Problem 60 - Calculus of a Single Variable

Question

Sketching a Graph In Exercises \(59-74\) , sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. Then use a graphing utility to verify your result. $$ y=\frac{x-4}{x-3} $$

Step-by-Step Solution

Answer

The y-intercept is at (0, \(\frac{4}{3}\)). The x-intercept is at (4,0). The vertical asymptote is at \(x = 3\). The slant asymptote is the line \(y = x - 1\).

Step 1: Compute the y-intercept

To find the y-intercept, we must set \(x = 0\) in the equation: \(y = \frac{0 - 4}{0 - 3} = \frac{4}{3}\). So the y-intercept is at (0, \(\frac{4}{3}\))

Step 2: Compute the x-intercept

To find the x-intercept, we must set \(y = 0\) in our equation and solve for \(x\). Given \(y = 0 , x - 4 = 0\). Therefore \(x = 4\). So, the x-intercept is at (4,0).

Step 3: Compute the asymptotes

A vertical asymptote occurs when the denominator of a rational function is equal to zero. Setting \(x - 3 = 0\), we find \(x = 3\) to be the vertical asymptote. There is no horizontal asymptote for this function because the degree of the numerator is equal to the degree of the denominator; instead, there is an oblique or slant asymptote. This can be found by performing the long division of the numerator by the denominator which results in \(y = x - 1\) as the equation of the slant asymptote.

Step 4: Check for symmetry

If the function is even, the graph will be symmetric with respect to the y-axis; if it is odd, it will be symmetric with respect to the origin. This function is not even (because it doesn’t satisfy \(f(x) = f(-x)\)) and it’s not odd (because it doesn’t satisfy \(f(x) = -f(-x)\)), so the graph will not be symmetric.

Step 5: Sketch the graph

Now, plot the x-intercepts and y-intercepts. Also draw the vertical asymptote at \(x = 3\) and the slant asymptote at \(y = x - 1\). Connect these points carefully, considering the asymptotes, to get the final sketch of the graph. Be sure to note that the graph approaches the asymptotes but never crosses them.

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches or crosses. They provide insights into the behavior of a function as the variable approaches specific values. For the function \( y = \frac{x-4}{x-3} \), there are two types of asymptotes to consider:

• Vertical Asymptote: This occurs when the denominator of the rational function is zero. Here, by setting \( x-3=0 \), we find \( x=3 \) is a vertical asymptote. As \( x \) approaches 3, the function tends towards infinity.


• Slant (Oblique) Asymptote: When the degrees of the numerator and denominator are the same, as in this function, there is no horizontal asymptote, but a slant asymptote exists. Performing long division on \( \frac{x-4}{x-3} \) results in \( y = x-1 \). The graph will tend close to this line as \( x \) goes to either extremely large positive or negative values.

Understanding these asymptotes helps you predict and sketch your graph. Remember, the graph will approach these lines but won't cross vertical ones.

Symmetry

Symmetry in graphing refers to whether a graph mirrors itself in some way. Symmetry can make graph sketching simpler by reducing the graphing work needed. For rational functions such as \( y = \frac{x-4}{x-3} \), we check for symmetry about the y-axis and the origin.

• Even Function (Symmetry with respect to the y-axis): If a function satisfies \( f(x) = f(-x) \), it is even. For \( y = \frac{x-4}{x-3} \), replace \( x \) with \( -x \) and the function becomes \( \frac{-x-4}{-x-3} \), which does not equal the original function, \( \frac{x-4}{x-3} \). Therefore, it is not symmetric about the y-axis.


• Odd Function (Symmetry with respect to the origin): If \( f(x) = -f(-x) \), the function is odd. Again testing this with \( y = \frac{x-4}{x-3} \), it fails to satisfy the condition for being an odd function as well. Thus, this function has no symmetry.

Without such symmetry, each section of the graph must be drawn separately, keeping in mind the behavior indicated by asymptotes and intercepts.

x-intercept

The x-intercept of a function is the point where the graph crosses the x-axis. It occurs when \( y = 0 \). For this rational function, \( y = \frac{x-4}{x-3} \), setting the numerator equal to zero determines this intercept.

• To find the x-intercept, solve \( x-4=0 \). Thus, \( x=4 \) is the x-intercept. This tells us that the graph will cross the x-axis at the point (4,0).

Knowing the x-intercept is vital for sketching the graph, as it provides a specific point through which the graph passes. Incorporate this point into your sketch, and ensure that this intercept aligns with the determination of asymptotes and symmetry, to create a clear representation of the function's behavior.

y-intercept

The y-intercept is a key feature of any graph. It's the point where the graph crosses the y-axis, meaning it occurs when \( x = 0 \). For your function \( y = \frac{x-4}{x-3} \), plugging in \( x=0 \) will lead to the y-intercept.

• Calculate the y-intercept by evaluating \( y = \frac{0-4}{0-3} = \frac{4}{3} \). Therefore, the y-intercept is (0, \( \frac{4}{3}\)).

This y-intercept provides another anchor point for your graph. When sketching, mark and label this point clearly on the y-axis. Both x- and y-intercepts give essential checkpoints to ensure the accuracy of your graph's shape and direction, helping in visualizing how the function behaves across different segments.