Problem 39 - College Algebra

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. $$ p(x)=\frac{2 x-3}{x+4} $$

Step-by-Step Solution

Answer

Horizontal intercept: \(\left(\frac{3}{2}, 0\right)\), Vertical intercept: \(\left(0, \frac{-3}{4}\right)\), Vertical asymptote: \(x = -4\), Horizontal asymptote: \(y = 2\).

Step 1: Finding the Horizontal Intercept

Find the horizontal intercept by setting the numerator of the function to zero and solving for \(x\). The equation is \(2x - 3 = 0\). Solving this, \(2x = 3\) and \(x = \frac{3}{2}\). Thus, the horizontal intercept is \(\left(\frac{3}{2}, 0\right)\).

Step 2: Finding the Vertical Intercept

Find the vertical intercept by evaluating the function at \(x = 0\). Substituting \(0\) into \(p(x)\), we have \(p(0) = \frac{2(0) - 3}{0 + 4} = \frac{-3}{4}\). Thus, the vertical intercept is \(\left(0, \frac{-3}{4}\right)\).

Step 3: Finding the Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, provided it does not cancel with the numerator. Set the denominator \(x + 4= 0\) and solve for \(x\). We find \(x = -4\), so there is a vertical asymptote at \(x = -4\).

Step 4: Finding the Horizontal Asymptote

For rational functions where the degrees of the numerator and denominator are the same, the horizontal asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients. Here, \(a = 2\) and \(b = 1\), so the horizontal asymptote is \(y = 2\).

Step 5: Sketching the Graph

On a coordinate plane, plot the horizontal intercept at \(\left(\frac{3}{2}, 0\right)\), the vertical intercept at \(\left(0, \frac{-3}{4}\right)\). Draw the vertical asymptote as a dashed line at \(x = -4\) and the horizontal asymptote as a dashed line at \(y = 2\). Use these intercepts and asymptotes to guide the curve of the graph. The graph approaches the asymptotes as \(x\) approaches the asymptote values.

Key Concepts

Horizontal Intercepts

Horizontal intercepts of a rational function are found by setting the numerator equal to zero and solving for the variable. This is because a horizontal intercept, which is also known as an x-intercept, occurs where the graph crosses the x-axis, meaning the y value at that point is zero. For example, in the given function, \(p(x) = \frac{2x - 3}{x + 4}\), you would set \(2x - 3 = 0\).

• Solving for \(x\), you get \(2x = 3\), leading to \(x = \frac{3}{2}\).
• This tells you the function crosses the x-axis at \(\left(\frac{3}{2}, 0\right)\).

The horizontal intercept signifies where the numerator reaches zero. It's an essential part of graphing as it helps pinpoint one of the graph's critical points.

Vertical Asymptotes

Vertical asymptotes occur at the values of \(x\) that make the denominator zero, causing the function to approach infinity. These asymptotes represent the points where the function is undefined, indicating sharp breaks in the graph where it doesn’t exist.In \(p(x) = \frac{2x - 3}{x + 4}\), the denominator \(x + 4 = 0\) gives the vertical asymptote when solved:

• Set \(x + 4 = 0\), resulting in \(x = -4\).

Place a dashed line on the graph at \(x = -4\) as a boundary the graph will approach but never touch or intersect. It is a crucial guide for graph sketching because these lines suggest areas where the graph might flip direction or trend steeply towards infinity.

Graph Sketching

Graph sketching using intercepts and asymptotes helps to accurately depict the behavior of a rational function visually. Start by plotting all known points and lines determined from calculations:

• Horizontal intercept at \(\left(\frac{3}{2}, 0\right)\).
• Vertical intercept at \(\left(0, \frac{-3}{4}\right)\).
• Vertical asymptote as a dashed line at \(x = -4\).
• Horizontal asymptote as a dashed line at \(y = 2\).

With these elements in place, draw a smooth curve that approaches the vertical asymptote from the left and the right without ever crossing it, and approaches the horizontal asymptote as \(x\) goes toward positive or negative infinity. This approach mimics real-world data projection by showing trends and limits in a rational function’s graph.

Horizontal Asymptotes

Horizontal asymptotes in rational functions indicate the limit the function approaches as \(x\) trends toward infinity or negative infinity. For functions where numerator and denominator have the same degree, the horizontal asymptote is calculated by the ratio of their leading coefficients.For the function \(p(x) = \frac{2x - 3}{x + 4}\), the horizontal asymptote is:

• Leading coefficient of the numerator: 2
• Leading coefficient of the denominator: 1

This results in a horizontal asymptote of \(y = 2\). On the graph, this is shown as a dashed line which the function will approach but not necessarily touch as \(x\) heads towards ±∞. Horizontal asymptotes provide guidelines for the graph's eventual behavior, indicating long-term trends.