Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=\frac{1}{x^{2}+1} $$

Step-by-Step Solution

Answer

The graph is symmetric about the y-axis, has a y-intercept at (0, 1), no x-intercepts, and approaches y=0 as asymptote.

Step 1: Check for Symmetries

To determine whether the function is symmetric about the y-axis, x-axis, or origin, substitute \( x = -x \) into the equation. \[ y = \frac{1}{(-x)^2 + 1} = \frac{1}{x^2 + 1} \]Thus, the equation is unchanged. This indicates that the function is symmetric about the y-axis.

Step 2: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation:\[y = \frac{1}{0^2 + 1} = 1\]So, the y-intercept is \((0, 1)\).

Step 3: Find the x-intercept(s)

To find the x-intercepts, set \( y = 0 \) and solve for \( x \):\[\frac{1}{x^2 + 1} = 0\]This equation implies that \( 1 = 0 \), which is not possible. Therefore, there are no x-intercepts for this function.

Step 4: Sketch the Graph

Using the information from the intercepts and symmetry, sketch the graph. The graph is symmetric about the y-axis due to the even powers. It approaches \( y = 0 \) as \( x \to \pm \infty \), but will never touch it, as it's a horizontal asymptote.

Key Concepts

Symmetry in Graphs

When analyzing a function's graph, we often seek to identify any symmetries. Symmetry can make graphing simpler, as it indicates a repetitive pattern. There are three main types of symmetries:

• Y-axis Symmetry: A graph has y-axis symmetry if, when you replace \( x \) with \( -x \) in the equation, the equation remains unchanged. This means that the left and right sides of the graph mirror each other across the y-axis.
• X-axis Symmetry: Occurs when replacing \( y \) with \( -y \) doesn't alter the equation. This type is less common in functions, as it implies that for every point \((x, y)\), there would also be a point \((x, -y)\).
• Origin Symmetry: A graph has origin symmetry if substituting both \( x = -x \) and \( y = -y \) leaves the equation unchanged. This means that rotating the graph 180 degrees about the origin results in an identical graph.

For the given function, \( y = \frac{1}{x^2 + 1} \), the symmetry about the y-axis is confirmed because substituting \( x = -x \) returns the same equation. This y-axis symmetry helps in understanding and sketching the graph, as the behavior on one side can be mirrored on the other.

Intercepts in Graphs

Intercepts are crucial in graphing as they mark the points where the graph intersects the axes. They provide key reference points:

• Y-intercepts are points where the graph crosses the y-axis. These occur where \( x = 0 \). For the given function, we substitute \( x = 0 \) into the equation to get \( y = \frac{1}{0^2 + 1} = 1 \). This tells us the y-intercept is at \((0, 1)\).
• X-intercepts occur where the graph crosses the x-axis, meaning \( y = 0 \). For \( y = \frac{1}{x^2 + 1} \), setting \( y = 0 \) leads to a non-existent situation (as \( \frac{1}{x^2 + 1} = 0 \) implies \( 1 = 0 \), which is impossible). Thus, there are no x-intercepts.

Understanding these intercepts helps graph the function accurately, displaying the key points where it interacts with the axes.

Asymptotic Behavior

Asymptotes are lines that a graph approaches but never touches. Understanding their role helps predict the function's behavior at extreme values. There are two main types:

• Horizontal Asymptotes: These occur when the value of \( y \) approaches a constant as \( x \) approaches infinity or negative infinity. For the function \( y = \frac{1}{x^2 + 1} \), as \( x \to \pm \infty \), \( y \) approaches 0. Thus, \( y = 0 \) is a horizontal asymptote.
• Vertical Asymptotes: These are lines \( x = c \) that the graph approaches as \( y \) heads towards infinity or negative infinity. Our function doesn't have vertical asymptotes as the denominator, \( x^2 + 1 \), is never zero.

Knowing these asymptotes clarifies that while the graph of \( y = \frac{1}{x^2 + 1} \) comes very close to the x-axis, it never crosses or touches it.