Transformation Of Quadratic Functions Worksheet

Transforming Quadratic Functions: A Comprehensive Worksheet Guide

Understanding quadratic functions and their transformations is crucial for success in algebra and beyond. This comprehensive guide will walk you through the key concepts, providing clear explanations and examples to help you master this essential topic. We'll delve into various transformations – translations, reflections, and stretches/compressions – and show you how to apply them to the parent function, y = x², to create a wide variety of quadratic graphs. This guide also serves as a complete worksheet, offering practice problems at various difficulty levels.

Introduction to Quadratic Functions

A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (typically x) is 2. The general form of a quadratic function is:

f(x) = ax² + bx + c

where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve. The parent function, the simplest quadratic function, is y = x². This parabola opens upwards, has its vertex at the origin (0,0), and its axis of symmetry is the y-axis (x=0).

Transformations of Quadratic Functions

Transformations alter the parent function's graph, shifting, reflecting, or stretching/compressing it. We'll explore each type:

1. Vertical Translations:

These shifts move the parabola up or down along the y-axis. They are represented by adding or subtracting a constant value, k, to the function:

f(x) = x² + k

• If k > 0, the parabola shifts upward by k units.
• If k < 0, the parabola shifts downward by |k| units.

Example: y = x² + 3 shifts the parent function 3 units upward. y = x² - 2 shifts it 2 units downward.

2. Horizontal Translations:

These shifts move the parabola left or right along the x-axis. They are represented by adding or subtracting a constant value, h, within the squared term:

f(x) = (x - h)²

• If h > 0, the parabola shifts to the right by h units.
• If h < 0, the parabola shifts to the left by |h| units. Note the seemingly counterintuitive nature: y = (x + 2)² shifts the graph 2 units to the left.

Example: y = (x - 4)² shifts the parent function 4 units to the right. y = (x + 1)² shifts it 1 unit to the left.

3. Vertical Stretches and Compressions:

These transformations change the parabola's vertical width. They are represented by multiplying the function by a constant value, a:

f(x) = ax²

• If |a| > 1, the parabola is vertically stretched (narrower).
• If 0 < |a| < 1, the parabola is vertically compressed (wider).
• If a < 0, the parabola is reflected across the x-axis (opens downwards).

Example: y = 2x² stretches the parabola vertically. y = (1/2)x² compresses it vertically. y = -x² reflects it across the x-axis.

4. Horizontal Stretches and Compressions:

These transformations change the parabola's horizontal width. They are represented by multiplying the x within the squared term by a constant value, b:

f(x) = (bx)²

• If |b| > 1, the parabola is horizontally compressed (narrower).
• If 0 < |b| < 1, the parabola is horizontally stretched (wider).

Example: y = (2x)² compresses the parabola horizontally. y = (1/2x)² stretches it horizontally. Note that this is less common than vertical stretches and compressions.

Combining Transformations

Quadratic functions often involve multiple transformations. The general form incorporating all these is:

f(x) = a(x - h)² + k

where:

• a affects vertical stretches/compressions and reflections.
• h affects horizontal translations.
• k affects vertical translations.

The vertex of this parabola is located at the point (h, k). The axis of symmetry is the vertical line x = h.

Worksheet Problems

Now, let's put your knowledge to the test! For each problem, identify the transformations applied to the parent function y = x² and sketch the graph.

Level 1 (Basic):

1. y = x² + 5
2. y = x² - 3
3. y = (x - 2)²
4. y = (x + 4)²
5. y = 3x²
6. y = (1/3)x²
7. y = -x²

Level 2 (Intermediate):

1. y = 2(x - 1)² + 3
2. y = - (x + 2)² - 1
3. y = (1/2)(x + 3)² - 4
4. y = -3(x - 2)² + 5
5. y = 0.5(x + 1)²
6. y = -2x² + 4
7. y = (x-3)² + 2

Level 3 (Advanced):

1. Describe the transformations of y = -2(x + 1)² + 4 compared to y = x² and find the vertex and axis of symmetry.
2. Write the equation of a parabola that opens downward, has a vertex at (-2, 3), and is vertically stretched by a factor of 2.
3. Write the equation of a parabola that is shifted 5 units to the right, 1 unit down, and reflected across the x-axis.
4. Find the vertex and axis of symmetry for y = 0.25(x - 6)² - 8. Describe the transformation compared to the parent function.
5. If a parabola has a vertex at (3, -1) and passes through the point (4, 2), what is its equation? (Hint: use the vertex form and substitute the point).
6. Graph and describe the transformations of y = -(x + 2)² + 5. Indicate the vertex, axis of symmetry and y-intercept.

Solutions (Level 1):

1. Up 5 units
2. Down 3 units
3. Right 2 units
4. Left 4 units
5. Vertically stretched by a factor of 3
6. Vertically compressed by a factor of 1/3
7. Reflected across the x-axis

Explanations and Further Exploration:

Remember to meticulously plot points to accurately graph each transformed function. You can use a table of x and y values to help with this. For instance, for y = 2(x - 1)² + 3, you might start by choosing x values around the vertex (1, 3) like 0, 1, 2, 3 etc., and calculating the corresponding y values. This will provide sufficient points to plot the parabola.

The advanced problems require you to work backward, using the given information to determine the equation of the parabola. Remember the vertex form is invaluable in these scenarios.

This comprehensive worksheet, covering basic to advanced levels, should significantly improve your understanding of transforming quadratic functions. Remember, practice is key! The more problems you work through, the more comfortable and proficient you will become with manipulating and interpreting quadratic equations and their graphs. Good luck!