Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$ f(x)=\frac{5 x+3}{3 x-7} $$

Step-by-Step Solution

Answer

The graph of \(f(x)\) has a vertical asymptote at \(x = \frac{7}{3}\), an oblique asymptote \(y = \frac{5}{3}\), and intercepts at \((0, -\frac{3}{7})\) and \((-\frac{3}{5}, 0)\).

Step 1: Identify the Asymptotes

A vertical asymptote occurs when the denominator is zero. Thus, set the denominator to zero: \[3x - 7 = 0\] Solving for \(x\): \[x = \frac{7}{3}\] This means there is a vertical asymptote at \(x = \frac{7}{3}\). There is no horizontal asymptote since the degrees of the numerator and denominator are the same, but there is an oblique asymptote. To find it, divide the coefficients of the highest degree terms: \[y = \frac{5}{3}\] This is the oblique asymptote.

Step 2: Find the Intercepts

For the y-intercept, set \(x = 0\) in \(f(x)\): \[f(0) = \frac{5(0) + 3}{3(0) - 7} = -\frac{3}{7}\] Thus, the y-intercept is \((0, -\frac{3}{7})\). For the x-intercept, set \(f(x) = 0\): \[\frac{5x + 3}{3x - 7} = 0\] This occurs when the numerator is zero: \[5x + 3 = 0\] Solving for \(x\): \[x = -\frac{3}{5}\] Thus, the x-intercept is \((-\frac{3}{5}, 0)\).

Step 3: Analyze the End Behavior

As \(x\) approaches infinity or negative infinity, the function \(f(x)\) approaches the oblique asymptote \(y = \frac{5}{3}\). This means that the graph will get closer to the line \(y = \frac{5}{3}\) as \(x\) moves away from the origin in either direction.

Step 4: Sketch the Graph

Plot the vertical asymptote at \(x = \frac{7}{3}\) using a dashed line and the oblique asymptote at \(y = \frac{5}{3}\) using a dashed line. Mark the intercepts at \((0, -\frac{3}{7})\) and \((-\frac{3}{5}, 0)\). The graph of \(f(x)\) will approach these asymptotes without touching or crossing them and will pass through the intercepts.

Key Concepts

Asymptotes

Asymptotes are invisible lines that rational functions approach but never actually touch or cross. They are very important in graphing rational functions as they help determine the overall shape and direction of the function.

• Vertical Asymptotes: These occur specifically where the denominator of a rational function is zero, making the function undefined. For the function given, the denominator is \(3x - 7\). Setting it to zero to find the vertical asymptote gives us: \[3x - 7 = 0\] Solving this equation, we find \(x = \frac{7}{3}\). So, there's a vertical asymptote at \(x = \frac{7}{3}\).
• Horizontal/Oblique Asymptotes: To determine the horizontal or oblique asymptote, we compare the degrees of the numerator and the denominator:
  • If the degree of the numerator is less than the degree of the denominator, there's a horizontal asymptote at \(y = 0\).
  • If the degrees are equal, the horizontal asymptote is at \(y = \text{coefficient of the numerator's highest power} \div \text{coefficient of the denominator's highest power}\), which for our function is \(y = \frac{5}{3}\).
  • If the numerator's degree is greater by one than the denominator's degree, like in our case, we find an oblique asymptote. This is further confirmed with the division: the oblique asymptote is \(y = \frac{5}{3}\).

These asymptotes guide us in predicting the behavior and the trajectory of the rational function as \(x\) values become very large or very small.

Intercepts

Intercepts are essential points where the function crosses the axes of a graph. Identifying intercepts provides crucial reference points for sketching the graph.

• Y-intercepts: These occur where the graph crosses the y-axis, meaning \(x = 0\). By substituting \(x = 0\) into the function \(f(x)\), we compute:\[f(0) = \frac{5(0) + 3}{3(0) - 7} = -\frac{3}{7}\]Therefore, the y-intercept is at the point \((0, -\frac{3}{7})\).
• X-intercepts: These happen where the graph touches or crosses the x-axis and occur when \(f(x) = 0\). To find the x-intercepts, set the numerator equal to zero because a fraction is zero only when its numerator is zero:\[5x + 3 = 0\]Solving for \(x\), we find:\[x = -\frac{3}{5}\]Thus, the x-intercept is at \((-\frac{3}{5}, 0)\).

These intercepts help us in effectively positioning the graph within its respective quadrants on the coordinate plane.

End Behavior

End behavior describes how the function behaves as \(x\) moves towards both positive and negative infinity. It shows us the graph's direction at its farthest edges, an integral part of sketching accurate graphs.

• Since the given function has an oblique asymptote at \(y = \frac{5}{3}\), the end behavior can be understood through this line. As \(x\) moves towards positive or negative infinity, the value of \(f(x)\) will approach \(y = \frac{5}{3}\) closely.
• Despite getting extremely near, the graph will never fully reach the asymptote, indicating that the function continues approaching the line throughout its extension in either direction.

This insight provides valuable information when drawing the full graph over large scales, ensuring it accurately reflects the function's overall tendency or path as it extends indefinitely.