Problem 494 - Precalculus

Question

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$q(x)=\frac{x-5}{3 x-1}$$

Step-by-Step Solution

Answer

Horizontal intercept at (5,0); vertical intercept at (0,5); vertical asymptote at \(x=\frac{1}{3}\); horizontal asymptote at \(y=\frac{1}{3}\).

Step 1: Find the Horizontal Intercept

To find the horizontal intercept of the function, set the numerator equal to zero and solve for \(x\): \(x-5=0\). Solving this gives \(x=5\). Thus, the horizontal intercept is at \((5,0)\).

Step 2: Find the Vertical Intercept

The vertical intercept is obtained by evaluating the function at \(x=0\). Substitute \(x=0\) into the function: \(q(0) = \frac{0-5}{3(0)-1} = \frac{-5}{-1} = 5\). Therefore, the vertical intercept is at \((0,5)\).

Step 3: Find the Vertical Asymptote

The vertical asymptote occurs where the denominator is zero. Set \(3x-1=0\) and solve for \(x\): \(3x=1\), so \(x=\frac{1}{3}\). Therefore, the vertical asymptote is the line \(x=\frac{1}{3}\).

Step 4: Find the Horizontal Asymptote

For rational functions where both numerator and denominator are linear, the horizontal asymptote is determined by the leading coefficients of the terms. The leading term of the numerator is \(x\) and of the denominator is \(3x\). The horizontal asymptote is the line \(y=\frac{1}{3}\), which is the ratio of the leading coefficients \(\frac{1}{3}\).

Key Concepts

Horizontal Intercept

When examining the graph of a rational function, one of the key intercepts to find is the horizontal intercept. To find the horizontal intercept of a rational function, such as \( q(x) = \frac{x-5}{3x-1} \), we set the numerator equal to zero and solve for \( x \).

• For \( q(x) \), this means solving \( x - 5 = 0 \).
• The solution is \( x = 5 \), meaning the graph crosses the x-axis at the point \((5,0)\).
• This point is valuable as it shows where the function equals zero.

The horizontal intercept is crucial as it helps us identify one of the locations where the y-value is zero, important for sketching the graph.

Vertical Intercept

Another vital detail for graph sketching is finding the vertical intercept. The vertical intercept indicates where the graph crosses the y-axis. This is found by evaluating the function at \( x = 0 \). For our example function, \( q(0) = \frac{0-5}{3(0)-1} \), simplifying to \( q(0) = 5 \).

• This gives a vertical intercept at \((0, 5)\), showing where the function touches the y-axis.
• The x-coordinate is always zero for the vertical intercept, while the y-coordinate is the calculated value.

Knowing the vertical intercept aids in understanding where the function begins in terms of y-values, contributing to our graph's shape.

Graph Sketching

Sketching the graph of a rational function involves determining several key features beyond intercepts. These include asymptotes which influence the graph's behavior. To accurately sketch \( q(x) \), consider both vertical and horizontal asymptotes:

• Vertical Asymptote: Set the denominator equal to zero, \( 3x-1 = 0 \), giving \( x = \frac{1}{3} \).
• Horizontal Asymptote: Compare the leading coefficients, \( y = \frac{1}{3} \).

These asymptotes indicate lines the graph approaches but never reaches, crucial for understanding the graph's end behavior. The interplay of intercepts and asymptotes helps sketch a curve that accurately represents \( q(x) \), offering insights into its growth or decay as x-values increase or decrease.

Rational Functions

Rational functions, like \( q(x) = \frac{x-5}{3x-1} \), have unique properties making them intriguing for study and graphing. They are ratios of polynomials, which means their behavior is deeply linked to the polynomials in the numerator and denominator.

• They can have discontinuities called asymptotes where the function is undefined.
• Key features include determining intercepts and asymptotes which frame the graph.
• Understanding how they behave near these points is essential for comprehensive graph analysis.

They provide valuable practice in polynomial division and equation solving, as each function's graph offers unique challenges and insight into broader mathematical concepts.