# How to Mastering the Art of Function Transformations

Transformations of functions are techniques used in mathematics to modify the graph of a function in various ways while preserving the overall shape and characteristics of the function. These transformations include shifting, stretching, compressing, and reflecting the graph of a function. Let's go through a step-by-step guide on how to apply these transformations, using the concept of a parent function as the starting point. ## 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.

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