Goals
1. Recognize and describe the unique features of a quadratic function.
2. Identify the inputs (x values) and outputs (y values) in a quadratic function.
Contextualization
Quadratic functions find application in numerous disciplines. From the physics of projectile motion to economics involving cost and profit analysis, these functions play a critical role in modelling and addressing real-world challenges. Grasping how to pinpoint the inputs (x values) and outputs (y values) of a quadratic function is vital for translating this knowledge into actionable insight.
Subject Relevance
To Remember!
Definition of Quadratic Function
A quadratic function is a type of polynomial function expressed in the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is non-zero. The graph represents a parabola, which can open upwards or downwards based on the value of 'a'.
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General Form: f(x) = ax² + bx + c
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Constants: 'a', 'b', and 'c' are real numbers, with 'a' ≠ 0
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Graph: The graph depicts a parabola
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Direction of the Parabola: The parabola opens upwards if 'a' > 0 and downwards if 'a' < 0
Identifying Inputs (x) and Outputs (y)
The inputs of a quadratic function refer to the 'x' values fed into the function, while the outputs are the resulting 'y' values. The connection between x and y is captured through the function formula.
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Input: The value of 'x' plugged into the function
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Output: The resulting value of 'y' after calculation of the function
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Relationship: The output 'y' is derived by substituting 'x' into the formula f(x)
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Example: For f(x) = x² - 4x + 3, if x = 2, then y = f(2) = 2² - 4*2 + 3 = -1
Calculating the Vertex of the Parabola
The vertex of a parabola signifies the point at which it achieves its maximum or minimum value. For a quadratic function f(x) = ax² + bx + c, one can determine the vertex using the formulas x_v = -b/(2a) and y_v = f(x_v).
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Vertex Formula: x_v = -b/(2a)
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Calculating y_v: y_v = f(x_v)
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Critical Point: The vertex indicates the maximum or minimum of the parabola
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Example: For f(x) = x² - 4x + 3, x_v = 2 and y_v = f(2) = -1
Practical Applications
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Aerospace Engineering: Predicting the pathway of rockets and satellites.
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Economics: Evaluating costs and profits to enhance business efficiency.
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Finance: Predicting stock performance and refining investment strategies.
Key Terms
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Quadratic Function: A polynomial function represented as f(x) = ax² + bx + c.
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Parabola: The graph associated with a quadratic function.
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Vertex: The peak or trough point of a parabola.
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Roots: The x-values at which f(x) = 0.
Questions for Reflections
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How can the understanding of quadratic functions assist in practical challenges like cost optimization in business?
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What instances can you identify where quadratic functions play a role in your day-to-day life?
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In what aspects do you find quadratic functions significant for your desired career path?
Modeling the Trajectory of a Rocket
In this mini-challenge, you will employ the principles of quadratic functions to craft a model of a rocket's trajectory. Working collaboratively, you will create a simplified rocket model and calculate its parabolic pathway using a quadratic function.
Instructions
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Form groups of 4-5 members.
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Utilize cardboard, scissors, tape, and a ruler to assemble a basic rocket model.
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Define a quadratic function that depicts the rocket's trajectory.
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Calculate the input values (x) and output values (y) of the function, including the vertex and roots.
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Illustrate the trajectory on a poster board or whiteboard.
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Each group should present their model and elucidate the calculations undertaken.
