Goals
1. Recognize how to represent a quadratic function using graphs and tables.
2. Distinguish between graphical and tabular formats.
3. Sketch a graph of a quadratic function.
Contextualization
Quadratic functions play a crucial role in mathematics and can be observed in numerous real-life scenarios. A typical example is the path of objects moving in a parabolic trajectory, like a ball thrown into the air. They're also applicable in other contexts such as tracking rocket trajectories, assessing maximum profit for businesses, and even estimating population growth.
Subject Relevance
To Remember!
Concept of Quadratic Function
A quadratic function is a polynomial of degree 2, typically written as y = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The term 'quadratic' comes from the fact that the variable x is raised to the power of 2.
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The general form is y = ax² + bx + c.
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The graph of a quadratic function takes the shape of a parabola.
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The value of 'a' influences the direction the parabola opens (upward if a > 0, downward if a < 0).
Graphical Representation of Quadratic Functions
A quadratic function is represented graphically as a parabola on the Cartesian plane. The shape of this parabola is influenced by the coefficients a, b, and c. The vertex of the parabola indicates the function's maximum or minimum based on whether 'a' is positive or negative.
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You can find the vertex with the formula (-b/2a, f(-b/2a)).
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The roots of the function are where the parabola crosses the x-axis.
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The parabola is symmetrical around the vertical line through the vertex.
Value Table for Quadratic Functions
A value table for a quadratic function contains pairs of x values and their corresponding y values (or f(x)). This table is vital for visualizing the function's behavior across various x values and is a key aid for sketching the function's graph.
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Select x values from a certain range to calculate the relevant y values.
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This table helps pinpoint important coordinates such as the roots and the vertex.
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It aids in grasping the shape of the parabola before creating the graph.
Practical Applications
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Engineering: Utilizing quadratic functions to determine the path of projectiles and enhance building materials.
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Economics: Modeling profit and cost calculations to pinpoint maximum profit or minimum costs.
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Data Science: Implementing machine learning algorithms that leverage quadratic functions to forecast trends and behaviors.
Key Terms
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Quadratic Function: A polynomial function of degree 2, written as y = ax² + bx + c.
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Parabola: The graphical representation of a quadratic function, characterized by its symmetrical curve.
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Vertex: The point of maximum or minimum of the parabola, calculated using the formula (-b/2a, f(-b/2a)).
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Roots: The x-intercepts of the parabola, also known as the zeros of the function.
Questions for Reflections
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How could the skills of creating and interpreting graphs of quadratic functions benefit your future careers?
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In what practical scenarios can quadratic functions be applied to resolve daily challenges?
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Why is it important to understand the shape and behavior of parabolas in various professional fields?
Exploring Quadratic Functions in the Real World
This mini-challenge invites students to apply their understanding of quadratic functions in practical, everyday contexts.
Instructions
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Form groups of 3-4 students.
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Select a real-world scenario where a quadratic function is applicable (like the path of a thrown ball, calculating a company's profits, etc.).
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Model the chosen scenario using a quadratic function. Find the coefficients a, b, and c that best represent the situation.
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Create a value table for the modeled function by selecting suitable x values for your scenario.
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Draw the graph of the quadratic function on graph paper, highlighting vital points such as the vertex and the roots.
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Prepare a brief presentation (3-5 minutes) to share with the class, detailing the chosen scenario, the modeled function, the corresponding value table, and the sketched graph.
