Parabola

Guide

A circle is formed when a right circular cone is 
cut by a plane that is parallel to the base of
the cone.  If the plane is tilted the points of 
intersection of the plane and the cone form a 
figure called a parabola.  An example of a parabola 
is people tossing a ball back and forth.

Basics

The standard form of a parabola is f[x]= a(x-h)¤+k (sometimes it is necessary to complete the square to get standard form). We will discover that in this form, we can determine the axis of symmetry, which is the line x=h.  The coordinates of the vertex are (h,k).  If "a" is greater than zero, the curve opens up.  If "a" is less than zero, the curve opens down.

Tutorial

What happens as the values of a,h,or k are changed

Note that with the function f[x]= a(x-h)¤+k as "a" changes the parabola opens up faster or slower.  The smaller the value the slower it opens.  The larger the value the faster it opens.  If "a" is negative the parabola will open down. "h" is zero therefore the axis of symmetry is x=0.  "k" is also zero therefore the vertex is (0,0). To see what changing "a" does to a parabola select the cells next to the graph and type open apple and y.  the parameter is "a".

Clear[f,x,a]
f[x_,a_]:= a(x-0)^2+0;
Do[Plot[f[x,a],{x,-10,10},PlotRange->{-10,10},
			PlotLabel->"= parameter"a,AspectRatio->Automatic,
			PlotStyle->RGBColor[0,1,0]],{a,-1,1,.25}];

With the equation f[x]= 1/2(x-h)¤+0
Notice with a change in the value of "h" the graph moves to the right or left.  With a positive "h" the graph moves to the right.  With a negative "h" the graph moves to the left. Since "k" equals zero the vertex has coordinates (h,0) the axis of symmetry is x=h. To see what changing "h" does to a parabola select the cells next to the graph and type open apple and y.  the parameter is "h".

Clear[f,x,h]
f[x_,h_]:=1/2(x-h)^2+0;
Do[Plot[f[x,h],{x,-10,10},PlotRange->{-10,10},
PlotLabel->"= parameter"h,AspectRatio->Automatic,
PlotStyle->RGBColor[1,0,0]],{h,-5,5,2.5}]

With the equation f[x]= 2(x-0)¤+k you can see 
that the axis of symmetry is x=0.  The vertex
is at (0,k).  Because "a" is positive the parabola
opens up. To see what changing "k" does to a parabola select the cells next to the graph and type open apple and y.  the parameter is "k".

Clear[f,x,k]
f[x_,k_]:= 2(x-0)^2+k;
Do[Plot[f[x,k],{x,-10,10},PlotRange->{-10,10},
		PlotLabel->"= parameter"k,AspectRatio->Automatic,
		PlotStyle->RGBColor[0,0,1]],{k,-5,5,2}];

How to graph a parabola

To graph a parabola set the squared term equal to qero and solve for x.  This is the axis of symmetry draw this line.  Then mark the point of the vertex.  Then select a few x values to the right or the left of the acis of symmetry substitute these in for x in the original equation and solve for f[x] plot these points and remember to plot another point an equal distance from the axis of symmetry on the opposite side.  Draw the parabola through the points you have plotted.
Example: f[x]=3(x-2)¤=6 -> x-2=0-> x=2 Axis of symmetry.  If x=3, f[x]=9; x=4,f[x]=18 by symmetry
x=2,f[x]=9; x=1,f[x]=18

Give it a try

Graph the function, give the coordinates of the vertex and the equation of the axis of symmetry for each of the following (sometimes you must complete the square).  Check your answers by double clicking on the rectangle Twice.

1.  f[x_]= (x-4)¤+2

Clear [f,x]
f[x_]= (x-4)^2 +2
Plot [f[x], {x,-10,10}, PlotRange->{-5,20},
AxesLabel->{"x","y"}, PlotStyle-> {RGBColor[0,0,1]}]

2.  f[x]= 4(x+2)¤-5

Clear [f,x]
f[x_]= 4(x+2)^2 -5
Plot [f[x], {x,-10,10}, PlotRange->{-5,20},
AxesLabel->{"x","y"}, PlotStyle-> {RGBColor[0,0,1]}]

3.  f[x]= 1/2(x-2)¤+3

Clear [f,x]
f[x_]= 1/2(x-2)^2 +3
Plot [f[x], {x,-10,10}, PlotRange->{-5,20},
AxesLabel->{"x","y"}, PlotStyle-> {RGBColor[0,0,1]}]

4.  f[x]= -4x¤

Clear [f,x]
f[x_]= -4x^2 
Plot [f[x], {x,-10,10}, PlotRange->{-20,5},
AxesLabel->{"x","y"}, PlotStyle-> {RGBColor[0,0,1]}]

5.  f[x]= -1/4(x+4)¤-4

Clear [f,x]
f[x_]= -1/4(x+4)^2 -4
Plot [f[x], {x,-10,10}, PlotRange->{-20,5},
AxesLabel->{"x","y"}, PlotStyle-> {RGBColor[.25,.25,1]}]

6.  f[x]= -3x¤-12x-8

f[x_]= -3(x+2)^2+4
Plot [f[x], {x,-10,10}, PlotRange->{-20,5},
AxesLabel->{"x","y"}, PlotStyle-> {RGBColor[.25,0,1]}]

Literacy

Graph the function, state the coordinates of the vertex 
and the equation of the axis of symmetry for each of 
the following functions.

1.  f[x]= -3(x-1)¤+5

7.  f[x]= -x¤-4x-7

3.  f[x]= 1/4(x-8)¤+8

4.  f[x]= -3x¤

5.  f[x]= 5x¤

6.  f[x]= x¤-16x+32

7.  f[x]= 4(x+2)¤-19