Original vs. Transformed Function Graphs

Understanding Transformations

Transformations alter the position or shape of a function’s graph. Understanding how different transformations affect the original function is crucial. The general transformed function can be represented as:

$$y = a f(b(x - h)) + k$$

Where:
* a: Vertical dilation (scaling) and/or reflection in the x-axis.
* b: Horizontal dilation (scaling) and/or reflection in the y-axis.
* h: Horizontal translation.
* k: Vertical translation.

Types of Transformations

1. Dilations (Scalings)

• Vertical Dilation (from the x-axis): $y = a f(x)$

  • If $a > 1$, the graph is stretched vertically (away from the x-axis).
  • If \$0 < a < 1$, the graph is compressed vertically (towards the x-axis).
  • If $a < 0$, the graph is also reflected in the x-axis.

• Horizontal Dilation (from the y-axis): $y = f(bx)$

  • If $b > 1$, the graph is compressed horizontally (towards the y-axis). The x-coordinates are divided by $b$.
  • If \$0 < b < 1$, the graph is stretched horizontally (away from the y-axis). The x-coordinates are divided by $b$.
  • If $b < 0$, the graph is also reflected in the y-axis.

2. Reflections

• Reflection in the x-axis: $y = -f(x)$

  • The y-coordinate of each point is multiplied by -1.
  • The graph is flipped over the x-axis.

• Reflection in the y-axis: $y = f(-x)$

  • The x-coordinate of each point is multiplied by -1.
  • The graph is flipped over the y-axis.

3. Translations

• Horizontal Translation: $y = f(x - h)$

  • If $h > 0$, the graph is translated h units to the right.
  • If $h < 0$, the graph is translated h units to the left.

• Vertical Translation: $y = f(x) + k$

  • If $k > 0$, the graph is translated k units upwards.
  • If $k < 0$, the graph is translated k units downwards.

Order of Transformations

The order in which transformations are applied matters. A common and reliable order is:

1. Reflections
2. Dilations
3. Translations

It is often easiest to consider transformations in the order of b, a, h, then k when reading from the equation.

Families of Transformed Functions

For a single transformation parameter, varying the parameter creates a family of transformed functions.

• Example (Vertical Translation): The family of functions $y = x^2 + k$ represents parabolas shifted vertically, where k is the parameter. Changing k shifts the parabola up or down.

• Example (Horizontal Dilation): The family of functions $y = (bx)^2$ represents horizontal dilations of a parabola, where b is the parameter.

Key Features and their Transformations

Understanding how key features of a graph (intercepts, turning points, asymptotes) transform is crucial for sketching.

• Intercepts:

  • x-intercepts are affected by horizontal dilations, horizontal translations, and reflections in the y-axis.
  • y-intercepts are affected by vertical dilations, vertical translations, and reflections in the x-axis.

• Turning Points: Affected by all transformations except reflections across the axis parallel to the coordinate.

• Asymptotes:

  • Vertical asymptotes are affected by horizontal dilations and horizontal translations.
  • Horizontal asymptotes are affected by vertical dilations and vertical translations.

Examples

1. $y = 2(x + 1)^2 - 3$: This is a transformation of $y = x^2$.

   • Vertical dilation by a factor of 2.
   • Horizontal translation 1 unit to the left.
   • Vertical translation 3 units down.

2. $y = -\sqrt{x - 4} + 1$: This is a transformation of $y = \sqrt{x}$.

   • Reflection in the x-axis.
   • Horizontal translation 4 units to the right.
   • Vertical translation 1 unit up.

Table of Transformations

Using Transformations for Graphing

1. Start with the basic function (e.g., $y = x^2$, $y = \sqrt{x}$, $y = \frac{1}{x}$).
2. Apply transformations step-by-step, considering the order of operations.
3. Track how key features (intercepts, turning points, asymptotes) are affected.
4. Sketch the final transformed graph.

Summary

Understanding transformations is essential for sketching graphs and analyzing functions. By recognizing the effects of dilations, reflections, and translations, you can efficiently graph complex functions and solve related problems. Remember the order of transformations. Practice applying these concepts to various functions to master the skill.