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Chapter 1: Problem 9

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

Short Answer

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

Step by step solution

01

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).
02

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.
03

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.
04

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.
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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