5.2: Vertical & Horizontal Shifts

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Objectives

Students will be able to

• Interpret what a shift does to a point on a function
• Vertically shift a function that is given either explicitly or graphically
• Horizontally shift a function that is given either explicitly or graphically
• Combine horizontal and vertical shifts to transform a function
• Interpret a shift in a real world setting

Important Items

Definitions:

vertical shift, horizontal shift, explicit formula, implicit formula

Notes:

Throughout all of Chapter 5, be sure to reinforce the ideas of \S5.5 (on combining function transformations and that the order of function transformations matter).

Use the language of inputs/outputs when talking about transformations; i.e., horizontal shifts result from a change to the input, whereas vertical shifts result from a change to the output. Do NOT refer to changes as occurring inside/outside the parentheses, as this can lead to confusion; for instance, $x^2+1=(x^2+1)$, but both represent a vertical shift.

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Lesson Guide

Introduction

Most students won’t have an issue getting the main idea of how shifts work. Horizontal shifts can be unintuitive to them at first, but they’ll still pick up on how it works quickly. However, the way this section is taught dictates how students will interpret transformations for the rest of this chapter. In later sections, if students don’t understand how specific types of transformations affect the coordinates of points on the graph, they will find these sections quite difficult and lose the physical interpretation for what transformations are doing to the points on a graph. In fact, when some function $g(x)$ is the transformation of $f(x)$, it would be beneficial if you wrote down sentences of the form "The height of the function $g$ at some input is somehow related to the height of $f$ at some input". If students can internalize how to discover these types of sentences, it will allow them to uncover what certain formulas should be. These sentences will be written in more detail in the following lesson plans.

One final thing is that students will struggle finding the explicit formula for functions. You'll want to remind them that these transformations are really just compositions. (Some students may actually not know what we mean by explicit and implicit functions -- don't take this for granted, you should define it for them.)

Warm-Up

Introduce the idea of function transformations by using ordered pairs.

• Example: Consider the point (1,2) on a coordinate plane (it may be helpful to draw this point on the board).

Ask students what point you would have if you moved this point up, down, left, and right by a certain number of units. It would be beneficial for you to talk about what coordinate is changing with every movement of the point.

Have students do Problem 1, which walks them through discovering these transformations on a larger scale.

Use these ordered pairs to discuss the relationship between the functions f(x) and g(x):

• How is the function $g(x)$ different from $f(x)$? Tell students to study the inputs and outputs of points, and ask them which has changed, and how has it changed.

* Write down with your students: "The height of $g$ at $x$ is one more than the height of $f$ at $x$". (Get students in the habit of writing these sentences from now, these sentences come in handy later.)
* Write down the implicit formula for $g(x)$ from the prior sentence: $g(x)=f(x)+1$. Write down that this is the implicit formula for $g(x)$ since it refers to another function.
* What has changed about the points on the graph, the inputs, or the outputs? By what value? 
* Note that $g(x) = x^2 + 1$. This is the explicit formula for $g(x)$, since it doesn't refer to any other functions. 

Go through a similar discussion of the relationship between f(x) and h(x).

Use Problem 1 to summarize the definitions for vertical and horizontal shifts.

If $f(x)$ is a function and $k$ is a positive constant, then the graph of 
* $y=f(x)+k$ is the graph of $y=f(x)$ vertically shifted up by $k$ units.
* $y=f(x)-k$ is the graph of $y=f(x)$ vertically shifted down by $k$ units.
When we do a vertical translation, the output value of the points change by $k$ units accordingly.

If $f(x)$ is a function and $h$ is a positive constant, then the graph of 
* $y=f(x-h)$ is the graph of $y=f(x)$ horizontally shifted right by $h$ units.
* $y=f(x+h)$ is the graph of $y=f(x)$ horizontally shifted left by $h$ units.
When we do a horizontal translation, the input value of the points change by $h$ units accordingly.

Have students work on Problems 2-7. It may be worthwhile to discuss Problems 4 and 6 to reiterate the implicit/explicit formulas. If you want them to get practice with the basic transformations, perhaps have them start with 4-7 and them go back to 2-3.

Note: You may consider breaking up class more by discussing vertical and horizontal shifts separately and having students work on problems in between topics.

Have students work on Problems 9-10 and use these to discuss how transformations affect domain and range. It may be useful to pull up graphs of these functions using something like Desmos.

Giving more examples of domain and range will be helpful as this shows up frequently on homework. For example, you could say something like "$f(x)$ has domain -2<x<=3 and range -4<=f(x)<10. What is the domain and range of f(x+3)-2?"

Combine horizontal and vertical shifts to transform a function

Say something like "We may also combine these two types of transformations. Just remember, vertical transformations change the output value of the points, the horizontal transformations change the input values."

• Example:

The graph of $y=f(x-2)+4$ is the graph of $y=f(x)$ shifted to the right by 2 units and up by 4 units. What transformations give the graph of $y=f(x+3) -2$?

It may also be helpful to apply these transformations to a point, and draw arrows representing how its being shifted while also keeping track of these changes of to the side. For example, if the point (1,1) was on f(x) and you wanted to see what happened to that point after shifting right two units and up four units, you could do this visually on the graph and then write down the following next to the graph: $(1,1)\rightarrow(3,1)\rightarrow(3,5)$. This will help emphasize that when you transform functions, you are really changing the inputs and outputs of the points.

Have students do Problems 8, 11-13.

Note: There is very little chance you will get through all of these problems, tell your students that you are going to pick problems that you think they need to work on the most. They can visit office hours or the MRC to work on the rest of the problems.

• Problems 4 \& 6: Students have a tendency to be able to describe this in words but not translate this to a graph. Look for opportunities to connect their words to graphs.
• Problems 5 \& 7: Many students may feel completely lost on these problems. Tell students to pick some key points and describe how those points are transformed.
• Problem 8: Tell students that this problem has shown up on many exams.
• Problem 12: Again, tell students to pick some key points and describe how those points are transformed.
• Problem 13: Make sure students use meaningful and complete sentences.