To analyze the rational function \( f(x) = \frac{x^2 + 3x + 2}{2x^2 - x - 2} \), we should first simplify it if possible, and then determine characteristics such as intercepts, asymptotes, and behavior at infinity to visualize what the graph should look like.
- Factoring the numerator and denominator:
- For the numerator \( x^2 + 3x + 2 \), we can factor it: \[ x^2 + 3x + 2 = (x + 1)(x + 2) \]
- For the denominator \( 2x^2 - x - 2 \), we need to factor it as well: \[ 2x^2 - x - 2 = (2x + 3)(x - 2) \]
Thus, the function can be rewritten as: \[ f(x) = \frac{(x + 1)(x + 2)}{(2x + 3)(x - 2)} \]
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Finding intercepts:
- x-intercepts: Set the numerator equal to zero: \[ (x + 1)(x + 2) = 0 \] This gives us \( x = -1 \) and \( x = -2 \) as x-intercepts.
- y-intercept: Set \( x = 0 \): \[ f(0) = \frac{(0 + 1)(0 + 2)}{(2(0) + 3)(0 - 2)} = \frac{1 \cdot 2}{3 \cdot (-2)} = \frac{2}{-6} = -\frac{1}{3} \] So the y-intercept is \( (0, -\frac{1}{3}) \).
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Vertical asymptotes: Set the denominator equal to zero: \[ (2x + 3)(x - 2) = 0 \] This gives \( x = -\frac{3}{2} \) and \( x = 2 \) as vertical asymptotes.
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Horizontal asymptote: Since the degrees of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients: \[ y = \frac{1}{2} \]
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End behavior: As \( x \to \pm \infty \), \( f(x) \to \frac{1}{2} \).
With this information, we can summarize:
- The x-intercepts are \( -1 \) and \( -2 \).
- The y-intercept is \( (0, -\frac{1}{3}) \).
- Vertical asymptotes at \( x = -\frac{3}{2} \) and \( x = 2 \).
- A horizontal asymptote at \( y = \frac{1}{2} \).
If you have specific graphs presented, look for one that includes the intercepts at \( (-1, 0) \) and \( (-2, 0) \), has vertical asymptotes at \( x = -\frac{3}{2} \) and \( x = 2 \), and approaches \( y = \frac{1}{2} \) as \( x \) approaches infinity.