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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?
- Horizontal shift
- Shape of the parabola (correct)
- The x-intercepts
- Vertical shift
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?
- Vertex form (correct)
- Factored form
- Intercept form
- Standard form
What does the vertex of a parabola represent in a quadratic function?
What does the vertex of a parabola represent in a quadratic function?
- The y-intercept
- The x-intercepts
- Minimum or maximum point (correct)
- The point of inflection
What is the axis of symmetry of a parabola?
What is the axis of symmetry of a parabola?
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?
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?
What is the factored form of a quadratic function?
What is the factored form of a quadratic function?
How can a table of values be created for a quadratic function?
How can a table of values be created for a quadratic function?
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?
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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