Lesson 3 — Function Transformations

Overview

Every function has a shape, a visual personality that tells a story.
The parent function f(x)=x2f(x) = x^2f(x)=x2 forms a parabola; f(x)=∣x∣f(x) = |x|f(x)=∣x∣ makes a “V”; f(x)=sin⁡(x)f(x) = \sin(x)f(x)=sin(x) oscillates like a wave.
When we transform a function, we change its position, size, or orientation — yet its core structure remains.

Transformations are the grammar of graphs. They allow you to predict how any algebraic tweak affects the visual outcome without plotting dozens of points.

In this lesson we’ll explore:

  1. Shifts (translations)

  2. Reflections (flips)

  3. Stretches and Compressions (scaling)

  4. Combining transformations efficiently

1. What Is a Transformation?

A transformation takes the basic rule of a function and modifies it through algebraic adjustments.
Given a base (parent) function f(x)f(x)f(x), we can create an infinite family of related graphs using:

g(x)=a f(b(x−h))+kg(x) = a \, f(b(x - h)) + kg(x)=af(b(x−h))+k

Each parameter changes the graph in a precise way:

SymbolEffectDirectionhhhHorizontal shiftright if +hhh, left if −hhhkkkVertical shiftup if +kkk, down if −kkkaaaVertical stretch / compression + reflectionstretch > 1, compress 0 <bbbHorizontal stretch / compression + reflectioncompress > 1, stretch 0 <

2. Shifts — Moving Without Changing Shape

Vertical Shifts

Add or subtract outside the function:

f(x)+kf(x) + kf(x)+k

Moves graph up kkk units if k>0k > 0k>0; down if k<0k < 0k<0.

Example:
f(x)=x2⇒g(x)=x2+3f(x)=x^2 \Rightarrow g(x)=x^2+3f(x)=x2⇒g(x)=x2+3
Graph moves 3 units up.

Horizontal Shifts

Add or subtract inside the parentheses:

f(x−h)f(x - h)f(x−h)

Moves right hhh units if h>0h > 0h>0; left if h<0h < 0h<0.

Example:
g(x)=(x−2)2g(x)=(x-2)^2g(x)=(x−2)2 moves f(x)=x2f(x)=x^2f(x)=x2 right 2 units.

Quick Tip:

Inside → opposite direction ; outside → same direction.

3. Reflections — Flipping the Graph

Flipping changes orientation but not shape.

Across the x-axis

Multiply the entire function by −1:

y=−f(x)y = -f(x)y=−f(x)

All y-values change sign.
If the original was above the x-axis, it now sits below.

Across the y-axis

Replace xxx with −xxx:

y=f(−x)y = f(-x)y=f(−x)

Every x-coordinate reflects horizontally.

Examples

  • f(x)=x3⇒−f(x)f(x)=x^3 \Rightarrow -f(x)f(x)=x3⇒−f(x) → flipped vertically

  • f(x)=2x⇒f(−x)f(x)=2^x \Rightarrow f(-x)f(x)=2x⇒f(−x) → mirror of exponential decay

Reflections reveal symmetry and are vital for recognizing even/odd behavior.

4. Stretches and Compressions — Resizing the Graph

These alter how steep or wide the graph appears.

Vertical Stretch/Compression

Multiply outside the function by aaa:

y=a f(x)y = a\,f(x)y=af(x)

Effect(a0 <aa < 0Also reflects across x-axis

Example:
f(x)=x2⇒g(x)=2x2f(x)=x^2 \Rightarrow g(x)=2x^2f(x)=x2⇒g(x)=2x2 → parabola twice as steep.

Horizontal Stretch/Compression

Multiply inside by bbb:

y=f(bx)y = f(bx)y=f(bx)

Effectb0 <bb < 0Also reflects across y-axis

Example:
f(x)=sin⁡(x)⇒g(x)=sin⁡(2x)f(x)=\sin(x) \Rightarrow g(x)=\sin(2x)f(x)=sin(x)⇒g(x)=sin(2x) → period halved (compressed).

5. Combining Transformations

All transformations can coexist. The order matters.

General Rule (from inside to outside):

  1. Horizontal shifts, stretches, reflections (inside xxx)

  2. Vertical reflections, stretches (multiplying the function)

  3. Vertical shifts (added at the end)

Example Walkthrough

Parent: f(x)=x2f(x)=x^2f(x)=x2
Transform: g(x)=−2(x+3)2+1g(x) = -2(x+3)^2 + 1g(x)=−2(x+3)2+1

Step 1 – (x+3)(x+3)(x+3) → shift left 3
Step 2 – ×−2× −2×−2 → reflect over x-axis + stretch by 2
Step 3 – +1+1+1 → shift up 1

The vertex (0, 0) of fff moves to (−3, 1), and the parabola opens downward twice as narrow.

Visual Intuition

Think of f(x)f(x)f(x) as an elastic curve pinned to the origin:

  • hhh slides it left/right

  • kkk slides it up/down

  • aaa stretches or flips it vertically

  • bbb stretches or flips it horizontally

Transformations are like combining zoom, flip, and drag operations on a digital graphing tool.

6. Efficient Graphing Strategy

Instead of re-plotting from scratch:

  1. Start with the parent shape
    Know base graphs: x2,∣x∣,x,1/x,sin⁡(x),exx^2, |x|, \sqrt{x}, 1/x, \sin(x), e^xx2,∣x∣,x​,1/x,sin(x),ex.

  2. Track the vertex or key points
    Transform only a few strategic points (e.g., vertex, intercepts).

  3. Apply transformations in order
    Inside → outside, maintain consistency.

  4. Use symmetry
    Even/odd patterns make half the work redundant.

  5. Annotate changes
    Label the new vertex, intercepts, and asymptotes for clarity.

7. Examples

Example 1:

f(x)=∣x∣,g(x)=−2∣x−3∣+4f(x) = |x|,\quad g(x) = -2|x-3| + 4f(x)=∣x∣,g(x)=−2∣x−3∣+4

Transformations:

  1. Shift right 3

  2. Reflect over x-axis

  3. Stretch by 2

  4. Shift up 4

Vertex: from (0, 0) → (3, 4)
Graph: inverted V, narrower, lifted up.

Example 2:

f(x)=x,g(x)=12−x+4−1f(x) = \sqrt{x},\quad g(x) = \tfrac{1}{2}\sqrt{-x+4} - 1f(x)=x​,g(x)=21​−x+4​−1

Steps:

  1. Inside −x+4-x+4−x+4 → reflect over y-axis then shift right 4

  2. Outside × ½ → vertical compression

  3. −1−1−1 → down 1

Starting point (0, 0) → new start (4, −1).

8. Summary Table

TransformationEquation FormEffectVertical Shiftf(x)+kf(x)+kf(x)+kUp (+), down (−)Horizontal Shiftf(x−h)f(x-h)f(x−h)Right (+), left (−)Vertical Stretch/Compressionaf(x)a f(x)af(x)TallerHorizontal Stretch/Compressionf(bx)f(bx)f(bx)NarrowerReflect x-axis−f(x)−f(x)−f(x)Flip verticallyReflect y-axisf(−x)f(−x)f(−x)Flip horizontallyCombine Allaf(b(x−h))+ka f(b(x−h)) + kaf(b(x−h))+kDo inside → outside

9. Real-World Connections

ContextFunctionTransformationEconomicsRevenue curves R(x)R(x)R(x)Scaling → price changesPhysicsProjectile h(t)=−16t2+v0t+h0h(t) = −16t^2 + v_0t + h_0h(t)=−16t2+v0​t+h0​Vertical shifts by initial heightSound WavesAsin⁡(ωt+φ)A\sin(ωt+φ)Asin(ωt+φ)Amplitude & phase → stretches and shiftsProgramming GraphicsSprite motionAll transformations applied as code translations

Transformations are universal — they let you model, predict, and animate everything from finance graphs to game physics.

10. Concept Reflection

Transformations are the language of change and comparison.
Once you understand how each algebraic symbol moves or reshapes a graph, you can sketch any function in seconds.
In later units, you’ll apply these same principles to trigonometric and exponential graphs — mastering how functions evolve and behave.