Problem 45 - Algebra and Trigonometry

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$ r(x)=\frac{18}{(x-3)^{2}} $$

Step-by-Step Solution

Answer

Y-intercept: (0, 2); Vertical asymptote at \( x = 3 \); Horizontal asymptote at \( y = 0 \); Domain: \( x \neq 3 \); Range: \( y > 0 \).

Step 1: Identify the Intercepts

For the y-intercept, set \( x = 0 \): \( r(0) = \frac{18}{(0-3)^2} = \frac{18}{9} = 2 \). So, the y-intercept is \((0, 2)\). Rational functions have no x-intercepts if the numerator is never zero, which is the case here.

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero. Set \((x-3)^2 = 0\), giving \( x = 3 \). Thus, there is a vertical asymptote at \( x = 3 \).

Step 3: Find Horizontal Asymptotes

Since the degree of the numerator (0) is less than the degree of the denominator (2), the horizontal asymptote is at \( y = 0 \).

Step 4: Determine the Domain

The domain is all real numbers except where the denominator is zero. Thus, the domain is \( x \in \mathbb{R} \setminus \{3\} \).

Step 5: Sketch the Graph

The graph is a hyperbola with a vertical asymptote at \( x = 3 \) and a horizontal asymptote at \( y = 0 \). It is reflected across the y-axis and will approach these asymptotes as it moves away from them. Use a graphing device to confirm this visual.

Step 6: State the Range

The range is all positive real numbers, \( y > 0 \), because the function never reaches 0 or negative values.

Key Concepts

Intercepts

Intercepts are points where the graph of a function crosses the axes. For rational functions like \( r(x) = \frac{18}{(x-3)^2} \), finding intercepts requires evaluating the function at specific values.

• Y-Intercept: To find where the graph crosses the y-axis, set \( x = 0 \). Here, \( r(0) = \frac{18}{(0-3)^2} = 2 \), yielding a y-intercept at the point \((0, 2)\).
• X-Intercepts: These occur where the graph crosses the x-axis. However, since the numerator (18) is never zero, the function \( r(x) \) has no x-intercepts.

Understanding these intercepts helps in sketching the graph and seeing where it interacts with the axes.

Vertical Asymptotes

Vertical asymptotes occur where a function approaches infinity as it nears a certain x-value. For the function \( r(x) = \frac{18}{(x-3)^2} \), vertical asymptotes are found where the denominator equals zero.
- Set \( (x-3)^2 = 0 \), which simplifies to \( x = 3 \). Therefore, there is a vertical asymptote at \( x = 3 \).This means that as x approaches 3, the values of \( r(x) \) grow indefinitely. Recognizing vertical asymptotes is crucial, as they highlight exploding behavior in the graph and must not be included in the domain.

Horizontal Asymptotes

Horizontal asymptotes indicate the value that a function approaches as x moves towards infinity or negative infinity. To find horizontal asymptotes in the function \( r(x) = \frac{18}{(x-3)^2} \), compare the degrees of the numerator and denominator.
- With the degree of the numerator being 0 (since it's a constant) and the denominator being 2, the horizontal asymptote is \( y = 0 \).This implies that as x becomes very large or very small, the value of \( r(x) \) gets closer to zero, but never actually reaches it. Horizontal asymptotes guide us in understanding the end-behavior of rational functions.

Domain

The domain of a function describes all possible x-values that can be input into the function without causing undefined expressions. In rational functions, a common cause of undefined values is division by zero.
- For \( r(x) = \frac{18}{(x-3)^2} \), the only point where the denominator equals zero is at \( x = 3 \).Thus, the domain of this function is all real numbers except for 3, expressed as \( x \in \mathbb{R} \setminus \{3\} \). Understanding the domain helps avoid errors in evaluating the function and in predicting the behavior of its graph.

Range

The range of a function is the set of possible output values (y-values) it can produce. For \( r(x) = \frac{18}{(x-3)^2} \), which is always positive and approaches infinity as x nears the vertical asymptote \( x = 3 \), the range can be determined as follows:
- Since the function values are only positive and never reach zero, the range is all positive real numbers, denoted as \( y > 0 \).This information about the range indicates that the graph is located entirely above the x-axis, helping visualize the graph's general shape and positioning.