1-2 Additional Practice Transformations Of Functions

Let's delve deeper into the fascinating world of function transformations, expanding beyond the basics to explore more nuanced and intricate manipulations. Mastering these advanced transformations will significantly enhance your ability to analyze, manipulate, and visualize functions effectively. We will be exploring two additional transformation techniques, focusing on how they impact the graph and equation of a function.

Understanding Function Transformations: A Quick Recap

Before tackling additional transformations, it's crucial to have a firm grasp on the foundational transformations:

• Vertical Shifts: Adding or subtracting a constant outside the function, f(x) + c or f(x) - c, shifts the graph vertically. +c moves the graph upwards, and -c moves it downwards.
• Horizontal Shifts: Adding or subtracting a constant inside the function, f(x + c) or f(x - c), shifts the graph horizontally. +c moves the graph to the left, and -c moves it to the right (counterintuitive, right?).
• Vertical Stretches/Compressions: Multiplying the function by a constant outside, a*f(x), stretches the graph vertically if |a| > 1 and compresses it if 0 < |a| < 1. If a is negative, it also reflects the graph across the x-axis.
• Horizontal Stretches/Compressions: Multiplying the input variable by a constant, f(bx), stretches the graph horizontally if 0 < |b| < 1 and compresses it if |b| > 1. If b is negative, it also reflects the graph across the y-axis.
• Reflections: Multiplying the entire function by -1, -f(x), reflects the graph across the x-axis. Replacing x with -x, f(-x), reflects the graph across the y-axis.

These core transformations serve as building blocks for more complex manipulations. Now, let's expand our repertoire.

1. Absolute Value Transformations: Introducing Symmetry

Absolute value transformations involve applying the absolute value function to either the entire function, |f(x)|, or only the input variable, f(|x|). These transformations introduce symmetry and alter the behavior of the function in predictable ways.

The Absolute Value of the Function: |f(x)|

This transformation takes the absolute value of the output of the function. This means any negative y-values are made positive, while positive y-values remain unchanged. Geometrically, this results in:

• The portion of the graph above the x-axis (where f(x) is positive) remains the same.
• The portion of the graph below the x-axis (where f(x) is negative) is reflected across the x-axis.

Essentially, the x-axis becomes a mirror, reflecting any part of the graph that lies below it.

Example: Let's consider f(x) = x^2 - 4.

1. Original Function: f(x) = x^2 - 4 is a parabola that opens upwards, with vertex at (0, -4), and x-intercepts at x = -2 and x = 2. The y-intercept is at y = -4.

2. Transformed Function: g(x) = |x^2 - 4|

   • For x < -2 and x > 2, x^2 - 4 is positive, so |x^2 - 4| = x^2 - 4. The graph remains unchanged in these regions.

   • For -2 < x < 2, x^2 - 4 is negative. Therefore, |x^2 - 4| = -(x^2 - 4) = 4 - x^2. The portion of the graph between x = -2 and x = 2 is reflected across the x-axis. The new "vertex" in this region is now at (0, 4), the reflection of the original vertex.

The resulting graph of g(x) = |x^2 - 4| is a parabola-like shape that touches the x-axis at x = -2 and x = 2, but now has a "hump" above the x-axis in the region between these points. The y-intercept is at y = 4. All y-values are now non-negative.

Equation Transformation: To represent this transformation algebraically, you'll need to consider piecewise notation:

g(x) = |f(x)| =
   { f(x)   if f(x) >= 0
   { -f(x)  if f(x) < 0

This shows that the transformed function g(x) is equal to the original function f(x) when f(x) is non-negative, and it's equal to the negative of f(x) when f(x) is negative.

Practice: Try applying this transformation to f(x) = sin(x). Notice how the negative portions of the sine wave are reflected above the x-axis, creating a series of "humps."

The Absolute Value of the Input: f(|x|)

This transformation takes the absolute value of the input variable, x. This means any positive x-value is used directly, while any negative x-value is transformed into its positive counterpart. The effect on the graph is to create symmetry about the y-axis. Specifically:

• The portion of the graph where x is positive (the right side of the y-axis) remains unchanged.
• The portion of the graph where x is negative (the left side of the y-axis) is replaced with a reflection of the right side across the y-axis. In other words, the function becomes an even function.

Important Note: This transformation only affects the graph if the original function is not already an even function (symmetric about the y-axis). If f(x) is even, then f(|x|) = f(x), and there's no visible change.

Example: Let's consider f(x) = x^3 - x.

1. Original Function: f(x) = x^3 - x is an odd function (symmetric about the origin). It has roots at x = -1, x = 0, and x = 1.

2. Transformed Function: g(x) = |x|^3 - |x| (substituting |x| for x in the original equation) or more concisely g(x) = f(|x|).

   • For x >= 0, |x| = x, so f(|x|) = f(x). The graph remains the same on the right side of the y-axis.

   • For x < 0, |x| = -x. The left side of the graph is replaced by a reflection of the right side. So, g(x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -f(x). However, since we're reflecting the right side to create the left side, it's more accurate to say that the left side now mirrors the right side.

The resulting graph of g(x) = f(|x|) is now an even function, symmetric about the y-axis. It looks like the right half of the original function has been copied and reflected to create the left half. The roots remain at x = -1, x = 0, and x = 1.

Equation Transformation:

g(x) = f(|x|)

This is the most straightforward representation. Just replace every instance of 'x' in the original function with '|x|'.

Practice: Try applying this transformation to f(x) = e^x. The exponential function is not even, so the transformation will create a function that looks like e^|x|, which is symmetric about the y-axis.

2. Piecewise Defined Transformations: Combining Functions

Piecewise defined transformations involve creating new functions by combining different function expressions over specific intervals of the domain. This allows for highly flexible and custom function behaviors. They aren't strictly "transformations" in the same way as shifts or stretches, but they transform the way a function is defined.

Understanding Piecewise Notation

A piecewise function is defined using the following notation:

f(x) =
  { expression_1  if condition_1
  { expression_2  if condition_2
  { ...
  { expression_n  if condition_n

Each line specifies a different function expression and the corresponding condition (an interval of x-values) for which that expression is valid.

Example:

f(x) =
  { x^2     if x < 0
  { x       if 0 <= x <= 1
  { 1       if x > 1

This function behaves like x^2 for negative x-values, like x for x-values between 0 and 1 (inclusive), and like the constant function 1 for x-values greater than 1.

Transformations within Piecewise Functions

The power of piecewise functions comes from the ability to apply transformations within each piece. You can shift, stretch, reflect, or apply any other transformation to a specific part of the function's domain.

Example: Let's create a piecewise function that combines a transformed sine wave with a linear function.

f(x) =
  { 2*sin(x) + 1   if x < pi
  { -x + 4          if x >= pi

1. 2*sin(x) + 1 for x < pi: This piece takes the sine function, stretches it vertically by a factor of 2, and shifts it upwards by 1 unit. It's only defined for x-values less than pi.

2. -x + 4 for x >= pi: This is a linear function with a slope of -1 and a y-intercept (if extended back to x=0) of 4. It's only defined for x-values greater than or equal to pi.

The graph of this function will show a transformed sine wave up to x = pi, and then a straight line with a negative slope for all x-values greater than pi. The two pieces will connect (or possibly not connect, creating a discontinuity) at x = pi. In this specific case, when x = pi, 2sin(pi) + 1 = 20 + 1 = 1. And -pi + 4 is approximately -3.14 + 4 = 0.86. So there is a discontinuity at x = pi.

Creating Discontinuities and Special Effects

Piecewise functions are excellent for creating functions with specific properties, such as discontinuities (jumps) or functions that behave differently in different regions.

Example: A function with a jump discontinuity at x = 2.

f(x) =
  { x + 1    if x < 2
  { x^2 - 2  if x >= 2

At x = 2, the first piece approaches 2 + 1 = 3, while the second piece starts at 2^2 - 2 = 2. There's a clear "jump" in the graph at x = 2.

Applications of Piecewise Functions

Piecewise functions are used extensively in:

• Modeling real-world phenomena: For example, tax brackets, where the percentage of income taxed changes based on income level.
• Computer graphics: For creating complex shapes and animations.
• Signal processing: For designing filters and processing audio or image signals.
• Control systems: For defining different control strategies based on system state.

Practice:

1. Create a piecewise function that is equal to x^3 for x < 0, equal to sqrt(x) for 0 <= x <= 4, and equal to 8 - x for x > 4. Graph the function and identify any points of discontinuity.

2. Design a piecewise function that represents the price of admission to a movie theater, where children under 12 pay $8, adults pay $12, and seniors (65+) pay $10.

Combining Transformations: The Order Matters

When applying multiple transformations to a function, the order in which you apply them is crucial. The general order to follow is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), but adapted to function transformations. Think of it as working from the inside out:

1. Horizontal Shifts: Deal with any additions or subtractions inside the function's argument, like f(x + a).

2. Horizontal Stretches/Compressions and Reflections about the y-axis: Address any multiplications or divisions inside the function's argument, like f(bx). This also includes reflections across the y-axis when b is negative.

3. Vertical Stretches/Compressions and Reflections about the x-axis: Handle any multiplications or divisions outside the function, like a*f(x). This also includes reflections across the x-axis when a is negative.

4. Vertical Shifts: Finally, take care of any additions or subtractions outside the function, like f(x) + c.

Why does the order matter?

Because each transformation affects the input or output of the previous transformation. Applying them in the wrong order can lead to incorrect results.

Example: Let's transform f(x) = x^2 into g(x) = 2(x - 1)^2 + 3.

1. Horizontal Shift: x^2 becomes (x - 1)^2 (shift right by 1 unit).

2. Vertical Stretch: (x - 1)^2 becomes 2(x - 1)^2 (stretch vertically by a factor of 2).

3. Vertical Shift: 2(x - 1)^2 becomes 2(x - 1)^2 + 3 (shift upwards by 3 units).

If you were to shift upwards before stretching vertically, the result would be different.

Practice Problems: Putting it All Together

Let's solidify your understanding with some comprehensive practice problems.

1. Transforming a Square Root Function:

   • Start with f(x) = sqrt(x).
   • Reflect it across the x-axis.
   • Shift it horizontally to the right by 3 units.
   • Shift it vertically upwards by 2 units.
   • Write the equation of the transformed function g(x).
   • Sketch both f(x) and g(x).

2. Transforming an Exponential Function:

   • Start with f(x) = e^x.
   • Compress it horizontally by a factor of 2.
   • Reflect it across the y-axis.
   • Stretch it vertically by a factor of 3.
   • Write the equation of the transformed function g(x).
   • Sketch both f(x) and g(x).

3. Absolute Value and Quadratic:

   • Start with f(x) = x^2 - 9.
   • Create g(x) = |f(x)|. Sketch g(x).
   • Create h(x) = f(|x|). Sketch h(x).
   • Explain the differences between the graphs of g(x) and h(x).

4. Piecewise Transformation:

   • Define a piecewise function:

     f(x) =
       { x      if x < 1
       { 2      if 1 <= x < 3
       { 6 - x  if x >= 3
     

   • Transform the second piece (2 for 1 <= x < 3) to 2 + sin(pi*(x-1)). This will make it a sine wave segment.
   • Write the new piecewise function g(x) with this modified second piece.
   • Sketch both f(x) and g(x).

Conclusion: The Art of Function Manipulation

Mastering function transformations is more than just memorizing rules; it's about developing a visual and algebraic intuition for how functions behave. By understanding the effects of different transformations, you gain the power to manipulate functions, create new ones, and model real-world phenomena with greater accuracy and insight. The ability to combine transformations, particularly within piecewise functions, unlocks a vast creative potential for designing functions with specific and nuanced behaviors. So keep practicing, experimenting, and exploring the fascinating world of function transformations!