Learn on PengienVision, Algebra 2Chapter 1: Linear Functions and Systems

Lesson 2: Transformations of Functions

In this Grade 11 enVision Algebra 2 lesson, students learn how to apply translations, reflections, stretches, and compressions to graph functions and write their equations. Using parent functions, they explore how vertical and horizontal translations shift a graph by analyzing equations in the form g(x) = f(x) + k and g(x) = f(x − h), and how reflections across the x- or y-axis change the signs of function values or inputs. By connecting changes in a function's equation to changes in its graph, students build a systematic understanding of transformations as part of Chapter 1's study of linear functions and systems.

Section 1

Graph Quadratic Functions of the form f(x) = x^2 + k

Property

The graph of f(x)=x2+kf(x) = x^2 + k shifts the graph of f(x)=x2f(x) = x^2 vertically kk units.

  • If k>0k > 0, shift the parabola vertically up kk units.
  • If k<0k < 0, shift the parabola vertically down ∣k∣|k| units.

Examples

  • To graph f(x)=x2+4f(x) = x^2 + 4, you start with the graph of f(x)=x2f(x) = x^2 and shift it vertically up 4 units because k=4k = 4.
  • To graph f(x)=x2−5f(x) = x^2 - 5, you start with the graph of f(x)=x2f(x) = x^2 and shift it vertically down 5 units because k=−5k = -5.

Section 2

Graph Quadratic Functions of the form f(x) = (x - h)^2

Property

The graph of f(x)=(x−h)2f(x) = (x - h)^2 shifts the graph of f(x)=x2f(x) = x^2 horizontally hh units.

  • If h>0h > 0, shift the parabola horizontally right hh units.
  • If h<0h < 0, shift the parabola horizontally left ∣h∣|h| units.

Examples

  • To graph f(x)=(x−3)2f(x) = (x - 3)^2, you shift the graph of f(x)=x2f(x) = x^2 to the right 3 units. The vertex moves from (0,0)(0, 0) to (3,0)(3, 0).
  • To graph f(x)=(x+4)2f(x) = (x + 4)^2, you rewrite it as f(x)=(x−(−4))2f(x) = (x - (-4))^2. This means you shift the graph of f(x)=x2f(x) = x^2 to the left 4 units.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Graph Quadratic Functions of the form f(x) = x^2 + k

Property

The graph of f(x)=x2+kf(x) = x^2 + k shifts the graph of f(x)=x2f(x) = x^2 vertically kk units.

  • If k>0k > 0, shift the parabola vertically up kk units.
  • If k<0k < 0, shift the parabola vertically down ∣k∣|k| units.

Examples

  • To graph f(x)=x2+4f(x) = x^2 + 4, you start with the graph of f(x)=x2f(x) = x^2 and shift it vertically up 4 units because k=4k = 4.
  • To graph f(x)=x2−5f(x) = x^2 - 5, you start with the graph of f(x)=x2f(x) = x^2 and shift it vertically down 5 units because k=−5k = -5.

Section 2

Graph Quadratic Functions of the form f(x) = (x - h)^2

Property

The graph of f(x)=(x−h)2f(x) = (x - h)^2 shifts the graph of f(x)=x2f(x) = x^2 horizontally hh units.

  • If h>0h > 0, shift the parabola horizontally right hh units.
  • If h<0h < 0, shift the parabola horizontally left ∣h∣|h| units.

Examples

  • To graph f(x)=(x−3)2f(x) = (x - 3)^2, you shift the graph of f(x)=x2f(x) = x^2 to the right 3 units. The vertex moves from (0,0)(0, 0) to (3,0)(3, 0).
  • To graph f(x)=(x+4)2f(x) = (x + 4)^2, you rewrite it as f(x)=(x−(−4))2f(x) = (x - (-4))^2. This means you shift the graph of f(x)=x2f(x) = x^2 to the left 4 units.