Goals
1. Grasp the concept of maxima and minima in quadratic functions.
2. Use max and min calculations to tackle real-world problems, such as figuring out the maximum area of a rectangle given a fixed perimeter.
3. Enhance analytical skills through identifying and solving mathematical issues linked to quadratic functions.
4. Promote teamwork through engaging group activities.
Contextualization
Quadratic functions are integral in modelling various real-world scenarios, from the path of a projectile to maximising profits for businesses, or optimising areas and volumes in engineering tasks. For instance, when determining how high a rocket can soar, we apply a quadratic function to depict its trajectory. Similarly, businesses use quadratic functions to optimise resources, helping to pinpoint the point of maximum efficiency or profit. Mastery in identifying the highest and lowest points of these functions is vital for addressing practical challenges efficiently.
Subject Relevance
To Remember!
Concept of Quadratic Function
A quadratic function is a polynomial of degree 2, usually expressed as f(x) = ax² + bx + c, where a, b, and c are real numbers, with a not equal to 0. The graphical form of a quadratic function is a parabola, which opens either upwards (if a > 0) or downwards (if a < 0).
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The graphical representation is a parabola.
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The coefficients a, b, and c shape and position the parabola.
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The coefficient 'a' dictates the concavity of the parabola (upward or downward).
Identification of Coefficients a, b, and c
Successfully solving quadratic function problems hinges on accurately identifying coefficients a, b, and c in the expression f(x) = ax² + bx + c. These coefficients directly affect the parabola's characteristics, including its concavity and positioning in the Cartesian plane.
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The coefficient 'a' impacts the width and orientation of the parabola.
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The coefficient 'b' affects where the vertex is located along the x-axis.
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The coefficient 'c' signifies where the parabola crosses the y-axis.
Vertex of the Parabola
The vertex of a parabola is the point at which it achieves its maximum or minimum value. For f(x) = ax² + bx + c, the vertex is found using the formulas x = -b/(2a) and y = f(-b/(2a)). The vertex is essential for identifying the maximum and minimum points of the function.
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The x-coordinate of the vertex is given by -b/(2a).
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The y-coordinate of the vertex is calculated by substituting x back into the function f(x).
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The vertex indicates the peak point (if a < 0) or bottom point (if a > 0) of the parabola.
Practical Applications
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Engineering: Ascertain the maximum height attained by a projectile or rocket, employing a quadratic function to model its flight path.
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Economics and Business: Use quadratic functions to maximise profits or minimise costs by modelling revenues and expenses.
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Architecture and Design: Optimise the area or volume of structures, such as calculating the maximum area of a rectangle with a given perimeter.
Key Terms
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Quadratic Function: A polynomial of degree 2, represented by f(x) = ax² + bx + c.
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Coefficients a, b, and c: Values that determine the shape and placement of the parabola in a quadratic function.
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Vertex: The maximum or minimum point of a parabola, calculated with x = -b/(2a) and y = f(-b/(2a)).
Questions for Reflections
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How does accurately identifying coefficients a, b, and c affect effective resolution of real-world problems?
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In what ways can the practical uses of maxima and minima in quadratic functions enhance efficiency within a company?
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What hurdles might you face when modelling real issues with quadratic functions and how might you address them?
Final Challenge: Resource Optimisation in a Company
Employ the concepts learnt about quadratic functions to address a practical resource optimisation problem within a company.
Instructions
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Form groups of 3 to 4 students.
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Each group must model the revenue function R(x) = -5x² + 50x - 80, where x represents units sold.
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Identify the maximum point of the function to ascertain the number of units that maximises revenue.
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Calculate the maximum revenue achievable by the company.
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Present the calculations and findings to the class, detailing the rationale behind your approach.
