Problem 10 - Precalculus with Calculus Previews

Question

Find the vertical and horizontal asymptotes for the graph of the given rational function. Find \(x\) - and \(y\) -intercepts of the graph. Sketch the graph of \(f\). $$ f(x)=\frac{2 x-3}{x} $$

Step-by-Step Solution

Answer

Vertical asymptote: \(x = 0\); horizontal asymptote: \(y = 2\); \(x\)-intercept: \(\left(\frac{3}{2}, 0\right)\); no \(y\)-intercept.

Step 1: Identify the Vertical Asymptote

To find the vertical asymptote, set the denominator of the function equal to zero and solve for \(x\). The function is \(f(x)=\frac{2x-3}{x}\). Set \(x = 0\) and solve: \[ x = 0 \]So, the vertical asymptote is \(x = 0\).

Step 2: Identify the Horizontal Asymptote

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Both are of degree 1, so the horizontal asymptote is found by dividing the leading coefficients. Here, both coefficients are 2 (numerator) and 1 (denominator):\[ y = \frac{2}{1} = 2 \]So, the horizontal asymptote is \(y = 2\).

Step 3: Determine the x-intercept

To find the \(x\)-intercept, set \(f(x) = 0\). Simplify:\[ \frac{2x - 3}{x} = 0 \]So, the numerator \(2x - 3 = 0\). Solve for \(x\):\[ 2x = 3 \quad \Rightarrow \quad x = \frac{3}{2} \]Thus, the \(x\)-intercept is \(\left(\frac{3}{2}, 0\right)\).

Step 4: Determine the y-intercept

To find the \(y\)-intercept, substitute \(x = 0\) in \(f(x)\). However, since \(x = 0\) makes the denominator zero, there is no \(y\)-intercept. The graph does not intersect the \(y\)-axis.

Step 5: Sketch the graph

Using the information from the previous steps, sketch the graph:1. Plot the vertical asymptote at \(x = 0\).2. Plot the horizontal asymptote at \(y = 2\).3. Mark the \(x\)-intercept at \(\left(\frac{3}{2}, 0\right)\).4. Since \(f(x)\) is not defined at \(x = 0\), the graph approaches the vertical asymptote without crossing it. The graph will move towards the horizontal asymptote as \(x\) goes to both \(+\infty\) and \(-\infty\).

Key Concepts

Vertical Asymptote

Vertical asymptotes in a rational function occur where the denominator equals zero, since division by zero is undefined. When you encounter a vertical asymptote, the graph of the function will approach this line but will never actually touch or cross it.

For the given function, \( f(x) = \frac{2x - 3}{x} \), we set the denominator \( x \) equal to zero to find the vertical asymptote:

• \( x = 0 \)

Thus, there is a vertical asymptote at \( x = 0 \). This means the graph will approach this line but will not intersect it.
Note that at \( x = 0 \), the function is undefined, adding a key characteristic of vertical asymptotes where the graph sharply turns upwards or downwards.

Horizontal Asymptote

Horizontal asymptotes give us insight into the behavior of a function as \( x \) approaches positive or negative infinity. It often reflects how the outputs (or \( y \)-values) stabilize, or level out, over vast ranges on the graph.

For the function \( f(x) = \frac{2x - 3}{x} \), we identify that both the numerator and the denominator have the same degree, which is 1. This means for horizontal asymptotes, we divide the leading coefficients:

• The leading coefficient in the numerator is 2.
• The leading coefficient in the denominator is 1.
• Thus, the horizontal asymptote is at \( y = \frac{2}{1} = 2 \).

The graph of the function will approach this line as \( x \) tends towards infinity in either direction. It's like a ceiling or floor that the graph gets close to but doesn't cross.

x-intercept

An \( x \)-intercept is the point where the graph of the function crosses the \( x \)-axis, indicating that at this point, \( y = 0 \). To find an \( x \)-intercept of a rational function, we set \( f(x) = 0 \), which translates to solving the equation where the numerator equals zero.

For the function \( f(x) = \frac{2x - 3}{x} \):

• Set numerator \( 2x - 3 = 0 \).
• Solve for \( x \): \( 2x = 3 \), resulting in \( x = \frac{3}{2} \).

Hence, the \( x \)-intercept is \( \left( \frac{3}{2}, 0 \right) \). On the graph, this point is where the curve crosses the \( x \)-axis, confirming where the output value is zero.

y-intercept

A \( y \)-intercept is the point on a graph where the function crosses the \( y \)-axis, or in other words, where \( x = 0 \). For many functions, this point is where we substitute \( x \) with zero in the equation \( f(x) \).

For the rational function \( f(x) = \frac{2x - 3}{x} \), let's substitute \( x = 0 \):

• Substitution: \( f(0) = \frac{2(0) - 3}{0} = \frac{-3}{0} \).
• This is undefined since division by zero is not permissible.

Therefore, there is no \( y \)-intercept for this function. The graph does not pass through the \( y \)-axis. This tells us that the function has a vertical asymptote at \( x = 0 \) instead.