Answer :
To solve this problem, let's evaluate each statement in the context of the given table of values.
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 6 \\ \hline -1 & 0 \\ \hline 0 & -4 \\ \hline 1 & -6 \\ \hline 2 & -6 \\ \hline 3 & -4 \\ \hline 4 & 0 \\ \hline 5 & 6 \\ \hline \end{array} \][/tex]
### Evaluating Statement A:
A. There are [tex]$x$[/tex]-intercepts at [tex]$(-1,0)$[/tex] and [tex]$(4,0)$[/tex].
An [tex]$x$[/tex]-intercept occurs where [tex]$f(x) = 0$[/tex]. According to the table:
- At [tex]$x = -1$[/tex], [tex]$f(x) = 0$[/tex].
- At [tex]$x = 4$[/tex], [tex]$f(x) = 0$[/tex].
So, there are indeed [tex]$x$[/tex]-intercepts at [tex]$(-1,0)$[/tex] and [tex]$(4,0)$[/tex]. Therefore, statement A is true.
### Evaluating Statement B:
B. There is a line of symmetry at [tex]$x=1.5$[/tex].
A line of symmetry would mean the points on either side of this line mirror each other in terms of both [tex]$x$[/tex] and [tex]$f(x)$[/tex]. Let's check the points:
- The midpoint of the table is at [tex]$x = 1.5$[/tex].
- Checking pairs of points:
- [tex]$(-2, 6)$[/tex] and [tex]$(5, 6)$[/tex], symmetric around [tex]$x = 1.5$[/tex].
- [tex]$(-1, 0)$[/tex] and [tex]$(4, 0)$[/tex], symmetric around [tex]$x = 1.5$[/tex].
- [tex]$(0, -4)$[/tex] and [tex]$(3, -4)$[/tex], symmetric around [tex]$x = 1.5$[/tex].
- [tex]$(1, -6)$[/tex] and [tex]$(2, -6)$[/tex], symmetric around [tex]$x = 1.5$[/tex].
Each pair of points is symmetric around [tex]$x = 1.5$[/tex]. Thus, statement B is true.
### Evaluating Statement C:
C. There is a [tex]$y$[/tex]-intercept at [tex]$(-4,0)$[/tex].
A [tex]$y$[/tex]-intercept occurs where [tex]$x = 0$[/tex]. From the table:
- At [tex]$x = 0$[/tex], [tex]$f(x) = -4$[/tex].
So, there is a [tex]$y$[/tex]-intercept at [tex]$(0, -4)$[/tex], not [tex]$(-4, 0)$[/tex]. Therefore, statement C is false.
### Evaluating Statement D:
D. In the interval from [tex]$x=0$[/tex] to [tex]$x=5, f(x)$[/tex] is increasing.
We should check if [tex]$f(x)$[/tex] is consistently increasing in this interval:
- From [tex]$x = 0$[/tex] to [tex]$x = 1$[/tex]: [tex]$f(x)$[/tex] changes from [tex]$-4$[/tex] to [tex]$-6$[/tex] (decreasing).
- Thus, even without further checking, [tex]$f(x)$[/tex] is not increasing in the entire interval from [tex]$x=0$[/tex] to [tex]$x=5$[/tex].
Therefore, statement D is false.
### Conclusion
After evaluating all the statements, we can conclude that the correct statement from the given options is:
- A. There are [tex]$x$[/tex]-intercepts at [tex]$(-1,0)$[/tex] and [tex]$(4,0)$[/tex].
So, the valid statements based on the table are:
[tex]\[ \boxed{1} \][/tex]
This result matches the previously provided answer.
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 6 \\ \hline -1 & 0 \\ \hline 0 & -4 \\ \hline 1 & -6 \\ \hline 2 & -6 \\ \hline 3 & -4 \\ \hline 4 & 0 \\ \hline 5 & 6 \\ \hline \end{array} \][/tex]
### Evaluating Statement A:
A. There are [tex]$x$[/tex]-intercepts at [tex]$(-1,0)$[/tex] and [tex]$(4,0)$[/tex].
An [tex]$x$[/tex]-intercept occurs where [tex]$f(x) = 0$[/tex]. According to the table:
- At [tex]$x = -1$[/tex], [tex]$f(x) = 0$[/tex].
- At [tex]$x = 4$[/tex], [tex]$f(x) = 0$[/tex].
So, there are indeed [tex]$x$[/tex]-intercepts at [tex]$(-1,0)$[/tex] and [tex]$(4,0)$[/tex]. Therefore, statement A is true.
### Evaluating Statement B:
B. There is a line of symmetry at [tex]$x=1.5$[/tex].
A line of symmetry would mean the points on either side of this line mirror each other in terms of both [tex]$x$[/tex] and [tex]$f(x)$[/tex]. Let's check the points:
- The midpoint of the table is at [tex]$x = 1.5$[/tex].
- Checking pairs of points:
- [tex]$(-2, 6)$[/tex] and [tex]$(5, 6)$[/tex], symmetric around [tex]$x = 1.5$[/tex].
- [tex]$(-1, 0)$[/tex] and [tex]$(4, 0)$[/tex], symmetric around [tex]$x = 1.5$[/tex].
- [tex]$(0, -4)$[/tex] and [tex]$(3, -4)$[/tex], symmetric around [tex]$x = 1.5$[/tex].
- [tex]$(1, -6)$[/tex] and [tex]$(2, -6)$[/tex], symmetric around [tex]$x = 1.5$[/tex].
Each pair of points is symmetric around [tex]$x = 1.5$[/tex]. Thus, statement B is true.
### Evaluating Statement C:
C. There is a [tex]$y$[/tex]-intercept at [tex]$(-4,0)$[/tex].
A [tex]$y$[/tex]-intercept occurs where [tex]$x = 0$[/tex]. From the table:
- At [tex]$x = 0$[/tex], [tex]$f(x) = -4$[/tex].
So, there is a [tex]$y$[/tex]-intercept at [tex]$(0, -4)$[/tex], not [tex]$(-4, 0)$[/tex]. Therefore, statement C is false.
### Evaluating Statement D:
D. In the interval from [tex]$x=0$[/tex] to [tex]$x=5, f(x)$[/tex] is increasing.
We should check if [tex]$f(x)$[/tex] is consistently increasing in this interval:
- From [tex]$x = 0$[/tex] to [tex]$x = 1$[/tex]: [tex]$f(x)$[/tex] changes from [tex]$-4$[/tex] to [tex]$-6$[/tex] (decreasing).
- Thus, even without further checking, [tex]$f(x)$[/tex] is not increasing in the entire interval from [tex]$x=0$[/tex] to [tex]$x=5$[/tex].
Therefore, statement D is false.
### Conclusion
After evaluating all the statements, we can conclude that the correct statement from the given options is:
- A. There are [tex]$x$[/tex]-intercepts at [tex]$(-1,0)$[/tex] and [tex]$(4,0)$[/tex].
So, the valid statements based on the table are:
[tex]\[ \boxed{1} \][/tex]
This result matches the previously provided answer.
