity.

As a result, the overall shape of the curve will rise from the origin, reach a maximum at \( (0, 0) \), and then approach the line \( y = 1 \) on both sides.

Final Thoughts

By combining all of these insights, you can accurately sketch the behavior of the function \( y = \frac{x^2}{1 + x^2} \). This function is a great example of how understanding its components can lead to a clearer picture of its overall behavior.