To determine which graph corresponds to the rational function \( f(x) = \frac{x^2 + 3x + 2}{2x^2 - x - 2} \), we need to analyze the function's characteristics.
-
Factor the Function:
- The numerator \( x^2 + 3x + 2 \) factors to \( (x + 1)(x + 2) \).
- The denominator \( 2x^2 - x - 2 \) can be factored by finding the roots using the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-2)}}{2(2)} = \frac{1 \pm \sqrt{1 + 16}}{4} = \frac{1 \pm \sqrt{17}}{4} \]
- Let's denote the roots as \( r_1 \) and \( r_2 \).
-
Vertical Asymptotes:
- The vertical asymptotes occur where the denominator is zero and the numerator is not zero. Thus, they will be at \( x = r_1 \) and \( x = r_2 \).
-
Horizontal Asymptotes:
- To find the horizontal asymptote, we compare the leading coefficients of the numerator and denominator. Since both the numerator and denominator have degree 2: \[ y = \frac{\text{leading coef of the numerator}}{\text{leading coef of the denominator}} = \frac{1}{2} \]
-
Finding Intercepts:
- The \( x \)-intercepts occur where \( f(x) = 0 \), which is where the numerator is zero: \( x = -1 \) and \( x = -2 \).
- The \( y \)-intercept occurs at \( f(0) = \frac{0 + 0 + 2}{0 - 0 - 2} = -1 \).
-
Analyzing Behavior:
- As \( x \to r_1 \) or \( x \to r_2 \), the function will go toward either positive or negative infinity (depending on the direction from which \( x \) approaches the vertical asymptotes).
- Additionally, near the horizontal asymptote at \( y = \frac{1}{2} \), the function will remain near that value as \( x \) moves away from 0.
Comparison with Graphs: From the descriptions given:
- The graph must have vertical asymptotes (dashed lines) at points determined by the roots of the denominator.
- It should also have a horizontal asymptote at \( y = \frac{1}{2} \).
- The x-intercepts at \( x = -1 \) and \( x = -2 \) should be visible.
- The behavior of the graph should match the increasing and decreasing nature around the calculated asymptotes.
Based on these characteristics, we can identify the correct graph among the provided options. The graph description that matches these characteristics should be selected.
Given the options you provided, the correct graph appears to be the one that features the vertical asymptotes at the correct x-values and the horizontal asymptote at \( y = \frac{1}{2} \).
Select the graph that best reflects these properties and closely approximates the analytical deductions made about the function \( f(x) = \frac{x^2 + 3x + 2}{2x^2 - x - 2} \).
1 answer