Question
End behavior of the graph shows a negative slope with three points:
(0, 5)
(3, 3)
(6, 0)
(0, 5)
(3, 3)
(6, 0)
Answers
GPT-4o mini
To analyze the end behavior of the graph of a function given the points \((0, 5)\), \((3, 3)\), and \((6, 0)\), we can observe the following:
1. **Slope Between Points**:
- The point \((0, 5)\) indicates that at \(x = 0\), the \(y\)-value is \(5\).
- The point \((3, 3)\) shows that at \(x = 3\), the \(y\)-value has decreased to \(3\).
- Finally, at the point \((6, 0)\), the \(y\)-value has decreased further to \(0\).
Looking at these points, as \(x\) increases from \(0\) to \(3\), the \(y\)-value decreases from \(5\) to \(3\). Continuing from \(3\) to \(6\), the \(y\)-value decreases again from \(3\) to \(0\). This indicates a negative slope between each of the points.
2. **End Behavior**:
- Since all the given points indicate a downward trend, we can assume the function has a negative slope as \(x\) increases beyond the points provided.
- Assuming the function is continuous and based on the points given, we would suggest that the graph continues to decline as \(x\) increases.
3. **Conclusion**:
- The end behavior indicates that as \(x\) approaches infinity (\(x \to +\infty\)), the \(y\)-value approaches negative infinity (\(y \to -\infty\)).
- In terms of mathematical notation, we can express this as:
\[
\lim_{x \to +\infty} f(x) = -\infty.
\]
In summary, the end behavior of the function suggested by the points provided shows a consistent negative slope indicating that as \(x\) increases, the function's output decreases without bound.
1. **Slope Between Points**:
- The point \((0, 5)\) indicates that at \(x = 0\), the \(y\)-value is \(5\).
- The point \((3, 3)\) shows that at \(x = 3\), the \(y\)-value has decreased to \(3\).
- Finally, at the point \((6, 0)\), the \(y\)-value has decreased further to \(0\).
Looking at these points, as \(x\) increases from \(0\) to \(3\), the \(y\)-value decreases from \(5\) to \(3\). Continuing from \(3\) to \(6\), the \(y\)-value decreases again from \(3\) to \(0\). This indicates a negative slope between each of the points.
2. **End Behavior**:
- Since all the given points indicate a downward trend, we can assume the function has a negative slope as \(x\) increases beyond the points provided.
- Assuming the function is continuous and based on the points given, we would suggest that the graph continues to decline as \(x\) increases.
3. **Conclusion**:
- The end behavior indicates that as \(x\) approaches infinity (\(x \to +\infty\)), the \(y\)-value approaches negative infinity (\(y \to -\infty\)).
- In terms of mathematical notation, we can express this as:
\[
\lim_{x \to +\infty} f(x) = -\infty.
\]
In summary, the end behavior of the function suggested by the points provided shows a consistent negative slope indicating that as \(x\) increases, the function's output decreases without bound.