Learn on PengienVision, Algebra 1Chapter 10: Working With Functions

Lesson 4: Translations of Functions

Grade 11 students in enVision Algebra 1 explore translations of functions in Chapter 10, Lesson 4, learning how adding a constant to the output produces vertical translations and subtracting a constant from the input produces horizontal translations. The lesson covers the general form g(x) = f(x - h) + k and shows how these transformations apply consistently across quadratic, exponential, square root, cube root, and absolute value functions. Students practice graphing combined horizontal and vertical translations and analyzing how the values of h and k shift any function's graph left, right, up, or down.

Section 1

General Translation Formulas for All Functions

Property

For any function $f(x)$ , translations follow these universal formulas:

Vertical translation: $g(x) = f(x) + k$
Horizontal translation: $g(x) = f(x - h)$
Combined translation: $g(x) = f(x - h) + k$

Examples

Section 2

Reference Point Tracking for Function Translations

Property

To track translations, identify key reference points on the original function $f(x)$ , then apply the transformation rule: if $(a, b)$ is on $f(x)$ , then $(a + h, b + k)$ is on $g(x) = f(x - h) + k$ .

Examples

Section 3

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

Property

The graph of $f(x) = x^2 + k$ shifts the graph of $f(x) = x^2$ vertically $k$ units.

If $k > 0$ , shift the parabola vertically up $k$ units.
If $k < 0$ , shift the parabola vertically down $|k|$ units.

Examples

To graph $f(x) = x^2 + 4$ , you start with the graph of $f(x) = x^2$ and shift it vertically up 4 units because $k = 4$ .

To graph $f(x) = x^2 - 5$ , you start with the graph of $f(x) = x^2$ and shift it vertically down 5 units because $k = -5$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

General Translation Formulas for All Functions

Property

For any function $f(x)$ , translations follow these universal formulas:

Vertical translation: $g(x) = f(x) + k$
Horizontal translation: $g(x) = f(x - h)$
Combined translation: $g(x) = f(x - h) + k$

Examples

Section 2

Reference Point Tracking for Function Translations

Property

Examples

Section 3

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

Property

The graph of $f(x) = x^2 + k$ shifts the graph of $f(x) = x^2$ vertically $k$ units.

If $k > 0$ , shift the parabola vertically up $k$ units.
If $k < 0$ , shift the parabola vertically down $|k|$ units.

Examples

To graph $f(x) = x^2 + 4$ , you start with the graph of $f(x) = x^2$ and shift it vertically up 4 units because $k = 4$ .

To graph $f(x) = x^2 - 5$ , you start with the graph of $f(x) = x^2$ and shift it vertically down 5 units because $k = -5$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

General Translation Formulas for All Functions

Property

For any function $f(x)$ , translations follow these universal formulas:

Vertical translation: $g(x) = f(x) + k$
Horizontal translation: $g(x) = f(x - h)$
Combined translation: $g(x) = f(x - h) + k$

Examples

Section 2

Reference Point Tracking for Function Translations

Property

Examples

Section 3

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

Property

The graph of $f(x) = x^2 + k$ shifts the graph of $f(x) = x^2$ vertically $k$ units.

If $k > 0$ , shift the parabola vertically up $k$ units.
If $k < 0$ , shift the parabola vertically down $|k|$ units.

Examples

To graph $f(x) = x^2 + 4$ , you start with the graph of $f(x) = x^2$ and shift it vertically up 4 units because $k = 4$ .

To graph $f(x) = x^2 - 5$ , you start with the graph of $f(x) = x^2$ and shift it vertically down 5 units because $k = -5$ .