This worksheet explores quadratic functions and their transformations. Understanding these concepts is crucial for success in algebra and beyond, as they form the foundation for many advanced mathematical topics. We'll cover key aspects, providing examples and explanations to solidify your understanding.
Understanding the Parent Function
The parent function for quadratic functions is f(x) = x². This simple equation represents a parabola that opens upwards, with its vertex at the origin (0,0). All other quadratic functions are transformations of this parent function.
Key Features of the Parent Function:
- Vertex: (0,0)
- Axis of Symmetry: x = 0
- Concavity: Opens upwards (positive leading coefficient)
- y-intercept: (0,0)
Transformations of Quadratic Functions
Transformations alter the parent function, shifting, stretching, or reflecting the parabola. We'll examine four key transformations:
1. Vertical Shifts
Adding or subtracting a constant 'k' to the function shifts the parabola vertically.
- f(x) + k: Shifts the graph upwards by 'k' units.
- f(x) - k: Shifts the graph downwards by 'k' units.
Example: f(x) = x² + 3 shifts the parabola three units upwards.
2. Horizontal Shifts
Adding or subtracting a constant 'h' within the parentheses shifts the parabola horizontally.
- f(x - h): Shifts the graph to the right by 'h' units.
- f(x + h): Shifts the graph to the left by 'h' units.
Example: f(x) = (x + 2)² shifts the parabola two units to the left. Remember, it's counter-intuitive: adding within the parentheses moves the graph left.
3. Vertical Stretches and Compressions
Multiplying the function by a constant 'a' stretches or compresses the parabola vertically.
- af(x), |a| > 1: Stretches the graph vertically.
- af(x), 0 < |a| < 1: Compresses the graph vertically.
- -f(x): Reflects the graph across the x-axis (opens downwards).
Example: f(x) = 2x² stretches the parabola vertically by a factor of 2. f(x) = -x² reflects the parabola across the x-axis.
4. Horizontal Stretches and Compressions
This transformation is less common but involves modifying the 'x' value inside the parentheses. This is generally less intuitive and often dealt with by analyzing the 'a' value in the general form.
The General Form of a Quadratic Function
The general form of a quadratic function is: f(x) = a(x - h)² + k
Where:
- a: Determines the vertical stretch/compression and reflection.
- h: Determines the horizontal shift.
- k: Determines the vertical shift.
- (h, k): Represents the vertex of the parabola.
Understanding this general form allows you to quickly identify the vertex and other key characteristics of any quadratic function.
Practice Problems
Now, let's test your understanding with some practice problems:
- Describe the transformations applied to the parent function f(x) = x² to obtain g(x) = 2(x - 1)² + 3.
- Find the vertex and axis of symmetry for the quadratic function h(x) = -(x + 4)² - 2.
- Write the equation of a parabola that opens downwards, has a vertex at (2, -5), and is vertically stretched by a factor of 3.
- Sketch the graph of f(x) = (x - 3)² - 1, labeling the vertex and axis of symmetry.
- Explain how the value of 'a' affects the shape and orientation of a parabola.
This worksheet provides a foundation for understanding quadratic functions and their transformations. Mastering these concepts will significantly improve your problem-solving abilities in algebra and beyond. Remember to practice regularly and consult additional resources if needed. Further exploration into completing the square and finding x-intercepts will deepen your understanding further.
