Problem 7 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) $$f(x)=\frac{3}{x-4}$$

Step-by-Step Solution

Answer

The graph has a vertical asymptote at \( x=4 \) and a horizontal asymptote at \( y=0 \), with no x-intercept and a y-intercept at \((0, -\frac{3}{4})\).

Step 1: Identify Asymptotes

The function \( f(x) = \frac{3}{x-4} \) is a rational function. To find the vertical asymptote, set the denominator equal to zero: \( x - 4 = 0 \). Thus, the vertical asymptote is at \( x = 4 \). Since there are no polynomials added or subtracted, the horizontal asymptote for this function, when dividing constants, is \( y = 0 \).

Step 2: Determine Intercepts

To find the y-intercept, evaluate the function at \( x = 0 \): \( f(0) = \frac{3}{0-4} = -\frac{3}{4} \). Hence, the y-intercept is \((0, -\frac{3}{4})\). There is no x-intercept as \( f(x) \) does not equal zero for any value of \( x \).

Step 3: Analyze the Sign of the Function

Analyze the sign of \( f(x) = \frac{3}{x-4} \) in the intervals divided by the vertical asymptote \( x=4 \). For \( x<4 \), the function is negative (since \( x - 4 < 0 \)); for \( x>4 \), the function is positive (since \( x - 4 > 0 \)).

Step 4: Sketch the Graph

Using the information from the previous steps, sketch the graph. The graph approaches the vertical asymptote \( x = 4 \) from both sides and the horizontal asymptote \( y = 0 \) as \( x \rightarrow \pm\infty \). The function decreases toward \( -\infty \) for \( x<4 \) and increases toward \( +\infty \) for \( x>4 \).

Step 5: Determine the Behavior at Asymptotes

The graph of \( f(x) = \frac{3}{x-4} \) behaves differently as it approaches the vertical asymptote. As \( x \to 4^+ \), \( f(x) \to \infty \); as \( x \to 4^- \), \( f(x) \to -\infty \). This confirms the function's end behavior observed in the sketch.

Key Concepts

Understanding Asymptotes

Rational functions, like \( f(x) = \frac{3}{x-4} \), often feature asymptotes, which are lines that the graph approaches but never quite touches. These are critical for understanding the behavior of such functions. Asymptotes can be both horizontal and vertical, and each type tells us something different about the graph.

For our function, the **vertical asymptote** occurs where the denominator equals zero. Here, setting \( x - 4 = 0 \) gives \( x = 4 \). This is where the function is undefined, and the graph shoots off towards infinity as it gets closer to this line. To see this visually, as \( x \) approaches 4 from the left (\( x \to 4^- \)), \( f(x) \to -\infty \), and from the right (\( x \to 4^+ \)), \( f(x) \to +\infty \).

For the **horizontal asymptote**, we consider the end behavior as \( x \to \pm\infty \). In this function, there are no variables in the numerator, just a constant. Therefore, as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 0 \). Thus, the horizontal asymptote is \( y = 0 \), indicating the function approaches this line as \( x \) gets very large or very small.

Sketching the Graph

Graph sketching is about bringing all the separate pieces of information together to see the whole picture. It's like connecting the dots using what you've discovered about the function.

Start with the asymptotes: draw a dashed line for the vertical asymptote at \( x = 4 \) and a horizontal line at \( y = 0 \). These lines won't be crossed by the graph.

Next, identify the y-intercept. For \( f(x) = \frac{3}{x-4} \), substituting \( x = 0 \) gives us \( f(0) = -\frac{3}{4} \), thus the y-intercept is at \( (0, -\frac{3}{4}) \). Place a point there.

Observe the behavior in each interval divided by the vertical asymptote. For \( x < 4 \), the function is negative, meaning it drops down towards both the x-axis (horizontal asymptote) and the vertical asymptote. For \( x > 4 \), \( f(x) \) is positive, climbing upward towards the vertical asymptote and tending downwards towards \( y = 0 \) as \( x \to \infty \). Finally, sketch smooth curves that decrease to \(-\infty\) (left of \( x=4 \)) and increase to \(+\infty\) (right of \( x=4 \)).

Function Analysis Made Easy

Analyzing rational functions involves looking at both their algebraic properties and their graphical behavior. This deep dive allows us to fully understand how the function behaves across its domain.

With our function \( f(x) = \frac{3}{x-4} \), it is valuable to note zones where the function is positive or negative. These are divided by the vertical asymptote, with the critical point at \( x = 4 \).

- **Sign Analysis:** For \( x < 4 \), \( x-4 \) is negative, thus \( f(x) \) is negative. Conversely, for \( x > 4 \), \( x-4 \) becomes positive, making \( f(x) \) positive.- **Intercepts:** Since there are no zeros in the numerator, there are no x-intercepts. However, analyzing where the function crosses the y-axis gives us the y-intercept at \( (0, -\frac{3}{4}) \).

Combining these observations with the asymptotic behavior, one gains a complete view of the function's tendencies. Understanding these aspects helps predict not only the path of the graph but also its acceleration, direction, and limits.