Problem 12 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

Graph has a vertical asymptote at \(x = \frac{7}{3}\), horizontal asymptote at \(y = \frac{5}{3}\), x-intercept at \(-\frac{3}{5}\), and y-intercept at \(-\frac{3}{7}\). Sketch based on these details.

Step 1: Identify Domain

First, determine the domain of the function. The function is undefined where the denominator equals zero. Setting the denominator \(3x-7 = 0\), solve for \(x\). \[3x = 7\] \[x = \frac{7}{3}\]. The domain is all real numbers except \(x = \frac{7}{3}\).

Step 2: Determine Asymptotes

**Vertical Asymptote:** The vertical asymptote occurs where the denominator is zero. So, there is a vertical asymptote at \(x = \frac{7}{3}\).**Horizontal Asymptote:** Check the degrees of the polynomial in the numerator and the denominator. Both are degree 1, so the horizontal asymptote can be found using the leading coefficients: \(y = \frac{5}{3}\).

Step 3: Calculate and Plot Intercepts

**Y-intercept:** Set \(x = 0\) to find the y-intercept. \[f(0) = \frac{5(0)+3}{3(0)-7} = -\frac{3}{7}\]**X-intercept:** Set \(f(x) = 0\) and solve for \(x\), which occurs when the numerator is zero: \[5x + 3 = 0\] \[x = -\frac{3}{5}\].

Step 4: Analyze End Behavior

As \(x\) approaches infinity or negative infinity, \(f(x)\) approaches the horizontal asymptote \(y = \frac{5}{3}\). This is due to the degrees of the numerator and the denominator being equal and dominated by their leading coefficients.

Step 5: Sketch the Graph

Combine all findings:- Draw a vertical dashed line at \(x = \frac{7}{3}\) for the vertical asymptote.- Draw a horizontal dashed line at \(y = \frac{5}{3}\) for the horizontal asymptote.- Plot the intercepts \((-\frac{3}{5}, 0)\) and \((0, -\frac{3}{7})\).- Draw the curve, approaching the vertical asymptote without crossing it, and ensuring it approaches the horizontal asymptote at the extreme ends of the graph.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are vertical lines that a graph approaches but never actually touches or crosses. They occur in a rational function at values of \(x\) which make the denominator zero, provided the numerator is not also zero at those points. When the function becomes undefined due to dividing by zero, you will likely find a vertical asymptote there.

For the function \(f(x) = \frac{5x+3}{3x-7}\), we identify the vertical asymptote by setting the denominator equal to zero:

• Find where \(3x - 7 = 0\).
• Solve for \(x\): \(3x = 7\) or \(x = \frac{7}{3}\).

This means there is a vertical asymptote at \(x = \frac{7}{3}\). As \(x\) approaches this value, the function's value will increase to infinity or decrease to negative infinity, depending on the direction of approach. Always approach vertical asymptotes in a smooth curve that gets infinitely closer but never touches the line.

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of a function approaches as \(x\) approaches positive or negative infinity. In rational functions, these asymptotes help indicate the end behavior of the function.

To determine the horizontal asymptote of \(f(x) = \frac{5x+3}{3x-7}\), observe the degrees of the polynomials in the numerator and denominator:

• The numerator, \(5x+3\), and the denominator, \(3x-7\), both have a degree of 1.
• When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
• Thus, the horizontal asymptote is \(y = \frac{5}{3}\).

As \(x\) increases or decreases without bound, the values of \(f(x)\) draw nearer to \(y = \frac{5}{3}\). The graph will flatten out and hover around this line at extreme values of \(x\). However, keep in mind it will never actually cross this horizontal line in a basic rational function graph.

Intercepts in Graphing

Intercepts are key points where a graph crosses the axes. There are two types: the y-intercept, where the graph crosses the y-axis, and the x-intercept, where it crosses the x-axis. These points are crucial for sketching an accurate graph of the function.

For the rational function \(f(x) = \frac{5x+3}{3x-7}\), we can find the intercepts as follows:

• Y-intercept: Set \(x = 0\) and solve for \(f(x)\):
• \(f(0) = \frac{5(0)+3}{3(0)-7} = -\frac{3}{7}\).
• This gives the point \((0, -\frac{3}{7})\), where the graph crosses the y-axis.
• X-intercept: Set \(f(x) = 0\) and solve for \(x\), which occurs when the numerator equals zero:
• \(5x + 3 = 0\).
• Solve for \(x\): \(x = -\frac{3}{5}\).
• This gives the point \((-\frac{3}{5}, 0)\), where the graph crosses the x-axis.

These intercepts provide anchor points for drawing the graph, serving as crucial references for the shape and position of the curve on the coordinate plane.