Goals
1. Understand and describe the key features of quadratic functions.
2. Identify the inputs (x values) and outputs (y values) of a quadratic function.
Contextualization
Quadratic functions are widely applicable in many areas of study. From physics, where they help explain projectile motion, to economics, where they're used to analyze costs and profits, these functions are crucial for modeling and resolving real-world issues. Learning to pinpoint the inputs (x values) and outputs (y values) of a quadratic function is essential for practical applications.
Subject Relevance
To Remember!
Definition of Quadratic Function
A quadratic function is a polynomial expressed in the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' cannot be zero. The graph of this function is a parabola, which opens either upwards or downwards based on the sign 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 is 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 are the 'x' values we plug into the function, and the outputs are the 'y' values we get out in return. The connection between x and y is described by the function formula.
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Input: The value of 'x' entered into the function
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Output: The value of 'y' calculated from the function
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Relationship: The output 'y' results from substituting 'x' back 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 is where it reaches its highest or lowest point. For a quadratic function f(x) = ax² + bx + c, the vertex can be determined with 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 marks 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: Calculating the flight path of rockets and satellites.
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Economics: Evaluating costs and profits to enhance business performance.
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Finance: Predicting stock trends and fine-tuning investment portfolios.
Key Terms
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Quadratic Function: A polynomial function in the form f(x) = ax² + bx + c.
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Parabola: The visual representation of a quadratic function.
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Vertex: The highest or lowest point on a parabola.
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Roots: The values of 'x' at which f(x) = 0.
Questions for Reflections
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How can understanding quadratic functions contribute to solving real-life issues like cost management in a business?
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In what ways do quadratic functions appear in your everyday experiences?
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How are quadratic functions relevant to the careers you aim to pursue?
Modeling the Trajectory of a Rocket
In this mini-challenge, you will use the principles of quadratic functions to model a rocket's trajectory. Work collaboratively to create a simplified model of a rocket and calculate its parabolic path using a quadratic function.
Instructions
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Form groups of 4-5 students.
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Utilize cardboard, scissors, tape, and a ruler to make a basic rocket model.
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Define a quadratic function that illustrates the rocket's flight path.
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Calculate the input values (x) and output values (y) of the function, along with the vertex and roots.
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Draw the trajectory on poster board or a whiteboard.
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Each group should present their model and explain the calculations involved.
