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Algebra 1

Transformations: Mini-Lesson Review

Algebra 1Transformations: Mini-Lesson Review

Transformations: Mini-Lesson Review

Transformations: Mini-Lesson Review

Mini Lesson Question

Question #3: Transformations

Name the transformations performed from f ( x ) to g ( x ) = f ( x − 3 ) + 4 .
  1. Right 3, Down 4
  2. Right 4, Down 3
  3. Left 3, Up 4
  4. Right 3, Up 4

Transforming Functions

A transformation is a move to a parent (or original) function. Three types of transformations include those that cause the graph to shift vertically or horizontally or reflect the function over the x x - or y y -axis.

Vertical Shift

Given a function f ( x ) f ( x ) , a new function g ( x ) = f ( x ) + k g ( x ) = f ( x ) + k , where k k is a constant, is a vertical shift of the function f ( x ) f ( x ) . All the output values change by k k units. If k k is positive, the graph will shift up. If k k is negative, the graph will shift down.

Horizontal Shift

Given a function f f , a new function g ( x ) = f ( x − h ) g ( x ) = f ( x − h ) , where h h is a constant, is a horizontal shift of the function f f . If h h is positive, the graph will shift right. If h h is negative, the graph will shift left.

How to Sketch a Graph Given a Function and Both a Vertical and a Horizontal Shift

Step 1 - Identify the vertical and horizontal shifts from the formula.

Step 2 - The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.

Step 3 - The horizontal shift results from a constant subtracted from the input. Move the graph right for a positive constant and left for a negative constant.

Step 4 - Apply the shifts to the graph in either order.

Reflections

Given a function f ( x ) f ( x ) , a new function g ( x ) = − f ( x ) g ( x ) = − f ( x ) is a vertical reflection of the function f ( x ) f ( x ) , sometimes called a reflection about (or over, or through) the x x -axis.

Given a function f ( x ) f ( x ) , a new function g ( x ) = f ( − x ) g ( x ) = f ( − x ) is a horizontal reflection of the function f ( x ) f ( x ) , sometimes called a reflection about the y y -axis.

Example

Name the transformations from f ( x ) f ( x ) to form g ( x ) = − f ( x + 4 ) − 8 g ( x ) = − f ( x + 4 ) − 8 .

  • Horizontal Shift
  • Left 4
  • Vertical Shift
  • Down 8
  • Reflection?
  • Yes, over the x x -axis.

Try it

Transforming Functions

Name the transformations from f ( x ) f ( x ) to form g ( x ) = − f ( x + 3 ) − 2 g ( x ) = − f ( x + 3 ) − 2 .

Check Your Understanding

Name all of the transformations from f ( x ) f ( x ) to form g ( x ) = − f ( x + 5 ) − 1 g ( x ) = − f ( x + 5 ) − 1 .

  1. Left 1, Up 5, Reflect over the x x -axis

  2. Left 5, Down 1, Reflect over the x x -axis

  3. Left 5, Down 1, Reflect over the y y -axis

  4. Right 5, Down 1, Reflect over the x x -axis

Video: Identifying Transformations

Watch the following video to see how to identify transformations, including shifts, of functions.

Khan Academy: Shifting Functions Introduction

Shifting functions introduction | Transformations of functions | Algebra 2 | Khan Academy

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