Problem 44 - 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. $$ s(x)=\frac{1-2 x}{2 x+3} $$

Step-by-Step Solution

Answer

x-intercept at \((\frac{1}{2}, 0)\), y-intercept at \((0, \frac{1}{3})\), vertical asymptote at \(x = -\frac{3}{2}\), horizontal asymptote at \(y = -1\); domain \(x \neq -\frac{3}{2}\), range \(y \neq -1\).

Step 1: Find the x-intercept

To find the x-intercept, set the numerator of the function equal to zero. The x-intercept occurs where \(1 - 2x = 0\). Solve for \(x\): \[ 1 - 2x = 0 \]\[ -2x = -1 \]\[ x = \frac{1}{2} \] Thus, the x-intercept is at \( (\frac{1}{2}, 0) \).

Step 2: Find the y-intercept

To find the y-intercept, substitute \(x = 0\) into the function. The y-intercept occurs at \( s(0) \):\[ s(0) = \frac{1 - 2(0)}{2(0)+3} = \frac{1}{3} \] Thus, the y-intercept is at \( (0, \frac{1}{3}) \).

Step 3: Determine Vertical Asymptote

To find vertical asymptotes, set the denominator equal to zero and solve for \(x\). The vertical asymptote occurs where \( 2x + 3 = 0 \):\[ 2x + 3 = 0 \]\[ 2x = -3 \]\[ x = -\frac{3}{2} \]Thus, there is a vertical asymptote at \( x = -\frac{3}{2} \).

Step 4: Determine Horizontal Asymptote

For rational functions, if the degree of the numerator is equal to the degree of the denominator (as is the case here), the horizontal asymptote is the ratio of the leading coefficients. Here, the leading coefficient for the numerator is \(-2\) and for the denominator is \(2\). Thus, the horizontal asymptote is \( y = -1 \).

Step 5: State the Domain

The domain of the function is all real numbers except where the denominator is zero, which we found to be \(x = -\frac{3}{2}\). Thus, the domain is \(x \in \mathbb{R}, x eq -\frac{3}{2}\).

Step 6: State the Range

The range is determined by the horizontal asymptote and the behavior of \(s(x)\). Since the horizontal asymptote is \(y = -1\) and no vertical restriction based on the numerator, the range is all real numbers except \(y = -1\). Thus, the range is \( y \in \mathbb{R}, y eq -1 \).

Key Concepts

Intercepts

Intercepts are the points where a graph crosses the axes. Rational functions like \( s(x) = \frac{1-2x}{2x+3} \) might have both x-intercepts and y-intercepts. - **X-Intercepts**: These are found by setting the numerator equal to zero. For our function, solve \( 1 - 2x = 0 \). This gives \( x = \frac{1}{2} \). Hence, the x-intercept is \( (\frac{1}{2}, 0) \). - **Y-Intercepts**: These are determined by substituting \( x = 0 \) in the function. For \( s(x) \), we get \( s(0) = \frac{1}{3} \). Therefore, the y-intercept is \( (0, \frac{1}{3}) \).Intercepts help us start understanding the behavior of the function on a graph.

Asymptotes

Asymptotes are lines that the graph of the function approaches but never touches. They are crucial for sketching the behavior of rational functions.- **Vertical Asymptotes** occur where the denominator is zero. In \( s(x) = \frac{1-2x}{2x+3} \), set \( 2x + 3 = 0 \) to find \( x = -\frac{3}{2} \). The graph will never touch this vertical line at \( x = -\frac{3}{2} \).- **Horizontal Asymptotes** suggest where a function stabilizes as \( x \) goes to infinity. Here, the degrees of numerator and denominator are the same. The horizontal asymptote is the ratio of leading coefficients, resulting in \( y = -1 \).Asymptotes behave like boundaries, guiding the direction of the graph without crossing them.

Domain and Range

The domain and range explain the limitations in values that \( x \) and \( y \) can take in the rational function.- **Domain**: This represents all potential x-values. For \( s(x) = \frac{1-2x}{2x+3} \), the only restriction is where the denominator zeroes out: \( x = -\frac{3}{2} \). Therefore, the domain is \( x \in \mathbb{R}, x eq -\frac{3}{2} \).- **Range**: This pertains to all possible y-values. Considering \( s(x) \)'s horizontal asymptote at \( y = -1 \), the function approaches this value but never attains it. Thus, the range is \( y \in \mathbb{R}, y eq -1 \).Comprehending domain and range is key to understanding the full scope and limitation of the function's graph.

Graphing Rational Functions

Graphing rational functions involves plotting various elements that paint a complete picture of the function's behavior.1. **Plot Intercepts**: Start with plotting x and y intercepts, which serve as anchor points on the graph.2. **Identify Asymptotes**: Draw the vertical asymptote \( x = -\frac{3}{2} \) and horizontal asymptote \( y = -1 \). These lines help in shaping the graph.3. **Behavior Analysis**: Observe how the graph approaches asymptotes, ensuring it never crosses.Using graphing tools or devices helps validate these plots, confirming your intercepts and asymptotes are accurately represented. Understanding each component makes the process of graphing rational functions systematic and intuitive.