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Questions and Answers
What does the value of 'a' determine in a quadratic function?
What does the value of 'a' determine in a quadratic function?
In which form is a quadratic function given by the equation y = a(x - h)² + k?
In which form is a quadratic function given by the equation y = a(x - h)² + k?
What does the vertex of a parabola represent in a quadratic function?
What does the vertex of a parabola represent in a quadratic function?
What is the axis of symmetry of a parabola?
What is the axis of symmetry of a parabola?
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How does the value of 'h' affect the graph of a quadratic function?
How does the value of 'h' affect the graph of a quadratic function?
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What is the significance of the sign of 'a' in a quadratic function?
What is the significance of the sign of 'a' in a quadratic function?
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What is the factored form of a quadratic function?
What is the factored form of a quadratic function?
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How can a table of values be created for a quadratic function?
How can a table of values be created for a quadratic function?
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What do the values of x and y represent in a table of values for a quadratic function?
What do the values of x and y represent in a table of values for a quadratic function?
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What does the vertex of a parabola represent?
What does the vertex of a parabola represent?
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Study Notes
Quadratic Functions and Their Representations
Quadratic Functions
Quadratic functions are mathematical functions that involve the square of a variable. They are important because they are used to model a wide range of real-world phenomena, such as projectile motion and the behavior of electric circuits. Quadratic functions are defined by the formula:
y = a(x - h)² + k
where a, h, and k are constants, and x is the independent variable. The value of a determines the shape of the parabola, with positive values resulting in an upward-opening parabola and negative values resulting in a downward-opening one. The value of h represents the horizontal shift of the parabola, and k represents the vertical shift.
Graphs
The graph of a quadratic function is a parabola, which is a "u"-shaped curve. The graph of the equation y = ax² + bx + c is a parabola with vertex at (-b/2a, c - b²/4a) and opening up if a > 0 and down if a < 0. The x-axis is the axis of symmetry, and the parabola is symmetric with respect to the x-axis.
Equations
Quadratic functions can be represented in various forms, including vertex form, standard form, and factored form. The vertex form is y = a(x - h)² + k, where a, h, and k are constants and (h, k) is the vertex of the parabola. The standard form is y = a(x - h)² + k, where a, h, and k are constants and (h, k) is the vertex of the parabola. The factored form is y = a(x - r)(x - s), where a, r, and s are constants and (r, s) is the vertex of the parabola.
Tables
Quadratic functions can be represented in a table of values. To create a table of values for a quadratic function, choose values for x and substitute them into the equation to find the corresponding values of y. For example, for the function y = x² - 2x + 1, a table of values could be:
x | y
---|---
1 | 0
2 | 1
3 | 4
4 | 9
5 | 16
Values
The values of a quadratic function can be found by plugging in values of x into the equation and simplifying. For example, for the function y = 3x² - 4x + 2, the value of y when x = 2 is:
y = 3(2)² - 4(2) + 2
y = 3(4) - 8 + 2
y = 12 - 8 + 2
y = 4
In conclusion, quadratic functions are an important part of mathematics, and their representations, including graphs, equations, tables, and values, are essential for understanding and working with these functions.
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Description
Explore the various representations and properties of quadratic functions, including equations, graphs, tables, and values. Learn how to identify key characteristics such as the vertex, axis of symmetry, and direction of opening for parabolas. Practice finding values of quadratic functions through calculations and table of values.
