To find the maximum value of a quadratic sequence, you can utilize the properties of its corresponding quadratic equation. A quadratic sequence can be represented by the quadratic equation y = ax² + bx + c.
Determining the Maximum Value
The maximum value of a quadratic sequence occurs at the vertex of the parabola represented by its equation. Here’s how to find it:
Using the Quadratic Equation
- Standard Form: If your quadratic sequence is represented by the equation in the form y = ax² + bx + c, you can directly calculate the maximum value.
- Formula: The maximum value is found using the formula: max = c - (b² / 4a)
Let's break this down with an example.
Example
Suppose you have a quadratic sequence with the equation y = -2x² + 8x + 5. Here's how to calculate the maximum:
- Identify a, b, and c:
- a = -2
- b = 8
- c = 5
- Apply the formula:
- max = c - (b² / 4a)
- max = 5 - (8² / (4 * -2))
- max = 5 - (64 / -8)
- max = 5 - (-8)
- max = 5 + 8
- max = 13
Therefore, the maximum value of the quadratic sequence represented by y = -2x² + 8x + 5 is 13.
Understanding the Concept
- Parabola: The graph of a quadratic equation is a parabola. If 'a' is negative (as in our example), the parabola opens downwards, having a maximum point at the vertex. If 'a' is positive, the parabola opens upwards, having a minimum point at the vertex.
- Vertex: The vertex of a parabola is the point where the curve changes direction (either the lowest or the highest point).
Practical Insights
- Vertex as Max/Min: The vertex gives you either the maximum or minimum value of the quadratic, depending on the sign of 'a'.
- Real-world Applications: Finding maximum values is crucial in various fields, including physics (projectile motion), business (maximizing profit), and engineering (optimizing structures).
