transformations of quadratic functions worksheet

Lemon

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04-02-2025

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This worksheet explores the fascinating world of quadratic functions and how their graphs can be manipulated through various transformations. Understanding these transformations is crucial for visualizing and analyzing quadratic equations effectively. We'll cover translations (shifts), reflections, and stretches/compressions. Let's dive in!

Understanding the Parent Function

Before we explore transformations, it's essential to understand the parent quadratic function: f(x) = x². This function forms a parabola with its vertex at the origin (0,0). All other quadratic functions are transformations of this parent function.

Key Transformations:

1. Vertical Translations: These shifts move the parabola up or down.

   • Upward Shift: f(x) = x² + k, where k is a positive constant. The parabola shifts k units upwards.
   • Downward Shift: f(x) = x² - k, where k is a positive constant. The parabola shifts k units downwards.

2. Horizontal Translations: These shifts move the parabola left or right.

   • Right Shift: f(x) = (x - h)², where h is a positive constant. The parabola shifts h units to the right.
   • Left Shift: f(x) = (x + h)², where h is a positive constant. The parabola shifts h units to the left.

3. Reflections: These transformations flip the parabola across an axis.

   • Reflection across the x-axis: f(x) = -x². The parabola flips upside down.
   • Reflection across the y-axis: Reflecting x² across the y-axis results in the same parabola, as it's symmetric.

4. Vertical Stretches and Compressions: These transformations alter the parabola's width.

   • Vertical Stretch: f(x) = ax², where a is a constant greater than 1. The parabola becomes narrower.
   • Vertical Compression: f(x) = ax², where a is a constant between 0 and 1. The parabola becomes wider.

Combining Transformations

The real power comes from combining these transformations. Consider a function like f(x) = a(x - h)² + k. This represents a parabola with:

• Vertex: (h, k)
• Vertical Stretch/Compression Factor: a
• Horizontal Shift: h units to the right (if h is positive), or h units to the left (if h is negative)
• Vertical Shift: k units upwards (if k is positive), or k units downwards (if k is negative)

If a is negative, the parabola opens downwards (a reflection across the x-axis).

Practice Problems

Now let's test your understanding with some practice problems. For each equation, identify the transformations applied to the parent function f(x) = x², and describe the resulting graph's vertex, direction of opening, and whether it's stretched or compressed.

1. g(x) = (x - 3)² + 2
2. h(x) = -2x² - 1
3. i(x) = 0.5(x + 1)²
4. j(x) = -(x + 2)² + 4
5. k(x) = 3(x - 1)² - 5

Solutions (Check your work!)

This section will provide detailed solutions to the practice problems above, helping you verify your understanding of quadratic function transformations. Remember to carefully consider each transformation individually and then combine them to analyze the complete graph. This approach will build your confidence and proficiency in working with quadratic functions.

This worksheet provides a solid foundation for understanding transformations of quadratic functions. Remember to practice regularly to master these concepts. Good luck!