To analyze the rational function \( f(x) = \frac{x^2 + 3x + 2}{2x^2 - x - 2} \), we first need to simplify it and find its characteristics, including vertical and horizontal asymptotes.
- Factoring the numerator and denominator:
- The numerator \( x^2 + 3x + 2 \) factors to \( (x + 1)(x + 2) \).
- The denominator \( 2x^2 - x - 2 \) factors to \( (2x + 3)(x - 2) \) (you can verify this by polynomial long division if necessary).
Thus, we can re-write \( f(x) = \frac{(x + 1)(x + 2)}{(2x + 3)(x - 2)} \).
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Finding vertical asymptotes:
- The vertical asymptotes occur where the denominator is zero (and the numerator is non-zero).
- Setting \( 2x + 3 = 0 \) gives \( x = -\frac{3}{2} \) and \( x - 2 = 0 \) gives \( x = 2 \).
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Finding horizontal asymptotes:
- Since the degree of the numerator (2) is equal to the degree of the denominator (2), we look at the leading coefficients. The horizontal asymptote is given by the ratio of the leading coefficients, which is \( \frac{1}{2} \).
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Behavior at critical points:
- As \( x \) approaches the vertical asymptote at \( x = 2 \), we can examine the behavior of the function. Since the numerator will be non-zero when \( x = 2 \) (as it factors are linear and won't be zero), the function heads to either positive or negative infinity depending on the direction from which \( x \) approaches 2.
- Analyzing limits directly on either side of the asymptote would confirm which direction the graph will go.
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Intercepts:
- Set \( f(x) = 0 \) for the x-intercepts: \( (x + 1)(x + 2) = 0 \) gives \( x = -1 \) and \( x = -2 \).
- Find \( f(0) \) for the y-intercept: \( f(0) = \frac{(0 + 1)(0 + 2)}{(2 \cdot 0 + 3)(0 - 2)} = \frac{2}{-6} = -\frac{1}{3} \).
Using all of this information, we can determine that the graph should exhibit the following characteristics:
- Vertical asymptote at \( x = 2 \) and \( x = -\frac{3}{2} \).
- Horizontal asymptote at \( y = \frac{1}{2} \).
- Curves that cross the x-axis at \( x = -1 \) and \( x = -2 \) and go towards the asymptotes as \( x \) approaches the vertical lines.
To match these characteristics with the descriptions of the graphs provided, you need to identify the graph that accurately depicts two curved branches, vertical asymptotes (at \( x = 2 \) and likely \( x = -1.5 \)), a horizontal asymptote at \( y = \frac{1}{2} \), and the correct intercepts.
From your description, the candidate graph with the correct features should be the one showing:
- Vertical dashed lines at the corresponding x-values,
- Curved branches approaching these asymptotes,
- The specific intercepts at the right coordinates.
Please check your descriptions again against these characteristics to find the correct response.