transformations with quadratic functions worksheet

2 min read 04-02-2025

transformations with quadratic functions worksheet

This worksheet delves into the fascinating world of quadratic functions and how their graphs are manipulated through various transformations. Understanding these transformations is crucial for grasping the behavior of parabolas and their applications in various fields, from physics to computer graphics. We'll explore vertical and horizontal shifts, reflections, and stretches/compressions, providing you with a solid foundation to master quadratic transformations.

Understanding the Parent Function

Before we dive into transformations, let's establish a baseline. The parent quadratic function is represented as:

f(x) = x²

This function forms a parabola with its vertex at the origin (0,0). All other quadratic functions are essentially transformations of this parent function.

Types of Transformations

We'll explore four key transformations:

1. Vertical Shifts

A vertical shift moves the parabola up or down along the y-axis. This is achieved by adding or subtracting a constant value, 'k', to the parent function:

f(x) = x² + k

k > 0: The parabola shifts upwards by 'k' units.
k < 0: The parabola shifts downwards by 'k' units.

2. Horizontal Shifts

A horizontal shift moves the parabola left or right along the x-axis. This is achieved by adding or subtracting a constant value, 'h', within the parentheses:

f(x) = (x - h)²

h > 0: The parabola shifts to the right by 'h' units.
h < 0: The parabola shifts to the left by 'h' units. Note the seemingly counter-intuitive nature here; subtracting 'h' shifts the graph to the right.

3. Reflections

A reflection flips the parabola across either the x-axis or the y-axis.

Reflection across the x-axis: This is achieved by multiplying the entire function by -1:

f(x) = -x²
Reflection across the y-axis: The standard quadratic function is symmetric about the y-axis, so reflecting it across the y-axis results in the same parabola.

4. Stretches and Compressions

Stretches and compressions alter the "width" or "narrowness" of the parabola. This is controlled by a constant 'a' multiplying the x² term:

f(x) = ax²

|a| > 1: The parabola is vertically stretched (narrower).
0 < |a| < 1: The parabola is vertically compressed (wider).
a < 0: This introduces a reflection across the x-axis in addition to stretching or compressing.

Combining Transformations

The real power of understanding quadratic transformations comes from combining these shifts, reflections, and stretches. A general form incorporating all these transformations is:

f(x) = a(x - h)² + k

Where:

'a' controls stretching/compression and reflection.
'h' controls horizontal shift.
'k' controls vertical shift.

By analyzing the values of 'a', 'h', and 'k', you can quickly determine the transformation applied to the parent function and sketch the resulting parabola.

Worksheet Exercises (Example)

Here are a few example problems to test your understanding. Remember to identify the transformations and sketch the graph:

f(x) = (x + 2)² - 3
f(x) = -2x² + 1
f(x) = 0.5(x - 1)²
f(x) = -(x + 3)² + 4

This worksheet provides a strong foundation for understanding transformations with quadratic functions. By practicing these examples and similar exercises, you'll master the ability to analyze and graph any quadratic function with ease. Remember to focus on identifying the individual transformations and their combined effect on the parent function's graph. Good luck!