1-2 Additional Practice Transformations Of Functions

Mastering Horizontal and Vertical Scaling: Your Next Step in Function Transformations

Understanding how to move and flip the graphs of basic functions—like shifting f(x) = x² left or reflecting f(x) = √x across an axis—is the essential first act in the play of function transformations. But to truly graph any function with confidence, you must command the next two powerful techniques: horizontal and vertical scaling. These transformations stretch or compress the graph away from or toward the axes, changing its shape and steepness. Mastering these concepts unlocks your ability to accurately sketch complex polynomials, rational functions, and trigonometric graphs, moving you from rote memorization to genuine analytical skill. This guide provides a deep, practice-focused exploration of these two critical transformations, complete with clear rules, common mistakes to avoid, and strategic problem-solving methods.

The Core Concept: Scaling vs. Shifting

Before diving in, it’s crucial to distinguish scaling from the translations (shifts) you already know.

• Translations (f(x - h) + k): Move the entire graph without altering its shape. The parabola y = x² remains a parabola; it just has a new vertex.
• Scaling (f(bx) or a*f(x)): Changes the shape and steepness of the graph by stretching or compressing it horizontally or vertically. The parabola y = x² can become wider or narrower, fundamentally altering its rate of growth.

Think of translations as moving a photograph on a wall. Scaling is like physically stretching or squeezing the photograph itself.

1. Horizontal Scaling: The f(bx) Transformation

This is often the most counterintuitive transformation for students. The rule is: For y = f(bx) where b > 0:

• If b > 1, the graph undergoes a horizontal compression by a factor of 1/b. It gets narrower and appears to move faster toward the axes.
• If 0 < b < 1, the graph undergoes a horizontal stretch by a factor of 1/b. It gets wider and appears to move slower.

The Golden Rule (and the source of confusion): The b in f(bx) affects the x-coordinates of the graph by the reciprocal factor 1/b. You divide the original x-values by b.

Why This Happens: A Step-by-Step Breakdown

Let’s use f(x) = x² and transform it to g(x) = f(2x) = (2x)² = 4x².

1. Identify Key Points: For f(x)=x², use points like (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4).
2. Apply the Transformation: To find the new x-coordinate for a point, solve bx = x_original. Here, 2x = x_original → x = x_original / 2.
3. Transform Points:
   • Original (-2, 4) → New x: -2 / 2 = -1 → New Point: (-1, 4)
   • Original (2, 4) → New x: 2 / 2 = 1 → New Point: (1, 4)
   • Original (1, 1) → New x: 1 / 2 = 0.5 → New Point: (0.5, 1)
4. Plot: The new points (-1,4), (0,0), (1,4) show the parabola is now narrower. It reached the y-value of 4 at x=±1 instead of x=±2. This is a horizontal compression by a factor of 1/2.

Practice Example with f(x) = |x|

Transform to h(x) = f(½x) = |½x|.

• Rule: b = ½ (which is between 0 and 1) means a horizontal stretch by factor 1/(½) = 2.
• Transform Points: Original (-2, 2), (-1, 1), (0,0), (1,1), (2,2).
  • New x: x_original / (½) = x_original * 2.
  • (-2, 2) → (-4, 2)
  • (2, 2) → (4, 2)
• Result: The V-shape of the absolute value graph is now wider. The point at y=2 is now at x=±4 instead of ±2.

Key Insight: Horizontal scaling changes how quickly the function reaches its y-values. A compression makes it happen sooner (smaller x for same y); a stretch makes it happen later (larger x for same y).

2. Vertical Scaling: The a*f(x) Transformation

This transformation is more intuitive, as it acts directly on the output, y. For y = a*f(x) where a > 0:

• If a > 1, the graph undergoes a vertical stretch by a factor of a. It gets taller.
• If 0 < a < 1, the graph undergoes a vertical compression by a factor of a. It gets shorter (or "squashed").

The Golden Rule: The a in a*f(x) affects the y-coordinates of the graph by the factor a. You multiply the original y-values by a.

Why This Happens: A Step-by-Step Breakdown

Use f(x) = x²