Transformations of Graphs:

Reflecting and Stretching 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 initial function?

What does the graph of the initial function look like?

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

Goals

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

parameters={a,c,d,x}

Students will identify what transformations occurred and in what order.

Concept Exploration

Reflecting Graphs

Plot the following two graphs on the same set of axes.

(1) f(x) = (x - ((4))^2)

(2) g(x) = -(x - ((4))^2)

💡 Remember to graph on the same set of axes, make sure to "Plot" the list, using {  } around the graphs.  Mathematica will automatically color the graphs differently.  You can use PlotStyle to modify these, as I have.

Plot[{(((x-4))^2),-(((x-4))^2)},{x,0,10},PlotRange->{-10,10},PlotStyle->{{Thick,Blue},{Thick,Red}}]

Multiplying the ENTIRE FUNCTION by a negative number reflects that number about the x-axis.

💡 I've "labeled" each graph below using the Mathematica function "Tooltip."  Move the mouse over each graph to see what happens.

Plot[{Tooltip[(((x-4))^2), "(x-4)^2"], Tooltip[-(((x-4))^2), "-(x-4)^2"]},{x,0,10},PlotRange->{-10,10},PlotStyle->{{Thick,Blue},{Thick,Red}}]

Stretching and Shrinking Graphs

Move the slider on the following model to see how multiplying the function by a number changes the appearance of the graph.

Manipulate[Plot[a*Sin[x],{x,-10,10},PlotRange->{-12,12},PlotLabel->a*sin[x],PlotStyle->{Thick,Orange}],{a,1,10}]

As a, the number multiplied to the function, increases, the graph stretches vertically.

What if we use this same demonstration, but, instead, multiply "a" to the "x"?

Manipulate[Plot[Sin[a*x],{x,-10,10},PlotRange->{-12,12},PlotLabel->sin[a*x],PlotStyle->{Thick,Orange}],{a,1,10}]

Very interesting...
The graph does not STRETCH horizontally, it SHRINKS.

Examples

Example 1

Run the following demonstration and give an explanation for what each variable does to the graph when changed.

Manipulate[Show[Plot[d*(2^(x+b))+c,{x,-10,10},PlotRange->{-10,10},PlotStyle->{Thick,Blue}],PlotLabel->d*(2^(x+b))+c],{b,-5,5},{c,-5,5},{d,{-1,1}}]

b :

c :

d :

Example 2

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

An absolute value graph reflected in the y-axis with a vertical stretch factor of 4 applied.

A greatest integer function with a horizontal stretch factor of 2 applied.

(Hint:  greatest integer function in Mathematica = Floor[x])

(Extra hint:  Multiplying the x by 2 causes a horizontal shrink to happen.  What do you think will cause a horizontal stretch?)

A square root function first stretched in the vertical direction by a factor of 2, then shifted up 1 unit, and finally reflected in the x-axis.

Example 3

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

y = -(3^x) - 8

y = 3 + 2|x-2|

y = 4 - sin(x - π)

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.

d : When d = -1, the graph is decreasing.  When d = 1, the graph is increasing.  Comparing the same graph, except the value of d, when changing from 1 to - 1, the
graph is reflected about a horizontal line, namely, y = c.

Example 2

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

An absolute value graph reflected in the x-axis with a vertical stretch factor of 4 applied.

Plot[-4*Abs[x],{x,-4,4},PlotRange->{-20,2},GridLines->{Range[-4, 4], Range[-20, 2]}, AspectRatio->Automatic,PlotStyle->{Thickness[0.01],Blue}]

A greatest integer function with a horizontal stretch factor of 2 applied.

Plot[Floor[x/2],{x,-6,6},PlotRange->{-5,5},GridLines->{Range[-6,6],Range[-5,5]},AspectRatio->Automatic,PlotStyle->{Thickness[0.01],Red}]

A square root function first stretched in the vertical direction by a factor of 2, then shifted up 1 unit, and finally reflected in the x-axis.

Plot[-(2(Sqrt[x])+1),{x,-6,6},AxesOrigin->{0,0},PlotRange->{-6,6},PlotStyle->{Thickness[0.01],Green}]

Example 3

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

y = -(3^x) - 8

Exponential graph with a base of 3 reflected in the x - axis, then shifted down 
8 units.  Or Exponential graph with a base of 3 shifted up 8 units, then 
reflected about the x - axis.

y = 3 + |x-2|

An absolute value graph shifted right 2 units, then stretched vertically by a
factor of 2, then shifted up 3 units.

y = 4 - sin(x - π)

A sine graph shifted right π units, reflected in the x - axis, and shifted
up 4 units.