91Ó°ÊÓ

In the world of rational functions, vertical asymptotes represent the values of \( x \) where the function tends toward infinity. Imagine them like invisible barriers that the graph can approach but never really touch. To find them, look at the denominator of the rational function. For our function, \( f(x) = \frac{x(x+2)}{2x-1} \), simply set the denominator equal to zero:
\( 2x - 1 = 0 \).
Once solved, this gives \( x = \frac{1}{2} \), so here lies the vertical asymptote. It’s crucial to ensure that at this \( x \)-value, the numerator isn't also zero; otherwise, it would result in a hole instead of an asymptote. This makes \( x = \frac{1}{2} \) an ideal spot for the asymptote in this function.

• Vertical asymptotes are found by setting the denominator equal to zero.
• Ensure the numerator is not zero at this value.
• They indicate where the function can "break" or become undefined.

Horizontal asymptotes give insight into the behavior of a graph as \( x \) approaches infinity or negative infinity. For any rational function like \( f(x) = \frac{x^2 + 2x}{2x - 1} \), it's all about comparing the degrees of the polynomials in the numerator and denominator. Here, both have the degree 2. In such cases, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
Therefore, the horizontal asymptote is \( y = \frac{1}{2} \). This signifies that as \( x \) heads towards positive or negative infinity, the graph will level off around this value.

• Compare the degrees of the numerator and denominator.
• If degrees are equal, divide the leading coefficients for the asymptote.
• Defines the boundary the function approaches but never fully reaches horizontally.

Intercepts offer principal points where the graph crosses the axes, providing essential markers for sketching. For \( x \)-intercepts, set the entire numerator to zero. In our function's case, \( x(x+2) = 0 \) boils down to \( x = 0 \) and \( x = -2 \). Thus, these are points where the graph intersects the \( x \)-axis.
For the \( y \)-intercept, substitute \( x = 0 \) into the function, yielding \( f(0) = 0 \). It touches the \( y \)-axis at \( (0,0) \), meaning the point where the function commences its journey.

• \( x \)-intercepts occur where the numerator equals zero.
• The \( y \)-intercept occurs by evaluating the function at \( x = 0 \).
• These points provide tangible touchpoints for the graph’s trajectory on axes.

Armed with the calculated asymptotes and intercepts, we can proceed to sketch the graph of our function \( f(x) = \frac{x^2 + 2x}{2x-1} \). Start by plotting the vertical asymptote at \( x = \frac{1}{2} \) and the horizontal asymptote at \( y = \frac{1}{2} \). Remember, these lines are like guidelines only—it is forbidden for the graph to cross them.
Next, place the \( x \)-intercepts at \( (0,0) \) and \( (-2,0) \), which display where the graph will touch the \( x \)-axis. Don’t miss locating the \( y \)-intercept also at \( (0,0) \), providing a clear starting point.
Sketch the graph, keeping in mind the behavior of the curves near the asymptotes—drawing closer yet never touching. The graph swoops smoothly between these key intercepts while respecting the boundary established by the asymptotes.

• Plot asymptotes and intercepts as reference points.
• Ensure the graph nears but refrains from crossing asymptotes.
• Use the intercepts to guide the curve's direction and shape.