Question
The table below represents a function.
x
1
2
3
4
5
y
1
16
64
256
1,024
Which statement would best describe the graph of the function?
The graph is a straight line that has a slope of 8.
The graph is a horizontal line at y = 16.
The graph starts flat but curves steeply upward.
The graph is a parabola that opens upward.
x
1
2
3
4
5
y
1
16
64
256
1,024
Which statement would best describe the graph of the function?
The graph is a straight line that has a slope of 8.
The graph is a horizontal line at y = 16.
The graph starts flat but curves steeply upward.
The graph is a parabola that opens upward.
Answers
GPT-4o mini
To determine the nature of the function represented by the given table, let's first analyze the values of \(y\) in relation to \(x\):
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 1 \\
2 & 16 \\
3 & 64 \\
4 & 256 \\
5 & 1,024 \\
\hline
\end{array}
\]
From the values:
- When \(x = 1\), \(y = 1 = 4^0\)
- When \(x = 2\), \(y = 16 = 4^2\)
- When \(x = 3\), \(y = 64 = 4^3\)
- When \(x = 4\), \(y = 256 = 4^4\)
- When \(x = 5\), \(y = 1,024 = 4^5\)
It appears that \(y\) is equal to \(4^x\). This function grows exponentially as \(x\) increases.
Now, regarding the descriptions provided:
1. **The graph is a straight line that has a slope of 8.** - This is incorrect, as the function is not linear.
2. **The graph is a horizontal line at y = 16.** - This is incorrect, as the values of \(y\) change significantly with \(x\).
3. **The graph starts flat but curves steeply upward.** - This accurately describes an exponential growth function, as \(y\) increases rapidly for higher \(x\) values.
4. **The graph is a parabola that opens upward.** - This is incorrect, as parabolic functions have a different form compared to exponential functions.
Therefore, the best statement to describe the graph of this function is:
**The graph starts flat but curves steeply upward.**
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 1 \\
2 & 16 \\
3 & 64 \\
4 & 256 \\
5 & 1,024 \\
\hline
\end{array}
\]
From the values:
- When \(x = 1\), \(y = 1 = 4^0\)
- When \(x = 2\), \(y = 16 = 4^2\)
- When \(x = 3\), \(y = 64 = 4^3\)
- When \(x = 4\), \(y = 256 = 4^4\)
- When \(x = 5\), \(y = 1,024 = 4^5\)
It appears that \(y\) is equal to \(4^x\). This function grows exponentially as \(x\) increases.
Now, regarding the descriptions provided:
1. **The graph is a straight line that has a slope of 8.** - This is incorrect, as the function is not linear.
2. **The graph is a horizontal line at y = 16.** - This is incorrect, as the values of \(y\) change significantly with \(x\).
3. **The graph starts flat but curves steeply upward.** - This accurately describes an exponential growth function, as \(y\) increases rapidly for higher \(x\) values.
4. **The graph is a parabola that opens upward.** - This is incorrect, as parabolic functions have a different form compared to exponential functions.
Therefore, the best statement to describe the graph of this function is:
**The graph starts flat but curves steeply upward.**