Question
A. Vertex: (0.5,9) B. Axis of symmetry : x=0.5 C. y-intercept: (0,5) C. x-intercepts from left to right: (-0.25, 0) and (1.25, 0).
Provide a real-world example that would be represented by this function and tell what the key features: vertex, x intercepts, y intercept and the axis, above represent.
Provide a real-world example that would be represented by this function and tell what the key features: vertex, x intercepts, y intercept and the axis, above represent.
Answers
GPT-4o mini
The function characterized by the given attributes—vertex, axis of symmetry, y-intercept, and x-intercepts—can represent various real-world scenarios. One possible example is the trajectory of a projectile, such as a ball thrown in the air.
### Real-World Example
Imagine a person throwing a ball from a height of 5 units (representing 5 meters off the ground). The ball reaches its maximum height of 9 units (9 meters) at a horizontal distance of 0.5 meters from the starting point. The ball then falls back to the ground, which is represented by the x-intercepts.
### Key Features Explained
1. **Vertex (0.5, 9)**:
- The vertex of the function indicates the highest point of the projectile's trajectory. In this context, the ball reaches its maximum height of 9 meters at a horizontal distance of 0.5 meters from where it was thrown.
2. **Axis of Symmetry (x = 0.5)**:
- The axis of symmetry is a vertical line passing through the vertex (0.5, 9). This line indicates that the projectile's path on the left side (before reaching the vertex) is mirrored perfectly on the right side (after reaching the vertex). It shows the point at which the trajectory of the ball is symmetric.
3. **Y-Intercept (0, 5)**:
- The y-intercept represents the initial height of the ball when it is thrown. In this case, the ball starts at a height of 5 meters. This is where the projectile intersects the y-axis, displaying the height when the horizontal distance is 0.
4. **X-Intercepts (-0.25, 0) and (1.25, 0)**:
- The x-intercepts indicate the points where the projectile hits the ground (height = 0). The ball reaches the ground at a distance of -0.25 meters (which may indicate a point behind the thrower, perhaps due to a calculation error or measurement conventions) and 1.25 meters away. The locations of these x-intercepts reveal the distances from the thrower's position where the ball lands.
### Summary
In summary, the function exemplified by these key features can describe the motion of a projectile, illustrating its maximum height, starting position, and where it will land. Understanding these points allows for analysis of the ball's flight path and performance in real-life scenarios.
### Real-World Example
Imagine a person throwing a ball from a height of 5 units (representing 5 meters off the ground). The ball reaches its maximum height of 9 units (9 meters) at a horizontal distance of 0.5 meters from the starting point. The ball then falls back to the ground, which is represented by the x-intercepts.
### Key Features Explained
1. **Vertex (0.5, 9)**:
- The vertex of the function indicates the highest point of the projectile's trajectory. In this context, the ball reaches its maximum height of 9 meters at a horizontal distance of 0.5 meters from where it was thrown.
2. **Axis of Symmetry (x = 0.5)**:
- The axis of symmetry is a vertical line passing through the vertex (0.5, 9). This line indicates that the projectile's path on the left side (before reaching the vertex) is mirrored perfectly on the right side (after reaching the vertex). It shows the point at which the trajectory of the ball is symmetric.
3. **Y-Intercept (0, 5)**:
- The y-intercept represents the initial height of the ball when it is thrown. In this case, the ball starts at a height of 5 meters. This is where the projectile intersects the y-axis, displaying the height when the horizontal distance is 0.
4. **X-Intercepts (-0.25, 0) and (1.25, 0)**:
- The x-intercepts indicate the points where the projectile hits the ground (height = 0). The ball reaches the ground at a distance of -0.25 meters (which may indicate a point behind the thrower, perhaps due to a calculation error or measurement conventions) and 1.25 meters away. The locations of these x-intercepts reveal the distances from the thrower's position where the ball lands.
### Summary
In summary, the function exemplified by these key features can describe the motion of a projectile, illustrating its maximum height, starting position, and where it will land. Understanding these points allows for analysis of the ball's flight path and performance in real-life scenarios.