Question

Select the correct answer from each drop-down menu.

Find the asymptote and determine the end behavior of the function from the graph.

Graph shows an exponential curve on a coordinate plane. The curve begins infinitely close to Y-axis at X equals 3 in the fourth quadrant, rises from left to right, concave down, intersects X-axis at 3.2, and goes through (4, 3).

The asymptote of the function is
. For very high x-values, y
.

Answer 1

Explanation:

From the given information, we can determine the asymptote and the end behavior of the function.

The asymptote of the function is a horizontal line that the graph of the function approaches but never touches. In this case, since the exponential curve rises from left to right and intersects the x-axis at 3.2, it means that the asymptote will be the x-axis (y = 0). As x approaches negative infinity, the curve will get closer and closer to the x-axis, but it will never touch it.

For very high x-values, we can determine the end behavior of the function by looking at the concavity of the curve. Since the curve is concave down, it means that as x approaches positive infinity, the function will decrease towards negative infinity.

To summarize:

1. The asymptote of the function is the x-axis, y = 0.

2. For very high x-values, as x approaches positive infinity, y approaches negative infinity.

It is important to note that without the specific equation or formula for the exponential function, we can make these conclusions based on the given information about the graph's behavior.

Answer 2

The asymptote of the function is the y-axis (x = 0).
As x approaches negative infinity, the y-values of the function decrease without bound.
As x approaches positive infinity, the y-values of the function increase without bound.

The asymptote of an exponential function is a line that the graph approaches but never touches. In this case, we can see that the exponential curve begins infinitely close to the y-axis at x = 3 in the fourth quadrant.

Since the curve rises from left to right and is concave down, the end behavior of the function is as follows:

- As x approaches negative infinity (x → -∞), the y-values of the function decrease without bound, meaning that the function approaches negative infinity. This can be seen as the curve goes downwards towards the left.
- As x approaches positive infinity (x → +∞), the y-values of the function increase without bound, meaning that the function approaches positive infinity. This can be observed as the curve goes upwards towards the right.

Given points on the graph:
- The points (3, -5), (3, -4), (3, -3), and (3, 0) all have the same x-coordinate, which is 3. This means that they lie on a vertical line parallel to the y-axis. However, these points do not provide any information about the asymptote or end behavior of the function.
- The point (4, 5) lies on the exponential curve and does not provide information about the asymptote or end behavior either.