Match the description of the end behavior with the graph of the function it describes. A B C y y y X X X 1. As x increases, y increases; as x decreases, y decreases. – 2. As x increases, y increases; as x decreases, y increases. – 3. As x increases, y decreases; as x decreases, y decreases. – Tu respuesta
The Correct Answer and Explanation is:
Correct Answer:
- As x increases, y increases; as x decreases, y decreases. — B
- As x increases, y increases; as x decreases, y increases. — A
- As x increases, y decreases; as x decreases, y decreases. — C
Explanation
To solve this problem, we must analyze the end behavior of each function’s graph. The end behavior of a function describes the trend of the y-values as the x-values approach positive infinity (x increases without bound) and negative infinity (x decreases without bound). We look at the far left and far right sides of each graph to determine this.
1. Analyzing Graph A:
This graph is a parabola that opens upward.
- As x increases (moving to the right): If we follow the graph to the far right, the arrow points upward. This indicates that the y-values are increasing and approaching positive infinity. So, as x increases, y increases.
- As x decreases (moving to the left): If we follow the graph to the far left, the arrow also points upward. This means that as x-values get smaller, the y-values are still increasing and approaching positive infinity. So, as x decreases, y increases.
This behavior, where y increases on both ends, matches the description in statement 2. This is characteristic of an even-degree polynomial with a positive leading coefficient.
2. Analyzing Graph B:
This graph resembles a cubic function.
- As x increases (moving to the right): Following the graph to the far right, the arrow points upward. This signifies that the y-values are increasing. So, as x increases, y increases.
- As x decreases (moving to the left): Following the graph to the far left, the arrow points downward. This signifies that the y-values are decreasing and approaching negative infinity. So, as x decreases, y decreases.
This behavior, where y increases on the right and decreases on the left, matches the description in statement 1. This is characteristic of an odd-degree polynomial with a positive leading coefficient.
3. Analyzing Graph C:
This graph is a polynomial function that opens downward on both ends.
- As x increases (moving to the right): As we trace the graph to the far right, the arrow points downward. This indicates that the y-values are decreasing and approaching negative infinity. So, as x increases, y decreases.
- As x decreases (moving to the left): As we trace the graph to the far left, the arrow also points downward. This means that the y-values are also decreasing and approaching negative infinity. So, as x decreases, y decreases.
This behavior, where y decreases on both ends, perfectly matches the description in statement 3. This is characteristic of an even-degree polynomial with a negative leading coefficient.
