POLYNOMIAL FUNCTIONS

Quadratic Functions

When the highest power of x is two, the graph is called a "parabola".
This means it changes direction ONCE (from up to down or from down to up).
How do you decide which way the parabola opens ??
How do you know where(if) the parabola crosses the axes ??
Experiment with editing the following commands.

Clear[f,x];
f[x_]=2 (x^2)+5 x-3;
Plot[f[x],{x,-10,10},AxesLabel->{x,y}]
Factor[f[x]]

Plot[2 (x^2)-5 x-3,{x,-10,10},AxesLabel->{x,y}]

Plot[-2 (x^2)+5 x-3,{x,-10,10},AxesLabel->{x,y}]

Plot[5 (x^2)+3 x-8,{x,-10,10},AxesLabel->{x,y}]

Plot[5 (x^2)+3 x-108,{x,-10,10},AxesLabel->{x,y}]

Cubic Functions

When the highest power of x is three, the graph can change direction TWICE (up-down-up or
down-up-down).
How do you decide which way the ends go? (right side goes up or down?)

Clear[f,x];
f[x_]=(x^3)-6 (x^2)-x+78;
Plot[f[x],{x,-10,10},AxesLabel->{x,y}]
Factor[f[x]]

Plot[-2 (x^3)-6 (x^2)-x+83,{x,-10,10},AxesLabel->{x,y}]

Plot[5 (x^3)-12 (x^2)+83,{x,-10,10},AxesLabel->{x,y}]

Plot[-22 (x^3)-3 (x^2)-x-100,{x,-10,10},AxesLabel->{x,y}]

Plot[-(x^3)-6 (x^2)-x-42,{x,-10,10},AxesLabel->{x,y}]

Plot[(x^3)+83,{x,-10,10},AxesLabel->{x,y}]

Quartic Functions

If the highest power of x is FOUR, then the graph can turn around
up to three times (up-down-up-down or down-up-down-up).
How do you decide which way they will go?

Plot[{(x^4)-20 (x^2)-x+83},{x,-10,10},AxesLabel->{x,y}]

Plot[{-(x^4)+36 (x^2)-18 x},{x,-10,10},AxesLabel->{x,y}]

Plot[{-(x^4)+36 (x^2)-77 x},{x,-10,10},AxesLabel->{x,y}]

Plot[{-(x^4)+36 (x^2)-180 x},{x,-10,10},AxesLabel->{x,y}]

Plot[{(x^4)-36 (x^2)+18},{x,-10,10},AxesLabel->{x,y}]

Plot[{-(x^4)-20 (x^2)-x+83},{x,-10,10},AxesLabel->{x,y}]

Plot[{(x^3)-6 (x^2)-x+83},{x,-10,10},AxesLabel->{x,y}]

Quintic Functions

Do some experimenting with functions of highest power 5 (and larger) ...
How many times can they change direction?

Plot[{24+2 x-29 (x^2)-3 (x^3)+5 (x^4)+(x^5)},{x,-6,6},AxesLabel->{x,y}]

Global Growth

If you take a large enough scale, a polynomial function will look just like a power function with the highest power of the polynomial.

Notice how the following two functions look different on a small scale, but become indistinguishable when the scale gets large enough.

Plot[{(x^3)-6 (x^2)-x+83,(x^3)},{x,-10,10},AxesLabel->{x,y}]

Plot[{(x^3)-6 (x^2)-x+83,(x^3)},{x,-300,300},AxesLabel->{x,y}]

Plot[{2 (x^6),(x^3)+3 (x^2)+2 (x^6)-75},{x,-2,2},AxesLabel->{x,y}]

Plot[{2 (x^6),(x^3)+3 (x^2)+2 (x^6)-75},{x,-10,10},AxesLabel->{x,y}]

What power function do you expect  x^2 + 2 x^5 - 5x  to look like on a large scale?
You can check your answer by playing with the command below (you will probably have to change the x-values to see the global growth patterns).

Plot[(x^2)+2 (x^5)-5 x,{x,-10,10},AxesLabel->{x,y},AxesOrigin->{0,0}]

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POWER FUNCTIONS

The rest of the graphs in this notebook are the ones Tami used in her lecture on power functions.  You can use them to practice for the test.

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N[(5^(1/3))]
N[(1/(5^3))]
(5^3)
(3^5)

Plot[{(1/x),(1/(x^3)),(1/(x^5))},{x,-5,5},PlotRange->{{-5,5},{-10,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{(1/(x^2)),(1/(x^4)),(1/(x^6))},{x,-5,5},PlotRange->{{-5,5},{-10,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{(Sqrt[x]),(x^(1/3)),(x^(1/10))},{x,0,5},PlotRange->{{0,5},{0,3}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{x,(x^3),(x^5),(x^7)},{x,-5,5},PlotRange->{{-5,5},{-10,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{(x^2),(x^4),(x^6)},{x,-5,5},PlotRange->{{-5,5},{-10,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{x,(x^2),(x^3),(x^4)},{x,-5,5},PlotRange->{{-5,5},{-10,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{x,(x^2),(x^3),(x^4)},{x,0,1.2`},PlotRange->{{0,1.2`},{0,2}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{x,(x^2),(x^3),(x^4)},{x,0,10},PlotRange->{{0,10},{0,10000}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{(x^3),(2^x)},{x,0,10},PlotRange->{{0,10},{0,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{(x^3),(2^x)},{x,0,10},PlotRange->{{0,10},{0,100}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{(x^3),(2^x)},{x,0,10},PlotRange->{{0,10},{0,1000}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{(x^3),(2^x)},{x,0,15},PlotRange->{{0,15},{0,10000}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Clear[a]
a=10;
(a^4)
30 (a^5)
(a^10)
(2^a)

Plot[{(x^3),(x^4),30 (x^5),(x^10),(2^x)},{x,0,10},AxesLabel->{x,y},AxesOrigin->{0,0}]

Clear[a]
a=50;
(a^4)
30 (a^5)
(a^10)
(2^a)

Plot[{(x^3),(x^4),30 (x^5),(x^10),(2^x)},{x,0,50},PlotRange->{{0,65},{0,97656250000000000}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Clear[a]
a=65;
(a^4)
30 (a^5)
(a^10)
(2^a)

Plot[{(x^3),(x^4),30 (x^5),(x^10),(2^x)},{x,0,65},PlotRange->{{0,65},{0,36993488147419103232}},AxesLabel->{x,y},AxesOrigin->{0,0}]

Plot[{(x^(1/3)),(1/(x^3)),(x^3),(3^x)},{x,-5,5},AspectRatio->1,AxesLabel->{x,y},AxesOrigin->{0,0}]

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Plot[x,{x,-5,5},PlotRange->{{-5,5},{-10,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]
Plot[(x^3),{x,-5,5},PlotRange->{{-5,5},{-10,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]
Plot[(x^5),{x,-5,5},PlotRange->{{-5,5},{-10,10}},AxesLabel->{x,y},AxesOrigin->{0,0}]

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Plot[3 (t^2),{t,-5,5}]

Plot[3 (t^4),{t,-5,5}]

Plot[3 (t^6),{t,-5,5}]

Plot[3 (t^8),{t,-5,5}]

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Plot[3 (t^1),{t,-5,5}]

Plot[3 (t^3),{t,-5,5}]

Plot[3 (t^5),{t,-5,5}]

Plot[3 (t^7),{t,-5,5}]

Plot[{3 (t^2),3 (t^4),3 (t^8)},{t,-5,5}]

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Plot[{3 (t^1),3 (t^3),3 (t^5),3 (t^7)},{t,-5,5}]

Plot[{3 (t^2),3 (t^4),3 (t^8)},{t,-1,1}]

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Plot[{300 (t^2),2 (t^4)},{t,-1,1}]

Plot[f,{x,xmin,xmax},PlotStyle->Automatic]

Plot[{300 (t^2),2 (t^4)},{t,-1,1}]

Plot[{300 (t^2),2 (t^4)},{t,-10,10}]

Plot[{300 (t^2),2 (t^4)},{t,-20,20}]

NSolve[300 (t^2)-2 (t^4)==0,t]

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Plot[{3 (t^1),3 (t^(1/3)),3 (t^(1/5)),3 (t^(1/7))},{t,0,1}]

Plot[{3 (t^1),3 (t^(1/3)),3 (t^(1/5)),3 (t^(1/7))},{t,-1,1}]

Plot[{3 (t^1),3 (t^(1/3)),3 (t^(1/5)),3 (t^(1/7))},{t,0,20}]

Plot[{3 (Sqrt[t]),3 (t^(1/4)),3 (t^(1/6)),3 (t^(1/8))},{t,0,1}]

Plot[{3 (Sqrt[t]),3 (t^(1/4)),3 (t^(1/6)),3 (t^(1/8))},{t,0,20}]

f[x_]:=(2^x)-(x^2)

Plot[f[x],{x,-20,20}]

Plot[f[x],{x,-5,5}]

Plot[f[x],{x,-1,0}]