Question
Lizzy is playing pool. She is trying to hit her ball so that it bounces off the wall of the pool table and rolls into the hole. This is shown in the graph below. The function that describes the path of the ball is y=|x−4|+k
.
(The ball starts at 0 and ends at 8)
(1 point)
Responses
4
4
-4
-4
8
8
-8
.
(The ball starts at 0 and ends at 8)
(1 point)
Responses
4
4
-4
-4
8
8
-8
Answers
GPT-4o mini
To determine the value of \( k \) in the equation \( y = |x - 4| + k \), we need to analyze the scenario described.
Since Lizzy's ball starts at x = 0 and ends at x = 8, we will check how the function behaves at these points:
1. **At \( x = 0 \)**:
\[
y = |0 - 4| + k = | - 4 | + k = 4 + k
\]
2. **At \( x = 8 \)**:
\[
y = |8 - 4| + k = |4| + k = 4 + k
\]
The condition is that the ball must roll into the hole at these positions. For it to be the same height (y value) at x = 0 and x = 8, we can observe that both calculations yield \( 4 + k \).
Now, when the ball hits the wall and bounces, we want the resulting trajectory to facilitate rolling into the hole.
The vertex of the absolute value function \( y = |x - 4| + k \) occurs at \( x = 4 \). The y-coordinate of the vertex (the lowest point of the graph) is:
\[
y = |4 - 4| + k = 0 + k = k
\]
For the ball to be able to bounce off the wall and go into the hole, the y-values at the starting and ending points should be aligned correctly.
If both ends are at the same height (which occurs when \( k = 0 \)), we must check the options given in your response:
- \( k = 4 \)
- \( k = -4 \)
- \( k = 8 \)
- \( k = -8 \)
Among these, **if we want the ball to be exactly in line (flat) at those points**, the ideal value for \( k \) is likely \( k = 0 \).
However, since the options provided do not include \( k = 0 \), and we are observing the height, we can determine from the vertex that \( k = -4\) would position the absolute value function so that it can indeed reflect and hit the holes correctly. Thus the best possible choice could relate to the lowest position.
Thus, a likely correct response based on function behavior regarding having potential \( k\) to represent heights of interest could be:
**Answer: -4**
Since Lizzy's ball starts at x = 0 and ends at x = 8, we will check how the function behaves at these points:
1. **At \( x = 0 \)**:
\[
y = |0 - 4| + k = | - 4 | + k = 4 + k
\]
2. **At \( x = 8 \)**:
\[
y = |8 - 4| + k = |4| + k = 4 + k
\]
The condition is that the ball must roll into the hole at these positions. For it to be the same height (y value) at x = 0 and x = 8, we can observe that both calculations yield \( 4 + k \).
Now, when the ball hits the wall and bounces, we want the resulting trajectory to facilitate rolling into the hole.
The vertex of the absolute value function \( y = |x - 4| + k \) occurs at \( x = 4 \). The y-coordinate of the vertex (the lowest point of the graph) is:
\[
y = |4 - 4| + k = 0 + k = k
\]
For the ball to be able to bounce off the wall and go into the hole, the y-values at the starting and ending points should be aligned correctly.
If both ends are at the same height (which occurs when \( k = 0 \)), we must check the options given in your response:
- \( k = 4 \)
- \( k = -4 \)
- \( k = 8 \)
- \( k = -8 \)
Among these, **if we want the ball to be exactly in line (flat) at those points**, the ideal value for \( k \) is likely \( k = 0 \).
However, since the options provided do not include \( k = 0 \), and we are observing the height, we can determine from the vertex that \( k = -4\) would position the absolute value function so that it can indeed reflect and hit the holes correctly. Thus the best possible choice could relate to the lowest position.
Thus, a likely correct response based on function behavior regarding having potential \( k\) to represent heights of interest could be:
**Answer: -4**