Horizontal and Vertical Shift Transformations

Identifying Vertical Shifts
Identifying Horizontal Shifts
Combining Vertical and Horizontal Shifts

Overview

Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.

Lesson

Identifying Vertical Shifts

One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function $LaTeX: g\left(x\right)=f\left(x\right)+k$ , the function $LaTeX: f\left(x\right)$ is shifted vertically k units.

To help you visualize the concept of a vertical shift, consider that $LaTeX: y=f\left(x\right)$ . Therefore, $LaTeX: f\left(x\right)+k$ is equivalent to LaTeX: y+k . Every unit of y is replaced by , so the y-value increases or decreases depending on the value of k. The result is a shift upward (+k) or downward (-k).

IDENTIFYING A VERTICAL SHIFT

Given the parent function $LaTeX: f\left(x\right)$ , a new function $LaTeX: g\left(x\right)=f\left(x\right)+k$ , where k is a constant, is a vertical shift of the function $LaTeX: f\left(x\right)$ . 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.

Identifying Horizontal Shifts

We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift, shown in figure 5 below .

For example, if $LaTeX: f\left(x\right)=x^2$ , then $LaTeX: g\left(x\right)=\left(x-2\right)^2$ is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in f.

IDENTIFYING A HORIZONTAL SHIFT

Given a function f, a new function $LaTeX: g\left(x\right)=f\left(x-h\right)$ , 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.

Combining Vertical and Horizontal Shifts

Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output y-axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input x- axis values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down and right or left.

How to: Given a function and both a vertical and a horizontal shift, sketch the graph.

1. 1. Identify the vertical and horizontal shifts from the formula.
  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.
  3. The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.
  4. Apply the shifts to the graph in either order.

Try these on your own and check your answers with Desmos!

#1

Given the function $LaTeX: f\left(x\right)=\sqrt{x}$ , graph the original function f(x)and the transformation $LaTeX: g\left(x\right)=f\left(x+2\right)$ $g$ ( x ) = f ( x + 2 ) on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

#2

Given $LaTeX: f\left(x\right)=\left|x\right|$ , sketch a graph of $LaTeX: h\left(x\right)=f\left(x-2\right)+4$ $h$ ( x ) = f ( x − 2 ) + 4.

#3

Write a formula for a transformation of the parent reciprocal function $LaTeX: f\left(x\right)=\frac{1}{x}$ that shifts the function's graph one unit to the right and one unit up.

Desmos Graphing Calculator

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What's next?

Next, you will complete the lessons for more types of transformations: reflections, scaling and combinations of all.

After completing all 1.1aa-d lessons, you will complete two related assignments due Monday, October 5 at 11:59pm.

Additional Resources

Desmos - Toolkit Functions