Translations of Graphs

Lesson Overview

Instructor Presentation

Introduction

It is easy to predict how a function's graph is going to behave by knowing the following items:

What is the basic function?

What does the graph of the basic function look like?

What transformation(s) have been applied to this function?

Goals

Students will be able to identify the basic function and graph it.

parameters={a,c,d,x}

Students will identify what transformations occurred and in what order.

Concept Exploration

Horizontal and Vertical Shifting

Move the sliders to find out what happens to the parabola f(x) = (x^2)+ c when c changes.

💡Note to the teacher: I have included a few Plot Options to make the graphs stand out.  If you want to learn more, click on Help, Documentation Center, and type Plot in the search box.

Manipulate[Plot[(x^2)+c,{x,-5,5}, PlotRange->{-5,5}, PlotLabel->(x^2)+c,PlotStyle->{Thick,Blue}],{c,-4,4}]

The graph moves c units vertically from the initial position of f(x) = (x^2).

Move the sliders to find out what happens function f(x) = (Sqrt[x-b]) when b changes.

Manipulate[Plot[(Sqrt[x-b]),{x,-5,5}, PlotRange->{-5,5}, PlotLabel->(Sqrt[x-b]),PlotStyle->{Thick,Blue}],{b,-4,4}]

Noting that the basic function is (Sqrt[x-b]), i.e. the subtraction sign inside the radical sign, The graph moves b units horizontally from the initial position of the basic function.

Now, look at the following model.  Move the sliders and click on the buttons.  Do the rules seem to change when the function changes?

Manipulate[Plot[function[x-b]+c,{x,-10,10},PlotRange->{-10,10},PlotStyle->{Thick,Blue},GridLines->{Range[-10, 10], Range[-10, 10]}, AspectRatio->Automatic],{b,-5,5},{c,-5,5},{function,{Sin,Sqrt,Abs,Exp}}]

The rules for shifting are true for all families of graphs.

Examples

Example 1

Move the sliders on the model below.   Give an explanation for what each variable does to the graph when changed.

Manipulate[Show[Plot[(2^(x+b))+c,{x,-10,10},PlotRange->{-10,10},PlotStyle->{Thick,Blue}],PlotLabel->(2^(x+b))+c],{b,-5,5},{c,-5,5}]

b :

c :

Example 2

Create and plot the following polynomial graphs.  Be sure to choose an appropriate window.

An cubic polynomial graph shifted 7 units to the right.

An quartic polynomial graph shifted up 3 units and left 5 units

A parabola shifted left 14 units, then down 1 unit.

Example 3

Describe the transformations, in the correct order, that occurred to create the given function.

f(x) = 7 + cos (x - 3π)

g(x) = -10 + ln x

Examples Answer Key

Example 1

b : Indicates the horizontal shift.  As b gets smaller, the graph shifts 
right b units.  As b gets larger, the graph shifts left b units.

c : Indicates the vertical shift.  As c increases, the graph shift up c units.  
  As c decreases, the graph shifts down c units.

Example 2

An cubic polynomial graph shifted 7 units to the right.

Plot[(((x-7))^3), {x,0,10},PlotRange->{-10,10},PlotStyle->Thick]

An quartic polynomial graph shifted up 3 units and left 5 units

Plot[(((x+5))^4)+3, {x,-10,5},PlotRange->{-1,12},PlotStyle->Thick]

A parabola shifted left 14 units, then down 1 unit.

Plot[(((x+14))^2)-1,{x,-100,50},PlotRange->{-5,1200},PlotStyle->Thick]

Example 3

Describe the transformations, in the correct order, that occurred to create the given function.

f(x) = 7 + cos (x - 3π)

A Cosine graph shifted to the right 3 π units and up 7 units.

g(x) = -10 + ln x

The natural logarithmic graph shifted down 10 units.