Learn on PengiCalifornia Reveal Math, Algebra 1Unit 10: Quadratic Functions

10-2 Transformations of Quadratic Functions

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn how to identify the effects of translations, dilations, and reflections on the graph of the parent quadratic function f(x) = x². The lesson covers vertex form g(x) = a(x − h)² + k, explaining how the constants h, k, and a produce horizontal and vertical shifts, stretches, compressions, and reflections across the x- or y-axis. Students practice describing and graphing these transformations as part of Unit 10 on Quadratic Functions.

Section 1

Vertical and Horizontal Translations of Quadratic Functions

Property

Starting from the parent function f(x)=x2f(x) = x^2, two types of translations shift the parabola without changing its shape:

Vertical shift:

g(x)=x2+kg(x) = x^2 + k
moves the graph up kk units when k>0k > 0 and down ∣k∣|k| units when k<0k < 0. Only the yy-coordinate of the vertex changes; the vertex moves from (0,0)(0, 0) to (0,k)(0, k).

Section 2

Vertical Stretches and Compressions of Quadratic Functions

Property

Given the parent quadratic function f(x)=x2f(x) = x^2, the transformed function is:

g(x)=ax2g(x) = ax^2

Section 3

Horizontal Dilations of Quadratic Functions

Property

For the function f(x)=x2f(x) = x^2, horizontal transformations are created using f(ax)=(ax)2f(ax) = (ax)^2 where a>0a > 0. When a>1a > 1, the graph compresses horizontally toward the y-axis by a factor of 1a\frac{1}{a}. When 0<a<10 < a < 1, the graph stretches horizontally away from the y-axis by a factor of 1a\frac{1}{a}.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Vertical and Horizontal Translations of Quadratic Functions

Property

Starting from the parent function f(x)=x2f(x) = x^2, two types of translations shift the parabola without changing its shape:

Vertical shift:

g(x)=x2+kg(x) = x^2 + k
moves the graph up kk units when k>0k > 0 and down ∣k∣|k| units when k<0k < 0. Only the yy-coordinate of the vertex changes; the vertex moves from (0,0)(0, 0) to (0,k)(0, k).

Section 2

Vertical Stretches and Compressions of Quadratic Functions

Property

Given the parent quadratic function f(x)=x2f(x) = x^2, the transformed function is:

g(x)=ax2g(x) = ax^2

Section 3

Horizontal Dilations of Quadratic Functions

Property

For the function f(x)=x2f(x) = x^2, horizontal transformations are created using f(ax)=(ax)2f(ax) = (ax)^2 where a>0a > 0. When a>1a > 1, the graph compresses horizontally toward the y-axis by a factor of 1a\frac{1}{a}. When 0<a<10 < a < 1, the graph stretches horizontally away from the y-axis by a factor of 1a\frac{1}{a}.

Examples