To analyze the given features of the parabola, let’s break down the information provided:
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As \( x \to \infty \), \( f(x) \to -\infty \): This indicates that the leading coefficient of the parabola is negative, implying that the parabola opens downwards.
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A root at \( x = 2 \): This tells us that \( f(2) = 0 \), which means \( (x - 2) \) is one of the factors of the polynomial.
Given these points, we can derive the implications for the other options:
Option A: The factor with a root of x = 2 has a positive slope.
Since the parabola opens downward (as we found from the first point), the factor corresponding to the root at \( x = 2 \) must be a linear factor that goes from above to below (meaning it crosses the x-axis), which gives it a negative slope. Therefore, Option A is incorrect.
Option B: The y-intercept is negative.
We cannot conclude this directly without additional information about the specific values of the polynomial, so we won't determine if this is true or false without further data. However, we would need to consider the polynomial's constant (y-intercept) in relation to the root and the behavior of the graph, which we cannot confirm definitively from the given data.
Option C: Both of the roots are positive.
Since we only know one root is positive (at \( x = 2 \)), and the other root can be either positive or negative, we cannot confirm this to be true. Therefore, Option C is incorrect.
Option D: One linear factor has a positive slope and the other linear factor has a negative slope.
Given that the parabola opens downwards, we have a downward-facing curve. If one root is at \( x = 2 \) (with a negative slope), the other root must also conform to the overall direction of the polynomial. For the total function to go downwards at both extremes, there must be another root that affects the slope. If the other root is negative, it will also create a situation where the polynomial can either have one factor rising towards the x-axis and the other falling away from it or combine in such a way that results in a factorization conforming to one positive and one negative slope overall.
Since the question states we have a parabola that opens downwards, this setup is indeed plausible.
Conclusion: Given the above plan of the polynomial, we lean towards Option D as the most plausible interpretation focusing on the features provided as they relate to slope directionalities.
The best answer is D. One linear factor has a positive slope and the other linear factor has a negative slope.