Problem 18 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) $$f(x)=\frac{2}{(x+1)^{2}}$$

Step-by-Step Solution

Answer

The graph is a hyperbola centered at \(x = -1\), symmetric about the vertical asymptote with a vertical stretch by factor 2.

Step 1: Identify the Basic Function

The function given is a transformation of the parent function \(f(x) = \frac{1}{x^2}\). The basic shape of the graph is hyperbolic and symmetric with respect to the y-axis.

Step 2: Apply Transformations

The given function is \(f(x) = \frac{2}{(x+1)^2}\). It involves a horizontal shift left by 1 unit and a vertical scaling by a factor of 2. This modifies both the position and the steepness of the graph.

Step 3: Determine Key Features

The function has a vertical asymptote at \(x=-1\) due to the term \((x+1)^2\) in the denominator. The horizontal asymptote is \(y=0\) because as \(x\) approaches \(\pm\infty\), \(f(x)\) approaches zero.

Step 4: Analyze Behavior Near Asymptotes

As \(x\to-1^-\), \(f(x)\to\infty\), indicating the function rises infinitely on the left of the asymptote. As \(x\to-1^+\), \(f(x)\to\infty\), indicating it also rises infinitely on the right.

Step 5: Plot Specific Points

Calculate a few points: \((0, 2)\), \((-2, 2)\). These points confirm the symmetrical nature of the graph about the vertical asymptote without touching the x-axis.

Step 6: Sketch the Graph

First draw the asymptotes, then plot the calculated points and finally sketch the curve, ensuring it approaches the asymptotes but never crosses them.

Key Concepts

Horizontal and Vertical Asymptotes

Rational functions often have asymptotes, which are lines that the graph approaches but never actually touches. There are two types of asymptotes we consider: horizontal and vertical.

Vertical asymptotes occur where the function is undefined or, in simpler terms, where the denominator equals zero. For the function given, \(f(x) = \frac{2}{(x+1)^2}\), the vertical asymptote is at \(x = -1\).

As \(x\) approaches this value from both the left and right, the function value \(f(x)\) increases indefinitely, it never crosses this line. This behavior is like a barrier the graph cannot pass.

Horizontal asymptotes tell us how the function behaves as \(x\) approaches very large numbers (positive or negative). For the function \(f(x)\), the horizontal asymptote is \(y = 0\). As \(x\) becomes very large or very small, the function value decreases towards zero. This indicates that no matter how large \(x\) becomes, the function value will never touch or cross this horizontal line.

Understanding these asymptotes helps in predicting the graph's behavior and how it "interacts" with these invisible boundaries.

Transformations of Functions

Transformations involve modifying a function's graph. For the function \(f(x) = \frac{2}{(x+1)^2}\), several transformation steps happen to the basic parent function \(g(x) = \frac{1}{x^2}\). Each transformation impacts the graph's position and shape.

Firstly, there is a horizontal shift. This occurs when we add or subtract inside the function. Here, adding 1 inside the denominator shifts the graph 1 unit to the left, making it \(x + 1\) instead of just \(x\).

Secondly, the function is vertically stretched. Notice the 2 in the numerator? That means every point on the graph is "pulled" away from the x-axis or, more simply, the graph's steepness increases. This vertical scaling modifies the way the graph looks compared to the original parent function's simpler curve.

Transformations make graphs flexible, allowing them to move and change in various ways. When you understand how each transformation alters the graph, you can better predict its shape and position.

Symmetry in Graphs

Symmetry is about balance and mirroring. A graph is symmetric if one part mirrors another. For rational functions, this typically means identifying if the graph looks the same on either side of a line.

For the parent function \(g(x) = \frac{1}{x^2}\), there's symmetry about the y-axis. This means if you fold the graph along the y-axis, both sides will overlap perfectly.

In our case with \(f(x) = \frac{2}{(x+1)^2}\), even with the transformation of shifting left, the function maintains a form of symmetry. Instead of a y-axis symmetry, it reflects this around the vertical asymptote at \(x = -1\).

This vertical line of symmetry signifies that the graph on the left side mirrors the graph on the right side of this vertical asymptote, attributing a balance to the entire graph.

Recognizing symmetry in graphs can simplify the process of sketching and understanding their layout, as knowing one half already gives you an idea of the other half.