Quadratic functions, those elegant parabolas, are fundamental to algebra and beyond. Understanding how to transform them—shifting, stretching, and reflecting—is key to mastering their behavior and applications. This worksheet will guide you through the core concepts and provide ample practice to solidify your understanding.
Understanding the Parent Function
Before delving into transformations, let's establish a baseline. The parent quadratic function is f(x) = x². This simple equation generates a parabola symmetric about the y-axis, with its vertex at the origin (0,0). All other quadratic functions are transformations of this parent function.
Key Transformations:
We can manipulate the parent function using various transformations, each impacting the graph in a predictable way:
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Vertical Shifts: Adding or subtracting a constant 'k' to the function shifts the parabola vertically.
f(x) = x² + kshifts upwards by 'k' units, andf(x) = x² - kshifts downwards by 'k' units. -
Horizontal Shifts: Adding or subtracting a constant 'h' inside the function, as in
f(x) = (x - h)², shifts the parabola horizontally. A positive 'h' shifts the parabola to the right, and a negative 'h' shifts it to the left. This can be confusing at first, but remember it's the opposite of what you might initially expect. -
Vertical Stretches and Compressions: Multiplying the function by a constant 'a' affects the parabola's vertical stretch or compression. If
|a| > 1, the parabola is stretched vertically (narrower). If0 < |a| < 1, the parabola is compressed vertically (wider). A negative 'a' reflects the parabola across the x-axis. -
Horizontal Stretches and Compressions: These transformations are less common but equally important. They involve modifying the 'x' value before it's squared. The function
f(x) = (bx)²horizontally compresses the parabola if|b| > 1and stretches it if0 < |b| < 1. A negative 'b' reflects the parabola across the y-axis.
General Form of a Transformed Quadratic Function
Combining these transformations, we arrive at the general form:
f(x) = a(x - h)² + k
Where:
- 'a' controls vertical stretches/compressions and reflections across the x-axis.
- 'h' controls horizontal shifts.
- 'k' controls vertical shifts.
- (h, k) represents the vertex of the parabola.
Practice Problems:
Now let's put your knowledge to the test! For each problem, identify the transformations applied to the parent function f(x) = x², describe the transformations, and sketch the graph (or use graphing software to verify your understanding).
f(x) = (x - 3)² + 2f(x) = -2x² - 1f(x) = (x + 1)² - 4f(x) = 0.5(x + 2)²f(x) = -(x - 1)² + 3f(x) = 3(x + 2)² + 1
Advanced Concepts (Optional):
- Finding the Equation from a Graph: Can you determine the equation of a quadratic function given its graph, identifying the vertex and a point on the curve?
- Applications: Quadratic functions model many real-world phenomena, like projectile motion. How can transformation knowledge help solve real-world problems?
This worksheet provides a foundation for understanding quadratic function transformations. Mastering these concepts will significantly improve your ability to analyze, interpret, and apply quadratic functions in various contexts. Remember to practice regularly to solidify your understanding and build confidence.
