Goals
1. Grasp the concept of quadratic functions.
2. Distinguish quadratic functions from linear functions.
3. Identify the key characteristics of quadratic functions.
Contextualization
Quadratic functions, often referred to as second-degree functions, are more prevalent in our everyday experiences than we might think. For instance, the path of a cricket ball when it’s hit into the air forms a parabola, which is a hallmark of quadratic functions. Additionally, these functions play a vital role across various fields like physics, engineering, and economics, where they are used to address and model intricate challenges.
Subject Relevance
To Remember!
Definition of Quadratic Function
A quadratic function, or second-degree function, is a polynomial function of degree 2 and has the general form y = ax² + bx + c, where a, b, and c are real coefficients and a ≠ 0. The presence of the ax² term is what sets it apart from linear functions, which do not contain this term.
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General Form: y = ax² + bx + c.
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Coefficient a: Must not be zero; dictates the shape of the parabola.
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Coefficients b and c: Can take any real number; influence the parabola's position and appearance.
Graph of Quadratic Function: The Parabola
The graph of a quadratic function is shaped like a parabola. This parabola can either open upwards (a > 0) or downwards (a < 0). The coefficients a, b, and c affect the position and form of the parabola.
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Concavity: Determined by the sign of coefficient a.
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Vertex: The peak or lowest point of the parabola, calculable using x = -b/(2a) and y = f(x).
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Roots: Points where the parabola crosses the x-axis (solutions to the equation ax² + bx + c = 0).
Coefficients a, b, and c and Their Effects on the Graph
The coefficients a, b, and c in the equation y = ax² + bx + c have distinct roles in shaping the parabola. The coefficient a determines the direction and width of the parabola, b influences the slope and horizontal position, and c indicates where the parabola intersects the y-axis.
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Coefficient a: Manifests the width and orientation of the parabola.
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Coefficient b: Influences the vertex's displacement along the x-axis.
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Coefficient c: Marks the point of intersection with the y-axis.
Practical Applications
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In civil engineering, quadratic functions are pivotal for determining the shapes of bridges and arches, which supports structural endurance and efficiency.
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In economics, these functions are applied to model costs and revenues, aiding firms in shaping strategies around production and pricing.
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In programming and data analysis, quadratic functions are utilized in optimization processes and predictive modelling, simplifying complex problem-solving.
Key Terms
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Quadratic Function: A polynomial function of degree 2 in the form y = ax² + bx + c.
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Parabola: The graphical representation of a quadratic function, defined by a symmetric arc.
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Coefficients a, b, c: Values that shape and position the parabola.
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Vertex: The highest or lowest point of the parabola.
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Roots: Points where the parabola touches the x-axis.
Questions for Reflections
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How can quadratic functions assist in resolving problems in sectors like engineering, economics, and technology?
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In what ways can a clear understanding of coefficients a, b, and c help us model everyday scenarios?
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What major challenges did you face while working with parabolas, and what strategies did you adopt to overcome them?
Practical Modeling with Quadratic Functions
Let’s solidify our grasp of quadratic functions through a hands-on challenge that involves designing and examining a parabola.
Instructions
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Form groups of 4 to 5 members.
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Using pliable wire, construct a parabola according to the provided coefficients a, b, and c.
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Secure the constructed parabola to graph paper and sketch the corresponding quadratic function.
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Calculate the vertex and roots of the parabola you’ve created.
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Present your physical model and the resulting graph to the class, mentioning how the selected coefficients affect the parabola's shape and location.
