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R64 • Lesson 94 of 105

Function Transformations

Shifts, stretches, reflections, and compressions of function graphs

Reserve & Extensions • K-12

Prerequisites: H29, H34, H45

Key Concepts

  • transformations
  • shifts
  • stretches
  • reflections
  • function families

Function Transformations

Once you know the shape of a basic function -- such as y = x2, y = |x|, or y = sqrt(x) -- you can shift, stretch, compress, and reflect it to create an entire family of related graphs. Understanding transformations lets you graph complex functions quickly without plotting dozens of points.

Vertical and Horizontal Shifts

A vertical shift moves the graph up or down:

y = f(x) + k → shifts UP by k (if k > 0) or DOWN by |k| (if k < 0)

A horizontal shift moves the graph left or right:

y = f(x − h) → shifts RIGHT by h (if h > 0) or LEFT by |h| (if h < 0)

Common Mistake: Horizontal Shift Direction

The horizontal shift goes the opposite direction of the sign inside the parentheses. y = (x − 3)2 shifts RIGHT 3 (not left). y = (x + 5)2 shifts LEFT 5 (not right). Think of it as: the value that makes the inside zero is where the new center lands.

Worked Example 1: Combining Shifts

Describe the transformation: y = |x + 2| − 4

Start with the parent function y = |x|, whose vertex is at (0, 0).

  1. x + 2 = x − (−2), so shift LEFT 2.
  2. The −4 outside shifts DOWN 4.

The new vertex is at (−2, −4). The V-shape is unchanged.

Vertical Stretch and Compression

y = a · f(x)

Worked Example 2: Stretching a Parabola

Compare y = x2, y = 3x2, and y = (1/2)x2.

At x = 2: y = 4, y = 12, and y = 2, respectively.

y = 3x2 is 3 times as tall at every point (vertical stretch by factor 3).

y = (1/2)x2 is half as tall at every point (vertical compression by factor 1/2).

Reflections

Reflection over x-axis
y = −f(x)

Flip the graph upside-down. Every y-value becomes its opposite.

Reflection over y-axis
y = f(−x)

Flip the graph left-to-right. Every x-value becomes its opposite.

Worked Example 3: Combining Transformations

Describe all transformations: y = −2(x − 1)2 + 5

Parent function: y = x2

  1. Horizontal shift: right 1 (from x − 1).
  2. Vertical stretch: by factor 2 (the coefficient 2).
  3. Reflection: over x-axis (the negative sign).
  4. Vertical shift: up 5 (the +5).

The vertex moves from (0, 0) to (1, 5), the parabola opens downward, and it is narrower than the standard parabola.

Order of Transformations

Apply transformations in this order: (1) Horizontal shift, (2) Horizontal stretch/reflection, (3) Vertical stretch/reflection, (4) Vertical shift. Alternatively, work "inside out" -- deal with what is happening to x first, then what is happening to y.

Practice Problems

1. Describe the transformation from y = x2 to y = (x − 4)2 + 7.

Solution

Shift right 4, shift up 7. The vertex moves from (0, 0) to (4, 7).

2. Write the equation if y = sqrt(x) is reflected over the x-axis and shifted left 3.

Solution

Reflect over x-axis: y = −sqrt(x). Shift left 3: replace x with (x + 3). Result: y = −sqrt(x + 3).

3. For y = 4|x + 1| − 2, identify the parent function, all transformations, and the vertex.

Solution

Parent: y = |x|. Transformations: shift left 1, vertical stretch by factor 4, shift down 2. Vertex: (−1, −2).

4. How does y = f(−x) differ from y = −f(x)? Give a specific example using f(x) = x3.

Solution

y = f(−x) = (−x)3 = −x3 reflects over the y-axis. y = −f(x) = −x3 reflects over the x-axis. For odd functions like x3, these happen to look the same, but in general they are different. For example, with f(x) = x2 + x: f(−x) = x2 − x, while −f(x) = −x2 − x -- these are different graphs.

5. Starting with y = x2, write the equation of a parabola that opens downward, is vertically compressed by 1/3, and has its vertex at (−2, 6).

Solution

Open downward: negative coefficient. Compressed by 1/3: |a| = 1/3. Vertex at (−2, 6): shift left 2 and up 6. Equation: y = −(1/3)(x + 2)2 + 6.

Summary

Overview