To determine the correct answer, we'll need to analyze the given values of the function [tex]\(g\)[/tex] and interpret the characteristics of the quadratic function.
Let's review the table of values:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -5 & -4 & -3 & -2 & -1 & 0 \\
\hline
g(x) & -1 & 0 & -1 & -4 & -9 & -16 \\
\hline
\end{array}
\][/tex]
### Understanding Quadratic Functions
A quadratic function generally has the form:
[tex]\[ g(x) = ax^2 + bx + c \][/tex]
Quadratic functions have a parabolic shape and can open upwards or downwards. If [tex]\(a > 0\)[/tex], the parabola opens upwards, and if [tex]\(a < 0\)[/tex], it opens downwards. The vertex of a quadratic function provides the maximum or minimum value of the function.
### Behavior of [tex]\(g(x)\)[/tex]
1. Based on the values of [tex]\(g(x)\)[/tex], let's observe how the function behaves:
- At [tex]\(x = -5\)[/tex], [tex]\(g(x) = -1\)[/tex]
- At [tex]\(x = -4\)[/tex], [tex]\(g(x) = 0\)[/tex]
- At [tex]\(x = -3\)[/tex], [tex]\(g(x) = -1\)[/tex]
- At [tex]\(x = -2\)[/tex], [tex]\(g(x) = -4\)[/tex]
- At [tex]\(x = -1\)[/tex], [tex]\(g(x) = -9\)[/tex]
- At [tex]\(x = 0\)[/tex], [tex]\(g(x) = -16\)[/tex]
2. Notice that as [tex]\(x\)[/tex] increases from [tex]\(-5\)[/tex] to [tex]\(0\)[/tex], [tex]\(g(x)\)[/tex] starts to decrease more rapidly, indicating that the parabola opens downwards.
3. The maximum value occurs at the vertex. By examining these points, the turning point (vertex) appears to be between [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex].
4. This means that the vertex occurs somewhere near these points. For our table's data, the exact vertex is not explicitly listed, but the pattern of the function suggests a downward opening parabola. In this case, for simplicity, let's infer that the approximate vertex is around an [tex]\(x\)[/tex] value of [tex]\(-2\)[/tex] and [tex]\(g(x)\)[/tex] at this point is [tex]\(-4\)[/tex].
### Evaluating the Statements
Now, let's evaluate each option:
- A. The minimum occurs at the function's [tex]\(x\)[/tex]-intercept.
- An [tex]\(x\)[/tex]-intercept is where the function crosses the x-axis ([tex]\(g(x) = 0\)[/tex]). This doesn't provide the minimum value for a downward opening parabola.
- B. The maximum occurs at the function's [tex]\(x\)[/tex]-intercept.
- An [tex]\(x\)[/tex]-intercept would not be the highest value of [tex]\(g\)[/tex]. Moreover, there isn't a clear x-intercept provided in this dataset contextually as maximum.
- C. The minimum occurs at the function's [tex]\(y\)[/tex]-intercept.
- At the [tex]\(y\)[/tex]-intercept ([tex]\(x = 0\)[/tex]), [tex]\(g(x) = -16\)[/tex]. This point is indeed a low point but not a minimum.
- D. The maximum occurs at the function's [tex]\(y\)[/tex]-intercept.
- The maximum is not at the [tex]\(y\)[/tex]-intercept.
Given these observations and calculations we can infer that none of the options are correct for this particular function's maximum or minimum. The vertex where the minimum or maximum lies is not accurately described in those specific terms. Thus, none of the options given are valid.
