On a coordinate plane, a curved line with minimum values of (negative 2, 0) and (1.05, negative 41), and a maximum value of (negative 0.5, 5), crosses the x-axis at (negative 2, 0), (0, 0), and (1.5, 0), and crosses the y-axis at (0, 0).

Question

Which statement is true about the end behavior of the graphed function?

As the x-values go to positive infinity, the function’s values go to positive infinity.
As the x-values go to zero, the function’s values go to positive infinity.
As the x-values go to negative infinity, the function’s values are equal to zero.
As the x-values go to negative infinity, the function’s values go to negative infinity.

Answer 1

Let's analyze the given information about the curved line (which seems to describe the graph of a polynomial function) and determine the end behavior of the function:

1. The function has minimum values at (-2, 0) and (1.05, -41), suggesting local minima at these points.
2. The function has a maximum value at (-0.5, 5), suggesting a local maximum at this point.
3. It crosses the x-axis at (-2, 0), (0, 0), and (1.5, 0). These are the roots of the function.
4. It crosses the y-axis at (0, 0), which is also one of the roots.

Given this information, let's construct a rough sketch of the polynomial:

- The function moves downward from some positive value, reaches a local minimum at (-2, 0), crosses the x-axis at (-2, 0), rises to a local maximum at (-0.5, 5), and then falls through the origin (0, 0).
- After the origin, the function falls to another local minimum at (1.05, -41) and rises again to cross the x-axis at (1.5, 0).
- Since the curve eventually must rise to cross the x-axis at (1.5, 0) after a deep fall, it is clear that both the positive and negative x-axis behaviors need to be considered.

To determine the end behavior, observe a polynomial tends to \( \pm \infty \) by analyzing the highest degree term in the polynomial.

Given how the roots and extrema are placed:

- The number of x-intercepts (roots) suggests a fourth-degree polynomial (or higher), since it appears to cross the x-axis three times and shows varying up-and-down behavior.
- A positive leading coefficient in even-degree polynomials (like the fourth degree) implies both ends rise to positive infinity,
- Conversely, if the leading coefficient is negative, both ends fall to negative infinity.

By visualizing general polynomial behavior and having multiple constraints, it's easy to conclude:

**The statement that's true about the end behavior is:**

**As the x-values go to positive infinity, the function’s values go to positive infinity.**

Given the behavior needs to reach up after falling back for - domains (leftwards negation) and \(+ve\) domains, the function will rise overall (if a case holder for positive leading term). Hence a graph sketched here confirms general tendency attributes overall leading behavior positive \( + +inf \) horizontally on moving right on \( xo +ve horizons +)