Problem 18 - Calculus Early Transcendentals

Question

Graphing rational functions Use the guidelines of this section to make a complete graph of \(f\) $$f(x)=\frac{2 x-3}{2 x-8}$$

Step-by-Step Solution

Answer

Answer: The vertical asymptote is at \(x=4\), the horizontal asymptote is at \(y=1\), the x-intercept is at \((\frac{3}{2}, 0)\), and the y-intercept is at \((0, \frac{3}{8})\).

Step 1: Determine the vertical asymptote

To find any vertical asymptotes, first set the denominator of the rational function equal to zero and solve for x: $$2x - 8 = 0$$ Now, solve for x: $$x = \frac{8}{2} = 4$$ There is one vertical asymptote at \(x=4\).

Step 2: Determine the horizontal or oblique asymptote

To find horizontal or oblique asymptotes, we compare the degrees of the numerator and the denominator: - If the degree of the numerator is less than the denominator, there is a horizontal asymptote at \(y=0\). - If the degrees are equal, divide the leading coefficients to find the horizontal asymptote. - If the degree of the numerator is greater than the denominator by 1, we have an oblique asymptote. In our case, both the numerator and the denominator have the same degree (degree 1). Therefore, we divide the leading coefficients: $$\frac{2}{2} = 1$$ There is a horizontal asymptote at \(y=1\).

Step 3: Find the x-intercept and y-intercept

To find the x-intercept, set \(f(x)\) equal to zero and solve for x: $$\frac{2x-3}{2x-8} = 0$$ Since any fraction equals zero only if the numerator is zero, we have: $$2x - 3 = 0$$ Now, solve for x: $$x = \frac{3}{2}$$ The x-intercept is at \((\frac{3}{2}, 0)\). To find the y-intercept, set \(x\) equal to zero in the function and solve for \(f(0)\): $$f(0) = \frac{2(0)-3}{2(0)-8} = \frac{-3}{-8} = \frac{3}{8}$$ The y-intercept is at \((0, \frac{3}{8})\).

Step 4: Use the information obtained to draw the complete graph

Now, we have all the information needed to graph the rational function: 1. Vertical asymptote at \(x=4\). 2. Horizontal asymptote at \(y=1\). 3. x-intercept at \((\frac{3}{2}, 0)\). 4. y-intercept at \((0, \frac{3}{8})\). Plot these points and asymptotes on the coordinate plane. Make sure to draw dashed lines for the asymptotes, which show the boundaries to which the graph approaches. Sketch the graph in each region between the asymptotes, making sure to pass through the x-intercept and y-intercept.

Key Concepts

Vertical Asymptote

In graphing rational functions, vertical asymptotes are essential as they show where the function goes towards infinity. To find a vertical asymptote, check where the denominator of your rational function equals zero. This is because a function becomes undefined when you try to divide by zero. For the function \( f(x) = \frac{2x-3}{2x-8} \), set the denominator \(2x-8\) to zero:

• Equation: \(2x - 8 = 0\)
• Solve for \(x: x = 4\)

Thus, there is a vertical asymptote at \(x = 4\). When graphing, use a dashed line to demonstrate the vertical asymptote, showing that the graph will shoot up or down as it gets close to \(x = 4\). This represents a boundary that the graph never crosses.

Horizontal Asymptote

Horizontal asymptotes describe the behavior of a function as \(x\) approaches infinity or negative infinity. They indicate the y-value that the function is approaching. For the rational function \( f(x) = \frac{2x-3}{2x-8} \), compare the degrees of the numerator and the denominator. Both have the same degree, which is 1:

• Numerator degree: 1 (\(2x\))
• Denominator degree: 1 (\(2x\))
• If degrees are equal, divide the leading coefficients.
• \(\frac{2}{2} = 1\)

Hence, there is a horizontal asymptote at \(y = 1\). On the graph, draw a dashed horizontal line at \(y = 1\), showing that as \(x\) heads towards large values, the function approaches but never quite reaches \(y = 1\).

X-Intercept

Finding the x-intercept is easy if you remember that this is where the function crosses the x-axis. At these points, \( y = 0 \). For the function \( f(x) = \frac{2x-3}{2x-8} \), set the entire function equal to zero and solve for \(x\):

• \(\frac{2x-3}{2x-8} = 0\)
• The fraction is zero only when the numerator is zero: \(2x - 3 = 0\)
• Solve for \(x: x = \frac{3}{2}\)

Thus, the x-intercept is at \(\left( \frac{3}{2}, 0 \right)\). On your graph, place a point directly on the x-axis at this coordinate. It's where the function touches the x-axis before making its way upward or downward towards an asymptote.

Y-Intercept

The y-intercept is crucial as it's the point where the graph crosses the y-axis. At this juncture, \( x = 0 \). To find the y-intercept for \( f(x) = \frac{2x-3}{2x-8} \), substitute \(x=0\) into the function:

• \(f(0) = \frac{2(0)-3}{2(0)-8} = \frac{-3}{-8}\)
• Simplify to find the y-intercept: \(\frac{3}{8}\)

Therefore, the y-intercept is at \((0, \frac{3}{8})\). Plot this point on the y-axis. This will show where the function begins its path on the graph before bending towards the asymptotes. With both intercepts marked, the trajectory of the rational function becomes clearer.