To determine how many zeros are indicated by the values in the table, we need to examine the changes in the sign of the [tex]\( y \)[/tex]-values as [tex]\( x \)[/tex]-values progress from [tex]\(-3.0\)[/tex] to [tex]\(3.0\)[/tex]. Whenever the [tex]\( y \)[/tex]-value changes from positive to negative or from negative to positive, it suggests that the graph of the function crosses the x-axis, indicating a zero.
Here are the given values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3.0 & 8.40 \\
\hline
-2.4 & 0.69 \\
\hline
-1.8 & -0.39 \\
\hline
-1.2 & 0.24 \\
\hline
-0.6 & 0.46 \\
\hline
0 & 0 \\
\hline
0.6 & -0.46 \\
\hline
1.2 & -0.24 \\
\hline
1.8 & 0.39 \\
\hline
2.4 & -0.69 \\
\hline
3.0 & -8.40 \\
\hline
\end{array}
\][/tex]
We can follow these steps to find the number of zeros:
1. Identify the zero at [tex]\( x = 0 \)[/tex]: Here, the table explicitly states that when [tex]\( x = 0 \)[/tex], [tex]\( y = 0 \)[/tex]. So, this is one zero.
2. Examine sign changes:
- From [tex]\( 8.40 \)[/tex] to [tex]\( 0.69 \)[/tex]: No sign change, still positive.
- From [tex]\( 0.69 \)[/tex] to [tex]\( -0.39 \)[/tex]: Positive to negative, one zero indicated.
- From [tex]\( -0.39 \)[/tex] to [tex]\( 0.24 \)[/tex]: Negative to positive, one zero indicated.
- From [tex]\( 0.24 \)[/tex] to [tex]\( 0.46 \)[/tex]: No sign change, still positive.
- From [tex]\( 0.46 \)[/tex] to [tex]\( 0 \)[/tex]: Positive to zero, already counted as a zero.
- From [tex]\( 0 \)[/tex] to [tex]\( -0.46 \)[/tex]: Zero to negative, already counted as a zero.
- From [tex]\( -0.46 \)[/tex] to [tex]\( -0.24 \)[/tex]: No sign change, still negative.
- From [tex]\( -0.24 \)[/tex] to [tex]\( 0.39 \)[/tex]: Negative to positive, one zero indicated.
- From [tex]\( 0.39 \)[/tex] to [tex]\( -0.69 \)[/tex]: Positive to negative, one zero indicated.
- From [tex]\( -0.69 \)[/tex] to [tex]\( -8.40 \)[/tex]: No sign change, still negative.
We notice four sign changes plus the zero explicitly mentioned at [tex]\( x = 0 \)[/tex]; thus, there is one more counting for that case:
Summing this up, we have:
- Zero at [tex]\( x = 0 \)[/tex]
- Zero from [tex]\(-2.4\)[/tex] to [tex]\(-1.8\)[/tex]
- Zero from [tex]\(-1.8\)[/tex] to [tex]\(-1.2\)[/tex]
- Zero from [tex]\(0.6\)[/tex] to [tex]\(1.2\)[/tex]
- Zero from [tex]\(1.8\)[/tex] to [tex]\(2.4\)[/tex]
Thus, there are a total of 5 zeros indicated by the values in the table.
