Welcome to the World of Transformations!

In your journey through Unit 1, you have explored polynomial and rational functions. You have learned how to find their zeros, identify their asymptotes, and describe their end behavior. But what if you already know everything about a "parent" function like \(f(x) = x^2\) and you want to create a new, slightly different function without starting from scratch?

That is where Transformations of Functions come in. Think of transformations as a set of instructions—like a filter on a photo—that tells you how to shift, stretch, or flip a graph to create a new one. By the end of these notes, you will be able to take any function and move it around the coordinate plane like a pro!

1. The Master Formula

Almost every transformation you will see in AP Precalculus can be summarized by this one general equation:

\(g(x) = a \cdot f(b(x - h)) + k\)

Each letter (\(a\), \(b\), \(h\), and \(k\)) performs a specific job. To make this easier, we split them into two groups:

  • The "Outside" (Vertical): Changes that happen outside the parentheses (\(a\) and \(k\)) affect the outputs (y-values). These do exactly what you’d expect them to do.
  • The "Inside" (Horizontal): Changes that happen inside the parentheses (\(b\) and \(h\)) affect the inputs (x-values). These are often "counter-intuitive"—they do the opposite of what you might expect!

2. Vertical Transformations (The "Outside" World)

Since these transformations affect the whole function \(f(x)\), they change the y-coordinates of the graph.

Vertical Translations (Shifting Up or Down)

Adding or subtracting a constant \(k\) outside the function moves the graph vertically.

  • \(f(x) + k\): Moves the graph up \(k\) units.
  • \(f(x) - k\): Moves the graph down \(k\) units.

Example: If \(f(x) = x^2\), then \(g(x) = x^2 + 3\) is the same parabola moved 3 units up.

Vertical Dilations (Stretching or Compressing)

Multiplying the outside by a constant \(a\) changes the height of the graph.

  • If \(|a| > 1\): The graph is vertically stretched (it gets taller/steeper).
  • If \(0 < |a| < 1\): The graph is vertically compressed (it gets shorter/flatter).

Vertical Reflection

  • If \(a\) is negative (\(-f(x)\)): The graph reflects over the x-axis (it flips upside down).

Quick Tip: Think of the "Outside" world as honest. If you add 5, it goes up 5. If you multiply by 2, it gets twice as tall!

3. Horizontal Transformations (The "Inside" World)

These transformations happen directly to the \(x\) variable inside the parentheses. They change the x-coordinates of the graph.

Horizontal Translations (Shifting Left or Right)

This is where it gets tricky! In the expression \(f(x - h)\):

  • \(f(x - h)\): Moves the graph right \(h\) units. (Even though it's minus!)
  • \(f(x + h)\): Moves the graph left \(h\) units. (Even though it's plus!)

Analogy: Imagine you have a deadline at 5:00 PM (\(x\)). If I say the deadline is now \((x - 2)\), you actually have 2 extra hours to finish because the clock "moved back." In math, subtracting from \(x\) moves the "action" later (to the right).

Horizontal Dilations (Stretching or Compressing)

Multiplying \(x\) by a constant \(b\) changes the width of the graph.

  • If \(|b| > 1\): The graph is horizontally compressed. (It moves "faster," so the graph looks skinnier).
  • If \(0 < |b| < 1\): The graph is horizontally stretched. (It moves "slower," so the graph looks wider).

Horizontal Reflection

  • If \(b\) is negative (\(f(-x)\)): The graph reflects over the y-axis (it flips left-to-right).

Quick Tip: Think of the "Inside" world as "Opposite Land." Adding moves left, and multiplying by a big number makes the graph smaller!

4. Summary of Transformation Effects

If you are given a point \((x, y)\) on the original function \(f(x)\), here is how the new point on \(g(x) = a \cdot f(b(x - h)) + k\) is calculated:

The New Point: \((\frac{x}{b} + h, a \cdot y + k)\)

Did you know? This same logic applies to all the functions you've studied, including the rational functions from Topic 1.9. For example, if a rational function has a vertical asymptote at \(x = 0\), the transformation \(f(x - 4)\) will shift that asymptote to \(x = 4\).

5. Common Pitfalls and How to Avoid Them

Pitfall #1: The Order of Operations

When applying multiple transformations, order matters! Generally, you should follow the PEMDAS logic from the perspective of the original \(y\)-value:

  1. Horizontal Shifts and Dilations (the "Inside").
  2. Vertical Dilations/Reflections (Multiplication).
  3. Vertical Shifts (Addition/Subtraction).

Pitfall #2: Factoring the "b" Value

If you see a function like \(f(2x - 6)\), you might think the horizontal shift is 6 units to the right. Wait! You must factor out the \(b\) value first:

\(f(2(x - 3))\)

Now you can see the true horizontal shift is actually 3 units to the right, and there is a horizontal compression by a factor of \(2\).

6. Step-by-Step: Constructing a Transformed Function

Suppose you are asked to take \(f(x) = x^3\) and write an equation for \(g(x)\) that is vertically stretched by 4, shifted left 2, and shifted down 5.

Step 1: Vertical Stretch. Multiply the outside by 4: \(4 \cdot f(x)\)

Step 2: Horizontal Shift Left. Replace \(x\) with \((x + 2)\): \(4 \cdot f(x + 2)\)

Step 3: Vertical Shift Down. Subtract 5 from the whole thing: \(4 \cdot f(x + 2) - 5\)

Final Result: \(g(x) = 4(x + 2)^3 - 5\)

Key Takeaways for the AP Exam

  • Practice 1.C: You will often be asked to construct new functions. Remember: \(y = \text{Vertical Changes} \cdot f(\text{Horizontal Changes}) + \text{Vertical Shifts}\).
  • Practice 2.A: Be ready to identify transformations from a graph. Look for the "key points" (like the vertex of a parabola or the center of a cubic) to see how far they moved.
  • Don't Forget: Inside transformations affect the Domain, and outside transformations affect the Range.

Note: For more on how these transformations affect specific features like zeros and holes, see Topic 1.10: Rational Functions and Holes.