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 multiple 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 multiple 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 mathematical properties you've described suggest a quadratic function (a parabola) with a vertex at (0.5, 9), an axis of symmetry of \(x = 0.5\), a y-intercept at (0, 5), and x-intercepts at \((-0.25, 0)\) and \((1.25, 0)\). Here are a few real-world examples that could be modeled by such a function along with explanations of the key features:
### 1. **Projectile Motion**
**Example:** A ball is thrown upward from a certain height, and its height can be modeled by a quadratic equation.
- **Vertex:** The vertex (0.5, 9) indicates the maximum height the ball reaches (9 units) at 0.5 seconds after it is thrown.
- **Y-intercept:** The y-intercept (0, 5) suggests that the ball was 5 units above the ground when it was thrown at time \(t = 0\).
- **X-intercepts:** The x-intercepts (-0.25, 0) and (1.25, 0) suggest that the ball hits the ground again at 1.25 seconds after being thrown, which could be the time of ascent and descent.
- **Axis of symmetry:** The axis of symmetry \(x = 0.5\) confirms that the peak height occurs midway through the time it takes for the ball to return to the ground.
---
### 2. **Revenue and Profit Maximization**
**Example:** A company analyzes its profit over a specific price range for a product.
- **Vertex:** The vertex (0.5, 9) represents the price point (0.5) at which profit is maximized (9 units).
- **Y-intercept:** The y-intercept (0, 5) indicates that even at a price of zero, the company earns a baseline of 5 units, perhaps from fixed revenues or sales.
- **X-intercepts:** The x-intercepts (-0.25, 0) and (1.25, 0) show the price points (negative prices do not make sense in this context, but could be interpreted as the plausible range for potential losses) where the company breaks even. The company would generate no profit either when the price is too low (negative) or too high (1.25).
- **Axis of symmetry:** The axis \(x = 0.5\) shows that the optimal price point for maximizing profits is at that value.
---
### 3. **Design of a Parabolic Arch Bridge**
**Example:** The shape of a parabolic arch bridge, where the curve describes the heights of the arch at various horizontal distances.
- **Vertex:** The vertex (0.5, 9) represents the highest point of the arch, indicating a maximum height of 9 units at the center of the bridge.
- **Y-intercept:** The y-intercept (0, 5) reflects the height of the bridge at the point it meets one side of the ground; the bridge has a base situation (like foundations) 5 units above the ground at point \(x = 0\).
- **X-intercepts:** The x-intercepts (-0.25, 0) and (1.25, 0) indicate where the bridge meets the ground at either end. These points help define the width and overall structure of the arch.
- **Axis of symmetry:** The axis \(x = 0.5\) shows that the bridge is symmetric around the mid-point, maintaining structural integrity and aesthetic appeal.
---
### Summary of Key Features:
- **Vertex (0.5, 9):** Represents the maximum point of the function (highest value).
- **Axis of Symmetry (x = 0.5):** Indicates that the left and right sides of the parabola mirror each other, defining balance or maximum effectiveness.
- **Y-Intercept (0, 5):** Supplies information about initial conditions or the starting point when the independent variable (like time, price, etc.) is zero.
- **X-Intercepts (-0.25, 0) and (1.25, 0):** Marks points of intersection with the x-axis, representing events like hitting ground level, breaching break-even points, or limits in a physical model.
These examples showcase how a parabolic function can model various real-world phenomena, with each feature providing meaningful context to the situation at hand.
### 1. **Projectile Motion**
**Example:** A ball is thrown upward from a certain height, and its height can be modeled by a quadratic equation.
- **Vertex:** The vertex (0.5, 9) indicates the maximum height the ball reaches (9 units) at 0.5 seconds after it is thrown.
- **Y-intercept:** The y-intercept (0, 5) suggests that the ball was 5 units above the ground when it was thrown at time \(t = 0\).
- **X-intercepts:** The x-intercepts (-0.25, 0) and (1.25, 0) suggest that the ball hits the ground again at 1.25 seconds after being thrown, which could be the time of ascent and descent.
- **Axis of symmetry:** The axis of symmetry \(x = 0.5\) confirms that the peak height occurs midway through the time it takes for the ball to return to the ground.
---
### 2. **Revenue and Profit Maximization**
**Example:** A company analyzes its profit over a specific price range for a product.
- **Vertex:** The vertex (0.5, 9) represents the price point (0.5) at which profit is maximized (9 units).
- **Y-intercept:** The y-intercept (0, 5) indicates that even at a price of zero, the company earns a baseline of 5 units, perhaps from fixed revenues or sales.
- **X-intercepts:** The x-intercepts (-0.25, 0) and (1.25, 0) show the price points (negative prices do not make sense in this context, but could be interpreted as the plausible range for potential losses) where the company breaks even. The company would generate no profit either when the price is too low (negative) or too high (1.25).
- **Axis of symmetry:** The axis \(x = 0.5\) shows that the optimal price point for maximizing profits is at that value.
---
### 3. **Design of a Parabolic Arch Bridge**
**Example:** The shape of a parabolic arch bridge, where the curve describes the heights of the arch at various horizontal distances.
- **Vertex:** The vertex (0.5, 9) represents the highest point of the arch, indicating a maximum height of 9 units at the center of the bridge.
- **Y-intercept:** The y-intercept (0, 5) reflects the height of the bridge at the point it meets one side of the ground; the bridge has a base situation (like foundations) 5 units above the ground at point \(x = 0\).
- **X-intercepts:** The x-intercepts (-0.25, 0) and (1.25, 0) indicate where the bridge meets the ground at either end. These points help define the width and overall structure of the arch.
- **Axis of symmetry:** The axis \(x = 0.5\) shows that the bridge is symmetric around the mid-point, maintaining structural integrity and aesthetic appeal.
---
### Summary of Key Features:
- **Vertex (0.5, 9):** Represents the maximum point of the function (highest value).
- **Axis of Symmetry (x = 0.5):** Indicates that the left and right sides of the parabola mirror each other, defining balance or maximum effectiveness.
- **Y-Intercept (0, 5):** Supplies information about initial conditions or the starting point when the independent variable (like time, price, etc.) is zero.
- **X-Intercepts (-0.25, 0) and (1.25, 0):** Marks points of intersection with the x-axis, representing events like hitting ground level, breaching break-even points, or limits in a physical model.
These examples showcase how a parabolic function can model various real-world phenomena, with each feature providing meaningful context to the situation at hand.