Problem 9 - 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. $$ y=x-\frac{1}{x} $$

Step-by-Step Solution

Answer

x-intercepts are at (1,0) and (-1,0); there is no symmetry.

Step 1: Find the x-intercept

The x-intercept is the point where the graph intersects the x-axis. For this, set the equation equal to zero and solve for x: \[ 0 = x - \frac{1}{x} \] Multiply both sides by x to eliminate the fraction: \[ 0 = x^2 - 1 \] This is a difference of squares, which factors to: \[ (x-1)(x+1) = 0 \] So, the solutions are \( x = 1 \) and \( x = -1 \). The x-intercepts are at the points (1,0) and (-1,0).

Step 2: Find the y-intercept

The y-intercept occurs where the graph intersects the y-axis, which is when \( x = 0 \). Substitute 0 for x in the equation: \[ y = 0 - \frac{1}{0} \] However, this expression is undefined due to division by zero. Thus, there is no y-intercept for the graph.

Step 3: Test symmetry with respect to the x-axis

A graph is symmetric with respect to the x-axis if replacing y with -y yields an equivalent equation. Substitute \( -y \) for y: \[ -y = x - \frac{1}{x} \] This new equation is not equivalent to the original equation \( y = x - \frac{1}{x} \), so the graph is not symmetric with respect to the x-axis.

Step 4: Test symmetry with respect to the y-axis

A graph is symmetric with respect to the y-axis if replacing x with -x yields an equivalent equation. Substitute \( -x \) for x: \[ y = -x + \frac{1}{-x} \] Simplifying the second term gives: \[ y = -x - \frac{1}{x} \] This is not equivalent to the original equation \( y = x - \frac{1}{x} \), so the graph is not symmetric with respect to the y-axis.

Step 5: Test symmetry with respect to the origin

A graph is symmetric with respect to the origin if replacing x with -x and y with -y yields an equivalent equation. Substitute \( -x \) for x and \( -y \) for y: \[ -y = -x - \frac{1}{x} \] Multiplying both sides by -1 gives: \[ y = x + \frac{1}{x} \] This is clearly not equivalent to the original equation \( y = x - \frac{1}{x} \), so the graph is not symmetric with respect to the origin.

Key Concepts

Understanding the X-Intercept

The x-intercept is a vital point on a graph where it crosses the x-axis. This means the value of y is 0 at this point. To find the x-intercept, you set the equation equal to zero.
This involves calculating where the graph intersects the x-axis by solving the equation:

• Start with the equation: \( 0 = x - \frac{1}{x} \)
• Multiply through by \( x \) to eliminate the fraction: \( 0 = x^2 - 1 \)
• This is a difference of squares, represented as \( (x-1)(x+1) = 0 \)
• Solving gives you \( x = 1 \) and \( x = -1 \)

Thus, the x-intercepts are at points (1, 0) and (-1, 0). These intercepts show where the line crosses the x-axis. Understanding this helps in visualizing how the graph behaves as it passes the x-axis.

Exploring the Y-Intercept

The y-intercept is where a graph crosses the y-axis. For this point, the value of x is zero. You find it by substituting 0 for x in the equation.
However, with this specific equation, when you plug in zero for x:

• The equation becomes: \( y = 0 - \frac{1}{0} \)
• Division by zero is undefined, which means there is no y-intercept for this graph.

This absence of a y-intercept is crucial. It indicates that the graph never touches or crosses the y-axis. When preparing to graph such equations, understanding situations where a y-intercept does not exist helps set the expectations for graph behavior.

Discovering Graph Symmetry

Symmetry in graphs can make them easier to interpret. It means that a graph looks the same on different sides of an axis or origin.
There are types of symmetry to check for:

• X-Axis Symmetry: Replacing y with -y
  \(-y = x - \frac{1}{x}\) is not equivalent to the original, so no x-axis symmetry.
• Y-Axis Symmetry: Replacing x with -x
  \ y = -x - \frac{1}{x}\ is not equivalent; thus, no y-axis symmetry.
• Origin Symmetry: Replacing x and y with -x and -y
  \ -y = -x - \frac{1}{x}\ simplifies to \ y = x + \frac{1}{x}\, which is not equivalent.

Symmetry helps in quickly sketching or understanding a graph's nature. Since this particular graph shows no symmetry in any usual sense (x-axis, y-axis, and origin), knowing these tests provides clarity on the unique behavior of this graph.