Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 17: Graphing Functions

Lesson 2: Transformations

In this Grade 4 AMC Math lesson from AoPS Introduction to Algebra, students explore function transformations, learning how vertical and horizontal shifts, scaling, and reflections affect the graph of y = f(x). Students practice applying rules such as adding a constant to the output for vertical shifts, replacing x with kx for horizontal scaling by a factor of 1/k, and negating the input or output to reflect a graph over the x- or y-axis. The lesson is part of Chapter 17 on Graphing Functions and emphasizes understanding how changes to a function's input versus output produce distinct graphical effects.

Section 1

Combined Horizontal and Vertical Shifts

Property

For a function f(x)f(x), the graph can be translated using both horizontal and vertical shifts simultaneously:

The graph of g(x)=f(x−h)+kg(x) = f(x - h) + k is the graph of f(x)f(x) shifted hh units horizontally and kk units vertically.

Section 2

Reflection Transformations

Property

Reflection over x-axis: y=−f(x)y = -f(x) reflects the graph across the x-axis by negating all y-coordinates.

Reflection over y-axis: y=f(−x)y = f(-x) reflects the graph across the y-axis by negating all x-coordinates.

Section 3

Horizontal Scaling: f(kx)

Property

For horizontal scaling: y=f(kx)y = f(kx) where k>0k > 0

  • If k>1k > 1: graph compresses horizontally by factor 1k\frac{1}{k}
  • If 0<k<10 < k < 1: graph stretches horizontally by factor 1k\frac{1}{k}

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Combined Horizontal and Vertical Shifts

Property

For a function f(x)f(x), the graph can be translated using both horizontal and vertical shifts simultaneously:

The graph of g(x)=f(x−h)+kg(x) = f(x - h) + k is the graph of f(x)f(x) shifted hh units horizontally and kk units vertically.

Section 2

Reflection Transformations

Property

Reflection over x-axis: y=−f(x)y = -f(x) reflects the graph across the x-axis by negating all y-coordinates.

Reflection over y-axis: y=f(−x)y = f(-x) reflects the graph across the y-axis by negating all x-coordinates.

Section 3

Horizontal Scaling: f(kx)

Property

For horizontal scaling: y=f(kx)y = f(kx) where k>0k > 0

  • If k>1k > 1: graph compresses horizontally by factor 1k\frac{1}{k}
  • If 0<k<10 < k < 1: graph stretches horizontally by factor 1k\frac{1}{k}