Question

Choose an x-value between two consecutive zeros on the graph. Solve the function at that value to determine if the graph is above or below the x-axis. Plot the point on the same graph that shows the zeros.

zeros are -3,1, and 2

pick -2

Answer 1

To determine whether the function is above or below the x-axis at \( x = -2 \), we first need to evaluate the function at that point.

Given the zeros of the function are at \( -3, 1, \) and \( 2 \), we can make some general assumptions about the behavior of the function based on these zeros. If we assume that the function has the general form
\[ f(x) = a(x + 3)(x - 1)(x - 2) \]
where \( a \) is a non-zero constant (positive or negative), we need to evaluate \( f \) at \( x = -2 \).

Using our general form:

1. Substitute \( x = -2 \):
\[
f(-2) = a(-2 + 3)(-2 - 1)(-2 - 2)
\]
\[
= a(1)(-3)(-4)
\]
\[
= a \cdot 12
\]

2. Since we do not have the value for \( a \), we can analyze the sign of \( f(-2) \):
- If \( a > 0 \), then \( f(-2) = 12a > 0 \), meaning the graph is above the x-axis.
- If \( a < 0 \), then \( f(-2) = 12a < 0 \), meaning the graph is below the x-axis.

Without knowing the specific value of \( a \), we cannot definitively say whether the graph is above or below the x-axis at \( x = -2 \), but we can conclude that:

- If \( a > 0\), the graph is above the x-axis at \( x = -2 \).
- If \( a < 0\), the graph is below the x-axis at \( x = -2 \).

If we are tasked with plotting the point on the graph of the function, we need to choose a value for \( a \) or depend on the context of the problem:

- If we assume \( a = 1 \) (common for basic polynomial representations), \( f(-2) = 12 \), indicating the point \( (-2, 12) \) lies above the x-axis.
- If we assume \( a = -1 \), \( f(-2) = -12\), indicating \( (-2, -12) \) lies below the x-axis.

Please clarify the value of \( a \) or provide the function for a more precise determination!