How to Mastering the Art of Function Transformations

Step-by-step Guide to Mastering the Art of Function Transformations

Here is a step-by-step guide to mastering the art of function transformations:

Understanding the Parent Function

A parent function is the simplest form of a function family, which serves as a template for graphing other functions within that family. Examples of parent functions include:

• Constant Function: \(y=c\)
• Linear Function: \(y=x\)
• Absolute Value Function: \(y=∣x∣\)
• Polynomial Function: \(y=x^n\)
• Rational Function: \(y=\frac{1}{x}​\)
• Radical Function: \(y=\sqrt{x​}\)
• Exponential Function: \(y=e^x\)
• Logarithmic Function: \(y=log(x)\)

Transformations of Functions

Horizontal Shift

• Right Shift \((y = f(x – k))\): If \(k>0\), the graph shifts \(k\) units to the right.
• Left Shift \((y = f(x + k))\): If \(k>0\), the graph shifts \(k\) units to the left.

Vertical Shift

• Upward Shift \((y = f(x) + k)\): If \(k>0\), the graph shifts \(k\) units upwards.
• Downward Shift \((y = f(x) – k)\): If \(k>0\), the graph shifts \(k\) units downwards.

Vertical Stretch and Compression

• Vertical Stretch \((y = kf(x))\): If \(k>1\), the graph stretches vertically away from the \(x\)-axis.
• Vertical Compression \((y = kf(x))\): If \(0<k<1\), the graph compresses towards the \(x\)-axis.

Horizontal Stretch and Compression

• Horizontal Compression \((y = f(kx))\): If \(k>1\), the graph compresses towards the \(y\)-axis.
• Horizontal Stretch \((y = f(kx))\): If \(0<k<1\), the graph stretches away from the \(y\)-axis.

Reflections

• Reflection Across the \(Y\)-Axis \((y = f(-x))\): The graph is mirrored across the \(y\)-axis.
• Reflection Across the \(X\)-Axis \((y = -f(x))\): The graph is mirrored across the \(x\)-axis.

Applying Transformations

1. Identify the Parent Function: Determine which parent function your function is based on.
2. Determine the Transformation(s): Look at the function and identify any modifications to the parent function. These can be shifts, stretches/compressions, or reflections.
3. Apply the Transformations Step-by-Step: If multiple transformations are present, apply them one at a time. Begin with reflections, followed by horizontal transformations (shifts and stretches/compressions), and finally vertical transformations.
4. Graph the Transformed Function: Using the transformations, graph the new function. Be mindful of key points, like the vertex of a parabola, which can help guide the transformation.

Final Word

Understanding and applying transformations to functions allow for a deeper comprehension of how changes in the function equation affect its graph. By mastering these techniques, one can predict and visualize the behavior of complex functions.

Example:

What is the parent graph of the following function and what transformations have taken place on it: \(y = 3(x – 4)^2 + 2\)

Solution:

1. Parent Graph: The parent graph for this function is \(y=x^2\), which is a basic quadratic function or a parabola.
2. Transformations:
   • Horizontal Shift: The term \((x−4)\) indicates a horizontal shift. Since it’s \((x−4)\), the shift is \(4 \) units to the right.
   • Vertical Stretch: The coefficient \(3\) in front of\((x−4)^2\) suggests a vertical stretch. The graph is stretched by a factor of \(3\).
   • Vertical Shift: The \(+2\) at the end of the function indicates a vertical shift upwards by \(2\) units.
3. Summary of Transformations:
   • The graph of \(y=x^2\) is shifted \(4\) units to the right, stretched vertically by a factor of \(3\), and then shifted upwards by \(2\) units.