How to Modify Parent Functions

I. The Transformation Formula

Every transformed function can be written in this general form:

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

Where f(x) is the parent function.

II. The Four Types of Transformations

1. Vertical Transformations

Outside the function → affect y-values

A. Vertical Shifts (k)

$$y=f(x)+k$$

• k>0: Shift UP k units

• k<0: Shift DOWN |k| units

Example: $$y=x^3+3$$ shifts parabola up 3

B. Vertical Stretch/Compression & Reflection (a)

$$y=a\codt f(x)$$

• ∣a∣>1: Vertical stretch by factor a

• 0<∣a∣<1: Vertical compression by factor a

• a<0: Reflection over x-axis (flips vertically)

Examples:
$$y=2x^2$$: Stretch vertically by 2
$$y=\frac{1}{3}x^2$$: Compress vertically by 1/3
$$y=-x^2$$: Reflect over x-axis

2. Horizontal Transformations

Inside the function → affect x-values
WARNING: Horizontal transformations behave counterintuitively!

A. Horizontal Shifts (h)

$$y=f(x-h)$$

• h>0: Shift RIGHT h units

• h<0: Shift LEFT ∣h∣ units

MEMORY TRICK: Inside changes are opposite of what they look like!
(x−3) means right 3, $(x+2)$ means left 2.

B. Horizontal Stretch/Compression & Reflection (b)

$$y=f(ax)$$

• ∣a∣>1: Horizontal compression by factor 1/a

• 0<∣a∣<1: Horizontal stretch by factor 1/a

• a<0: Reflection over y-axis (flips horizontally)

MEMORY TRICK: Horizontal stretches/compressions are reciprocals!

Examples:
$$y=(2x)^2$$: Compress horizontally by 1/2
$$y=(\frac{1}{2})^2$$: Stretch horizontally by 2

III. Order of Transformations

When you have multiple transformations, apply them in this order:

1. Horizontal shift (h) : f(x)→f(x−h)

2. Horizontal stretch/compression & reflection (a) : f(x−h)→f(a(x−h))

3. Vertical stretch/compression & reflection (b) : f(a(x−h))→b⋅f(a(x−h))

4. Vertical shift (k) : b⋅f(a(x−h))→b⋅f(a(x−h))+k

IV. Step-by-Step Examples

Example 1: $$y=2(x-3)^2+4$$

Parent:$$f(x)=x^2$$

1. Inside: (x−3) → right 3

2. No horizontal stretch (x coefficient 1)

3. Outside: 2 → vertical stretch by 2

4. Outside: +4 → up 4

Example 2: $$y=-\frac{1}{2}\sqrt{x+5}$$​

Parent:$$f(x)=\sqrt{x}$$​

1. Inside: (x+5) → left 5

2. No horizontal stretch (x coefficient 1)

3. Outside: -$$\frac{1}{2}$$ → reflect over x-axis and compress vertically by 1/2

Example 3: $$y=3|2x-4|-1$$

First, rewrite in standard form:
Factor inside: 2x−4=2(x−2)
So: y=3∣2(x−2)∣−1

Parent: $$f(x)=|x|$$

1. Inside: (x−2) → right 2

2. Inside: 2 → compress horizontally by 1/2

3. Outside: 3 → stretch vertically by 3

4. Outside: −1 → down 1

V. Special Considerations

Multiple Reflections

• Both a<0 AND b<0 → equivalent to 180° rotation

• Example: y=−f(−x) rotates graph 180° about origin

Domain & Range Changes

• Horizontal shifts change domain

• Vertical shifts change range

• Reflections may swap domain/range intervals

Invariant Points

Points that don’t move under certain transformations:

• x-intercepts stay fixed during vertical stretches

• y-intercepts stay fixed during horizontal stretches

• Reflection points (on axis of reflection) stay fixed

VI. Transformation Effects by Function Type

For periodic functions:
Horizontal stretch/compression changes the period
If y=sin⁡(bx), period = 2π/∣b∣​

VII. Common AP Exam Questions

Type 1: Describe Transformations

“Describe the transformations from $$f(x)=x^3$$ to $$g(x)=-2(x+1)^3+4$$

Answer:

1. Left 1 unit

2. No horizontal stretch

3. Vertical stretch by 2

4. Reflection over x-axis

5. Up 4 units

Type 2: Write Equation from Description

“Write the equation after shifting $$f(x)=\sqrt{x}$$​ right 3, reflecting over y-axis, and stretching vertically by 2″

Answer:

$$y=2\sqrt{-x+3}$$ or $$y=2\sqrt{-(x-3)}$$​