Problem 49 - A Graphical Approach to College Algebra

Question

Sketch a graph of each rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{-4}{3 x+9}$$

Step-by-Step Solution

Answer

The graph has a vertical asymptote at \(x = -3\), a horizontal asymptote at \(y = 0\), and a y-intercept at \((0, -\frac{4}{9})\).

Step 1: Identify Vertical Asymptote

A vertical asymptote exists where the denominator of the function equals zero. For the function \(f(x) = \frac{-4}{3x + 9}\), set the denominator to zero: \[3x + 9 = 0\]Solve for \(x\):\[3x = -9\]\[x = -3\]Thus, the vertical asymptote is at \(x = -3\).

Step 2: Identify Horizontal Asymptote

For rational functions where the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). In \(f(x) = \frac{-4}{3x + 9}\), the numerator's degree is 0 (constant) and the denominator's degree is 1, hence the horizontal asymptote is \(y = 0\).

Step 3: Find Intercepts

To find the y-intercept, set \(x = 0\):\[f(0) = \frac{-4}{3(0) + 9} = \frac{-4}{9}\]This means the y-intercept is \((0, -\frac{4}{9})\). The function does not have an x-intercept as the numerator, \(-4\), never equals zero.

Step 4: Analyze the Sign of the Function

Since \(f(x) = \frac{-4}{3x + 9}\) includes a constant negative numerator, the sign of the function depends only on the denominator's sign. For \(x < -3\), \(3x + 9 < 0\), thus, \(f(x) > 0\).For \(x > -3\), \(3x + 9 > 0\), thus, \(f(x) < 0\).

Step 5: Sketch the Graph

Use the asymptotes, intercepts, and sign analysis to sketch the function:- Draw a vertical asymptote at \(x = -3\).- Draw a horizontal asymptote at \(y = 0\).- Mark the y-intercept at \((0, -\frac{4}{9})\).- Note that for \(x < -3\), the function is positive, approaching the vertical asymptote from the upper side.- For \(x > -3\), the function is negative, descending towards the horizontal asymptote.

Key Concepts

Understanding Vertical Asymptotes

A vertical asymptote is a vertical line that indicates where a function becomes unbounded or undefined. For rational functions like \(f(x)=\frac{-4}{3x+9}\), vertical asymptotes occur where the denominator is zero. This function's vertical asymptote is at \(x=-3\). To find this, solve the equation \(3x + 9 = 0\), leading to \(x = -3\). Here, the function does not produce a finite output, and as \(x\) approaches \(-3\), \(f(x)\) tends towards infinity or negative infinity depending on the side from which \(x\) approaches.

• If \(x\) is slightly less than \(-3\), the function tends to positive infinity.
• If \(x\) is slightly more than \(-3\), the function tends to negative infinity.

Grasping Horizontal Asymptotes

Horizontal asymptotes illustrate the behavior of a function as \(x\) approaches positive or negative infinity. For the function \(f(x) = \frac{-4}{3x + 9}\), the numerator has a degree of 0 (since it’s a constant, \(-4\)) and the denominator has a degree of 1. When the degree of the numerator is less than that of the denominator, the horizontal asymptote is \(y = 0\). As \(x\) moves towards infinity, the function \(f(x)=\frac{-4}{3x+9}\) approaches 0, which means it levels off along the horizontal asymptote. No matter how large or small \(x\) gets, \(f(x)\) will get closer and closer to zero:

• For very large positive \(x\), \(f(x)\) becomes a very small negative number.
• For very large negative \(x\), \(f(x)\) behaves similarly but on the positive side.

Graph Sketching Essentials

Graphing rational functions involves marking where the function crosses axes and approaches asymptotes. For \(f(x) = \frac{-4}{3x + 9}\), start by plotting the vertical asymptote at \(x = -3\) and the horizontal asymptote at \(y = 0\). These asymptotes are indicators of major changes in the behavior of the function. Next, find the intercepts:

• The y-intercept is at \((0, -\frac{4}{9})\) by setting \(x = 0\) in the function. This is where the graph touches the y-axis.
• There are no x-intercepts because the numerator \(-4\) is a constant and will not equal zero.

When sketching:

• As \(x\) is less than \(-3\), the graph exists in the upper quadrant, approaching the vertical asymptote from above.
• As \(x\) is greater than \(-3\), the graph exists in the lower quadrant and falls toward the horizontal asymptote.

These steps ensure that the major features and behavior of the function are accurately represented on the graph.