Problem 50 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{x^{2}+3 x+2}{x^{2}-9}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{x^{2}+3 x+2}{x^{2}-9}\) will have vertical asymptotes at \(x = -3\) and \(x = 3\), a horizontal asymptote at \(y = 1\), x-intercepts at \(x = -1\) and \(x = -2\), and a y-intercept at \(y = -2/9\).

Step 1: Find the Vertical Asymptotes

To find the vertical asymptotes of a rational function, we need to find the values of \(x\) that make the denominator equal to zero. As \(x^2 - 9 = 0\) when \(x = -3\) or \(x = 3\), the graph will have vertical asymptotes at \(x = -3\) and \(x = 3\).

Step 2: Find the Horizontal Asymptotes

A horizontal asymptote is determined by the degree of the numerator polynomial relative to the degree of the denominator. In this case, both the numerator and the denominator are of degree 2. Thus, the horizontal asymptote is at \(y = a/b\) where \(a\) is the coefficient of highest degree term of numerator and \(b\) is the coefficient of highest degree term of denominator, hence \(y = 1/1 = 1\).

Step 3: Find the x-intercepts

To find the x-intercepts of the graph of a function, we set \(y = 0\), and solve for \(x\). In the case of our function \(f(x)\), this gives us the equation \(0 = (x^2+3x+2)/(x^2-9)\). This equation is only true when the numerator is zero, i.e. when \(x = -1\) or \(x = -2\). Therefore, the function will have x-intercepts at \(x = -1\) and \(x = -2\).

Step 4: Find the y-intercept

To find the y-intercept, we plug in \(x = 0\) into our function and solve for \(y\). Substituting \(x = 0\) into \(f(x)\) yields \(-2/9\). Therefore, the function will have a y-intercept at \(y = -2/9\).

Step 5: Sketch the Graph

Now that we have the vertical asymptotes at \(x = -3\) and \(x = 3\), the horizontal asymptote at \(y = 1\), the x-intercepts at \(x = -1\) and \(x = -2\), and the y-intercept at \(y = -2/9\), we can plot these on a graph and sketch the function from these.

Key Concepts

Understanding Vertical Asymptotes

When dealing with rational functions like \(f(x)=\frac{x^{2}+3x+2}{x^{2}-9}\), vertical asymptotes are lines that the graph approaches but never actually touches.

To find these asymptotes, we set the denominator equal to zero and solve for \(x\). In our example, the denominator is \(x^2 - 9\).

Setting this equal to zero gives us:

• \(x^2 - 9 = 0\)
• \((x - 3)(x + 3) = 0\)
• \(x = 3\) or \(x = -3\)

So, the vertical asymptotes are located at \(x = -3\) and \(x = 3\). These lines indicate where the function "breaks" or where the graph tends to infinity. These are crucial for understanding the graph's behavior near these \(x\)-values.

Exploring Horizontal Asymptotes

Horizontal asymptotes help us understand the behavior of a rational function as \(x\) approaches infinity.

To identify these asymptotes, we compare the degrees of the numerator and the denominator. For \(f(x)=\frac{x^2+3x+2}{x^2-9}\), both the numerator \(x^2 + 3x + 2\) and the denominator \(x^2 - 9\) are degree 2.

This implies that the horizontal asymptote is determined by the ratio of the leading coefficients (the coefficients of the highest power of \(x\)).

• The leading coefficient of the numerator is 1.
• The leading coefficient of the denominator is also 1.

Thus, the horizontal asymptote is a constant line at \(y = 1/1 = 1\). As \(x\) moves towards positive or negative infinity, the value of \(f(x)\) will get closer and closer to this line.

Finding Intercepts: X and Y

Intercepts are key features of the graph because they pinpoint where the graph crosses the axes.

**X-Intercepts**: To find where the function crosses the \(x\)-axis, set \(y = 0\) and solve for \(x\). This means solving

• \(0 = \frac{x^2+3x+2}{x^2-9}\)
• This can only happen when the numerator equals zero: \(x^2+3x+2 = 0\)

Factoring gives

• \((x+1)(x+2) = 0\)
• \(x = -1\) or \(x = -2\)

Therefore, the x-intercepts are at \(x = -1\) and \(x = -2\).

**Y-Intercept**: To find the y-intercept, substitute \(x = 0\) into the function:

• \(f(0) = \frac{0^2+3(0)+2}{0^2-9}\)
• \(f(0) = \frac{2}{-9} = -\frac{2}{9}\)

So, the y-intercept is \(y = -\frac{2}{9}\). These intercepts help form the basic framework of the graph.

Tips for Graph Sketching

Graph sketching combines all previously discussed concepts to create a visual representation of the function.

First, mark the identified intercepts and asymptotes on the graph. These serve as reference points and boundaries.

• Plot the x-intercepts at \(x = -1\) and \(x = -2\).
• Draw the vertical asymptotes as dashed lines at \(x = -3\) and \(x = 3\).
• Mark the horizontal asymptote at \(y = 1\).
• Place the y-intercept at \(y = -\frac{2}{9}\).

Next, consider how the function behaves as it approaches the asymptotes:

• As \(x\) draws near the vertical asymptotes from either side, the graph will rise or fall sharply.
• As \(x\) becomes very large (either positively or negatively), the graph will level off and approach the horizontal asymptote.

This process gives a rough sketch of the behavior of the function, helping you understand and visualize potential changes in curvature and slope.