Let's analyze the problem step-by-step based on the given table:
[tex]\[\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-2 & 6 \\
\hline
-1 & 0 \\
\hline
0 & -4 \\
\hline
1 & -6 \\
\hline
2 & -6 \\
\hline
3 & -4 \\
\hline
4 & 0 \\
\hline
5 & 6 \\
\hline
\end{tabular}\][/tex]
A. There is a [tex]\(y\)[/tex]-intercept at [tex]\((-4,0)\)[/tex]:
- The [tex]\(y\)[/tex]-intercept is where the graph crosses the [tex]\(y\)[/tex]-axis. This occurs when [tex]\(x = 0\)[/tex].
- From the table, at [tex]\(x = 0\)[/tex], the value of [tex]\(f(x)\)[/tex] is [tex]\(-4\)[/tex]. Thus, the [tex]\(y\)[/tex]-intercept should be at [tex]\((0, -4)\)[/tex], not [tex]\((-4, 0)\)[/tex].
Hence, Statement A is false.
B. There are [tex]\(x\)[/tex]-intercepts at [tex]\((-1,0)\)[/tex] and [tex]\((4,0)\)[/tex]:
- The [tex]\(x\)[/tex]-intercepts are where the graph crosses the [tex]\(x\)[/tex]-axis. This occurs when [tex]\(f(x) = 0\)[/tex].
- From the table, at [tex]\(x = -1\)[/tex] and [tex]\(x = 4\)[/tex], [tex]\(f(x) = 0\)[/tex].
Therefore, Statement B is true.
C. There is a line of symmetry at [tex]\(x=1.5\)[/tex]:
- For there to be a line of symmetry, the values of the function should be mirror images around the line [tex]\(x = 1.5\)[/tex].
- By observation and comparison, the table suggests that the values are symmetric around [tex]\(x = 1.5\)[/tex]:
- [tex]\((-2, 6)\)[/tex] and [tex]\((5, 6)\)[/tex]
- [tex]\((-1, 0)\)[/tex] and [tex]\((4, 0)\)[/tex]
- [tex]\((0, -4)\)[/tex] and [tex]\((3, -4)\)[/tex]
- [tex]\((1, -6)\)[/tex] and [tex]\((2, -6)\)[/tex].
Hence, Statement C is true.
D. In the interval from [tex]\(x=0\)[/tex] to [tex]\(x=5\)[/tex], [tex]\(f(x)\)[/tex] is increasing:
- To determine if the function is increasing in this interval, we need to check if [tex]\(f(x)\)[/tex] continuously increases as [tex]\(x\)[/tex] moves from 0 to 5.
- By reviewing the given values:
- [tex]\(f(0) = -4\)[/tex]
- [tex]\(f(1) = -6\)[/tex]
- [tex]\(f(2) = -6\)[/tex]
- [tex]\(f(3) = -4\)[/tex]
- [tex]\(f(4) = 0\)[/tex]
- [tex]\(f(5) = 6\)[/tex].
- The function is not continuously increasing in this interval. It decreases from [tex]\(x = 0\)[/tex] to [tex]\(x = 1\)[/tex] and is constant from [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex].
Thus, Statement D is false.
Therefore, based on our detailed analysis:
- Statement B is true: There are [tex]\(x\)[/tex]-intercepts at [tex]\((-1,0)\)[/tex] and [tex]\((4,0)\)[/tex].
- Statement C is true: There is a line of symmetry at [tex]\(x=1.5\)[/tex].
