91Ó°ÊÓ

A vertical asymptote is a line where a graph approaches but never quite touches or crosses. For rational functions, vertical asymptotes occur where the denominator equals zero. In our example, the function is \( f(x) = \frac{4}{2x - 3} \).
To find the vertical asymptote, we set the denominator \(2x - 3\) to zero because this is where the function "blows up" or becomes undefined:
\[ 2x - 3 = 0 \]
Solving this simple algebraic equation results in:
\[ x = \frac{3}{2} \]
This means that at \( x = \frac{3}{2} \), the graph of \( f(x) \) will have a vertical asymptote.
An important thing to note is the behavior of the graph near a vertical asymptote. As \( x \) approaches \( \frac{3}{2} \) from the left, the graph will trend towards negative infinity, and as it approaches from the right, it trends towards positive infinity.

A reciprocal function is a type of rational function represented by \( f(x) = \frac{k}{x-a} \), where \(k\) is a constant and \(x - a\) forms the denominator.
For our function, \( f(x) = \frac{4}{2x - 3} \), the constant \(k\) is 4, and the function denominator is \(2x - 3\).
This puts us in a typical reciprocal function format with \(a = \frac{3}{2}\).
Reciprocal functions have important characteristics such as vertical asymptotes, which we've found at \(x = \frac{3}{2}\).
Additionally, such functions may also have horizontal asymptotes, often determined by comparing the degrees of numerator and denominator.
For this specific case, since the numerator is a constant and the denominator a simple linear term, the horizontal asymptote is the x-axis.

Graphing calculators are powerful tools that help visualize complex functions like reciprocal functions.
To graph \( f(x) = \frac{4}{2x - 3} \), you’ll need a graphical calculator that can input rational expressions.
Simply enter the function exactly as displayed in the calculator’s function input area.

• Observe how the graph approaches the vertical line at \(x = \frac{3}{2}\).
• Your graph will show a curve in different directions around this line.
• You can visually understand the behavior of \(f(x)\) around this critical point well.
• The graph will be in two sections: decreasing towards the vertical asymptote from the left and increasing from the right.

When studying this graph, note where the graph is above or below the x-axis, which aligns with the function's positivity and negativity discussed earlier.
This visual will confirm that \( f(x) > 0 \) for \( x > \frac{3}{2} \). The descriptive power of graphing calculators makes seeing mathematical concerns like asymptotic behavior straightforward.