Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Mini Lesson Question

Question #3: Transformations

Name the transformations performed from

f(x)

to

g(x)=f(x-3)+4

.

Right 3, Down 4
Right 4, Down 3
Left 3, Up 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 $x$ - or $y$ -axis.

Vertical Shift

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

Horizontal Shift

Given a function $f$ , a new function $g (x) = f (x - h)$ , where $h$ is a constant, is a horizontal shift of the function $f$ . If $h$ is positive, the graph will shift right. If $h$ 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 $f (x)$ , a new function $g (x) = - f (x)$ is a vertical reflection of the function $f (x)$ , sometimes called a reflection about (or over, or through) the $x$ -axis.

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

Example

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

Horizontal Shift
Left 4

Vertical Shift
Down 8

Reflection?
Yes, over the $x$ -axis.

Try it

Try It: Transforming Functions

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

Here are the transformations made from $f (x)$ to $g (x)$ :

Horizontal Shift
Left 3

Vertical Shift
Down 2

Reflection?
Yes, over the $x$ -axis.

Check Your Understanding

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

Multiple Choice:

Left 1, Up 5, Reflect over the $x$ -axis
Left 5, Down 1, Reflect over the $x$ -axis
Left 5, Down 1, Reflect over the $y$ -axis
Right 5, Down 1, Reflect over the $x$ -axis

Check yourself: The negative in front of the function makes the function reflect over the $x$ -axis. The constant in $f (x + 5)$ causes a shift left 5, and the constant -1 causes a shift down 1.

Try again. Take a moment to think about what you learned in the mini-lesson review.

Video: Identifying Transformations

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

Khan Academy: Shifting Functions Introduction

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Transformations: Mini-Lesson Review