Problem 56 - 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. $$f(x)=\frac{2 x^{2}+3}{x-4}$$

Step-by-Step Solution

Answer

Plot the vertical asymptote \(x = 4\), the y-intercept at \((0, -\frac{3}{4})\), and consider end behavior resembling \(f(x) = x\). The graph rises to the right of the asymptote and falls to the left.

Step 1: Identify Asymptotes

To find the vertical asymptote, set the denominator equal to zero and solve: \(x - 4 = 0\). So, the vertical asymptote is \(x = 4\). For the horizontal asymptote, compare the degrees of the numerator and the denominator. The numerator has degree 2, and the denominator has degree 1, so no horizontal asymptote exists.

Step 2: Find Intercepts

Determine the y-intercept by evaluating \(f(0)\): \(f(0) = \frac{2(0)^2 + 3}{0 - 4} = \frac{3}{-4} = -\frac{3}{4}\). For x-intercepts, set the numerator equal to zero and solve: \(2x^2 + 3 = 0\). This gives no real solutions, meaning there are no x-intercepts.

Step 3: Analyze End Behavior

Since the degree of the numerator is greater than the degree of the denominator, this function's end behavior will resemble \(f(x) = x\). Specifically, as \(x \to \infty\), \(f(x) \to \infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).

Step 4: Sketch the Graph

Draw the vertical asymptote at \(x = 4\). Plot the y-intercept at \( (0, -\frac{3}{4}) \). Since there are no x-intercepts, the graph will not cross the x-axis. The graph approaches the vertical asymptote but doesn't cross it and follows the end behavior analysis, rising on the right side of the asymptote and falling on the left side.

Key Concepts

Asymptotes

When dealing with rational functions, asymptotes are lines that the graph of the function cannot cross. They help us understand the shape and behavior of the graph. For the function \( f(x) = \frac{2x^2 + 3}{x - 4} \), we identify two types of asymptotes: vertical and horizontal.

**Vertical Asymptotes:**
To find these, we set the denominator to zero and solve the equation. In this case, \( x - 4 = 0 \) gives us the vertical asymptote at \( x = 4 \). The graph will approach this line but never touch or cross it.

**Horizontal Asymptotes:**
These depend on the degrees of the numerator and the denominator. Here, the numerator's degree (2) is greater than the denominator's degree (1), indicating that there is no horizontal asymptote for this function.

Understanding asymptotes is crucial as they define boundaries and behavior patterns the graph will follow. They play a significant role in shaping the graph of any rational function.

Intercepts

Intercepts are the points where the graph crosses the axes. They provide key reference points for sketching the graph. Let's explore how to find both y-intercepts and x-intercepts for \( f(x) = \frac{2x^2 + 3}{x - 4} \).

**Y-intercept:**
To find the y-intercept, substitute \( x = 0 \) into the function. This gives us \( f(0) = \frac{3}{-4} = -\frac{3}{4} \). So the y-intercept is at \( (0, -\frac{3}{4}) \). This tells us the point where the graph crosses the y-axis.

**X-intercepts:**
These occur where the numerator of the function equals zero. Solve \( 2x^2 + 3 = 0 \); however, this equation has no real solutions. Thus, there are no x-intercepts for this graph. The absence of x-intercepts means the graph will not touch or cross the x-axis at any point.

Intercepts are essential as they help anchor the graph in the coordinate system and provide crucial points of reference.

End Behavior

End behavior describes how the graph behaves as the input values become very large (positive or negative). This helps predict the overall trend of the graph for the function \( f(x) = \frac{2x^2 + 3}{x - 4} \).

**Understanding the End Behavior:**
Since the degree of the numerator (2) is greater than that of the denominator (1), the graph's end behavior mirrors the leading term in the numerator when simplified. In this case, the end behavior resembles that of \( f(x) = x \) as \( x \to \infty \) and \( x \to -\infty \).

• As \( x \) approaches infinity, \( f(x) \) also approaches infinity, meaning the graph will rise indefinitely.
• Conversely, as \( x \) approaches negative infinity, \( f(x) \) will also tend toward negative infinity, causing the graph to fall without bound.

Recognizing end behavior is key to understanding and predicting the function's behavior far away from the origin, ensuring you can sketch more accurate graphs without a calculator.