Absolute Value Graphs

Lesson Overview

The student will learn about

Translating (shifting) absolute value graphs

Reflections across axes.

Stretching

parameters={a,c,d,x}

Concept Exploration

Plot the line y = x - 5

Plot[x-5,{x,-10,10},PlotRange->{-10,10},AspectRatio->Automatic,PlotStyle->{Thick,Blue}]

Now, plot f(x)=x-5 and f(x)= |x - 5| together on the same graph.

Plot[{x-5,Abs[x-5]},{x,-10,10},AspectRatio->Automatic,PlotRange->{-10,10},PlotStyle->{{Thick,Blue},{Thick,Green}}]

Above the x-axis, when the output is positive |f(x)|=f(x) or green = blue.

Below the x-axis, when the output is negative |f(x)|=-f(x) or green = - blue

Journal: Where do the graph begin to differ and will this always be true?

Coefficients

Slide the control a and visualize how it changes the function f(x)=a|x|.

Manipulate[Plot[a*Abs[x],{x,-10,10},PlotRange->{-10,10},AspectRatio->Automatic,PlotStyle->{Thick,Blue}],{{a,1},-10,10,.25,Appearance->"Labeled"}]

Journal: In your own words explain how a changes the function?

Placement of a

Now visualize the graph of f(x)=|ax| where the a is inside the absolute value.

Manipulate[Plot[Abs[a*x],{x,-10,10},PlotRange->{-10,10},AspectRatio->Automatic,PlotStyle->{Thick,Blue}],{{a,1},-10,10,.5,Appearance->"Labeled"}]

Journal: Explain the difference between the graphs of a|x| and |ax|. When are they equivalent?

Horizontal Shifts

Now that we have seen that a relates directly to the slope, lets see how h in f(x)=a|x-h| changes the graph.

Manipulate[Plot[a*Abs[x+h],{x,-10,10},PlotLabel->Style[a*Abs[x+h],Large],PlotRange->{-10,10},AspectRatio->Automatic,PlotStyle->{Thick,Blue},Background->LightYellow],{{a,1},-10,10,1,Appearance->"Labeled"},{{h,0},-10,10,1,Appearance->"Labeled"}]

Journal: In your own words come up with a generalization for horizontal shifts.

Pulling out a GCF

Journal: Is a|x+h|=|ax+ah| always true? Come up with a hypothesis then compare the graphs below to see. 
	a|x+h| is dark blue, while |ax+ah| is light blue.

Manipulate[Plot[{a*Abs[x+h],Abs[a*x+a*h]},{x,-10,10},PlotRange->{-10,10},AspectRatio->Automatic,PlotStyle->{Thick,Blue}],{{a,3},-10,10,1,Appearance->"Labeled"},{{h,0},-10,10,1,Appearance->"Labeled"}]

Hopefully you were able to realize that this is the same as the situation a|x|  vs. |ax| that we saw earlier.
When a is negative the graphs are reflections across the x-axis and are not equivalent.

Therefore, if we want to manipulate the function into standard form when x has a coefficient: pull out the coefficient, but only the absolute value of a. For example:
 |2x-8|= 2|x-4|  is true, but |-2x-8|=-2|x+4| is not.

In order to solve the second example think about it like this: 

|-2x-8|= |-2(x+4)|=|-2|*|(x+4)|=2|x+4|.

Journal: Put y=2|-3x+12| into standard form then plot the function given what you have learned thus far.

Vertical Shifts

A vertical shift is introduced with a variable we will call k in y=a|x+h|+k.

Adjust the controls so that you have a vertex at (-3,-4) and a slope of 5.

Manipulate[Plot[a*Abs[x+h]+k,{x,-10,10},PlotLabel->Column[
{
Style[Row[{"     y = ",ToString[a] ," |x",If[h<0," - "," + "], ToString[Abs[h]],"|",If[k<0," - "," + "], ToString[Abs[k]]}]],Style[Row[{"Vertex = (" ,If[h<0,"","-"], ToString[Abs[h]],",",If[k<0," - ",""], ToString[Abs[k]],")   ","Slope = ",ToString[a]}]]
},
Background->Lighter[Yellow]],PlotRange->{-10,10},AspectRatio->Automatic,PlotStyle->{Thick,Blue},Background->LightYellow,Frame->True],{{a,1},-10,10,1,Appearance->"Labeled"},{{h,3},-10,10,1,Appearance->"Labeled"},{{k,-2},-10,10,1,Appearance->"Labeled"}]

Answer: a=5, h=3, k=-4

Journal: Give a generalization of how k changes the function.

Exploring other functions

The following descriptions are graphed on the same set of axes.

1)  y = (x^2) - 8,

The parabola as is in blue.

The absolute value of the parabola... y=|(x^2)-8| is in orange.

Plot[{(x^2)-8,Abs[(x^2)-8]},{x,-10,10},AspectRatio->Automatic,PlotRange->{-10,10},PlotStyle->{{Thick,Blue},{Thick,Orange}}]

Journal: What do you notice about the location of the sharp points or edges of the orange graph? Why is this the case?

Journal: Design a function that would look like the letter M.

You try...

Remember that f(x) just tells us that it is a function named f with x as its variable.

1. Graph f(x)=2|x-1|+2

2. Graph g(x)=((-1)/2)|x+2|-3

3. Graph p(x)=2|-4x+20|-1...return to slide 6 if necessary.

4. Graph q(x) =(1/4)|8x-4|+3

5. Graph t(x)=|(x^3)|