Problem 10 - Calculus with Concepts in Calculus

Question

Determine all intercepts of the graph of the equation. Then decide whether the graph is symmetric with respect to the \(x\) axis, the \(y\) axis, or the origin. $$ y=\frac{x}{1+x^{2}} $$

Step-by-Step Solution

Answer

The intercept is at (0, 0); the graph is symmetric about the origin.

Step 1: Identify the Y-Intercept

To find the y-intercept of a graph, set \(x = 0\) in the equation and solve for \(y\). Substitute \(x = 0\) into the equation: \[ y = \frac{0}{1 + (0)^2} = 0 \]Therefore, the y-intercept is at the point \((0, 0)\).

Step 2: Identify the X-Intercepts

To find the x-intercepts of a graph, set \(y = 0\) in the equation and solve for \(x\). Set \(y = 0\):\[ 0 = \frac{x}{1 + x^2} \]The only solution to this equation is \(x = 0\). Therefore, the x-intercept is also at the point \((0, 0)\).

Step 3: Check Symmetry with Respect to the X-Axis

Replace \(y\) with \(-y\) in the original equation and see if the resulting equation is equivalent to the original:\[ -y = \frac{x}{1 + x^2} \]Since the equation \(-y = \frac{x}{1 + x^2}\) is not equivalent to \(y = \frac{x}{1 + x^2}\), the graph is not symmetric with respect to the x-axis.

Step 4: Check Symmetry with Respect to the Y-Axis

Replace \(x\) with \(-x\) in the equation and see if the resulting equation is equivalent to the original:\[ y = \frac{-x}{1 + (-x)^2} = \frac{-x}{1 + x^2} \]Since \(\frac{-x}{1 + x^2}\) is not equivalent to \(\frac{x}{1 + x^2}\), the graph is not symmetric with respect to the y-axis.

Step 5: Check Symmetry with Respect to the Origin

Replace \((x, y)\) with \((-x, -y)\) and see if we get the original equation back:\[ -y = \frac{-x}{1 + (-x)^2} = \frac{-x}{1 + x^2} \]Rewriting yields \(y = \frac{x}{1 + x^2}\), which is the original equation. Thus, the graph is symmetric with respect to the origin.

Key Concepts

X-Intercepts

X-intercepts are crucial points where the graph intersects the x-axis. At these points, the value of the function equals zero. To find x-intercepts, you need to set the function equal to zero and solve for the x-values.In the exercise, the equation provided is:\[ y = \frac{x}{1 + x^2} \]To find the x-intercepts, we substitute \(y = 0\) into the equation. This leads us to:\[ 0 = \frac{x}{1 + x^2} \]From this, it's evident that the only solution is \(x = 0\). Therefore, the single x-intercept is at the point \((0, 0)\).Remember, the x-intercepts are the spots where the graph touches or crosses the x-axis. Knowing how to find these points helps in sketching the graph accurately.

Y-Intercepts

Y-intercepts occur where the graph meets the y-axis. This happens when the value of x equals zero. To find the y-intercepts of a function, you simply substitute \(x = 0\) into the function and solve for y.Considering the given equation:\[ y = \frac{x}{1 + x^2} \]Substituting \(x = 0\) leads to:\[ y = \frac{0}{1 + 0^2} = 0 \]Therefore, the y-intercept is at the point \((0, 0)\).Y-intercepts are pivotal in understanding where the graph crosses the vertical y-axis, providing a foundational point for graphing the function.

Symmetry with Respect to the Origin

Graph symmetry offers insight into the balance and pattern of the graph. A graph is symmetric with respect to the origin if the transformation of each point \((x, y)\) to \((-x, -y)\) results in the same graph.For our given equation:\[ y = \frac{x}{1 + x^2} \]Checking for origin symmetry involves substituting \(x\) and \(y\) with their negative counterparts, leading to:\[ -y = \frac{-x}{1 + (-x)^2} = \frac{-x}{1 + x^2} \]Rewriting it gives us:\[ y = \frac{x}{1 + x^2} \]This is equivalent to the original equation, confirming the graph's symmetry with respect to the origin.Understanding symmetry helps in predicting graph behavior, as having origin symmetry often signifies certain aesthetic or theoretical properties of the graph.