Problem 44 - Algebra 2

Question

Graph each rational function. $$ f(x)=\frac{1}{(x+2)^{2}} $$

Step-by-Step Solution

Answer

Plot a vertical asymptote at \(x = -2\); the curve approaches \(y = 0\) as \(x\to\pm\infty\), and has a \(y\)-intercept at \((0, \frac{1}{4})\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero. Set \((x+2)^2 = 0\). Solving this gives \(x + 2 = 0\) or \(x = -2\). Thus, there is a vertical asymptote at \(x = -2\).

Step 2: Identify Horizontal Asymptotes

For large \(x\), the behavior of \(f(x)\) is dominated by its leading terms. Since the numerator's degree is less than the denominator's degree, the horizontal asymptote is \(y = 0\).

Step 3: Determine Intercepts

There are no \(x\)-intercepts since the numerator, 1, is never zero. The \(y\)-intercept occurs by evaluating \(f(0)\): \(f(0) = \frac{1}{(0+2)^2} = \frac{1}{4}\). So, the \(y\)-intercept is at \((0, \frac{1}{4})\).

Step 4: Analyze the Graph

Since the factor \((x+2)^2\) is squared, the graph will be symmetric concerning the vertical asymptote at \(x = -2\) and will approach infinity on both sides. Additionally, the function will be above the x-axis everywhere because the fraction is always positive.

Step 5: Sketch the Graph

Based on the information from previous steps, sketch the graph showing: 1) vertical asymptote at \(x = -2\), 2) horizontal asymptote at \(y = 0\), and 3) the positive curve that decreases towards \(y = 0\) as \(x\) moves away from \(-2\) and is above the x-axis.

Key Concepts

Vertical Asymptotes in Rational Functions

Vertical asymptotes in rational functions occur when the denominator equals zero, leading to undefined points in the graph. In the function \(f(x) = \frac{1}{(x+2)^2}\), set the denominator \((x+2)^2\) equal to zero. Solve the equation: \((x+2)^2 = 0\) gives us \(x + 2 = 0\). Consequently, the vertical asymptote is at \(x = -2\).

This asymptote represents a line the graph approaches but never touches. As \(x\) gets closer to \(-2\), \(f(x)\) increases or decreases drastically toward infinity or negative infinity.

• Vertical asymptotes signify how the function behaves near certain critical x-values.
• When the graph approaches these values, it makes sharp turns towards positive or negative infinity.
• Vertical asymptotes provide valuable insight into the function's behavior and discontinuities.

Understanding vertical asymptotes is essential for sketching clear and accurate graphs of rational functions.

Horizontal Asymptotes of Rational Functions

In rational functions, horizontal asymptotes describe the function's end behavior as \(x\) approaches positive or negative infinity. For the function \(f(x) = \frac{1}{(x+2)^2}\), compare the degrees of the polynomial in the numerator and the denominator.

Here, the numerator's degree (0) is less than the denominator's degree (2). When the degree of the denominator is greater, the horizontal asymptote is on the x-axis, specifically \(y = 0\).

• Horizontal asymptotes tell us what the outputs of the function will look like as the inputs become extremely large or small.
• Unlike vertical asymptotes, the function can cross horizontal asymptotes at finite points on the graph, but the end behavior still approaches this line.
• Knowing horizontal asymptotes helps predict the function's behavior as \(x\) moves towards infinity.

Thus, understanding horizontal asymptotes is critical for predicting long-term behavior of rational functions.

Intercepts of Rational Functions

Rational functions can have both vertical and horizontal intercepts. Intercepts are points where the graph intersects the axes, providing useful reference points for graphing.

**X-intercepts**:
The x-intercept occurs where \(f(x) = 0\), which means the numerator must equal zero. However, in \(f(x) = \frac{1}{(x+2)^2}\), the numerator is 1 and never zero, resulting in no x-intercepts.

**Y-intercepts**:
The y-intercept occurs when \(x = 0\). To find it, simply evaluate the function at zero: \(f(0) = \frac{1}{(0+2)^2} = \frac{1}{4}\). Therefore, the y-intercept is at \((0, \frac{1}{4})\).

• Y-intercepts are always found by substituting zero for \(x\) in the function.
• X-intercepts, if any, require setting the function equal to zero and solving for \(x\).
• Intercepts provide starting points for graphing and showing where the graph crosses the axes.

Considering intercepts ensures a more complete and accurate graph of the rational function, revealing key intersection points with the axes.