How do I graph x^3 - 4x^2 + 2 and 8x^3 - 16x^2 + 6? PLEASE EXPLAIN

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Oddman Profile
Oddman answered
You graph these the way you graph any functions. You pick a few values for x, find the corresponding values of the expressions (y), plot the (x, y) points on a graph, and draw a smooth curve through the points.

It is convenient to have a method for finding values of x that might be of interest. When you examine these functions, you see that each is in the form
  ax^2(x-b) + c

This tells you that the expression value is equal to "c" when the value of x is 0 or "b". It also tells you the function will tend toward negative infinity for values of x below 0. And it tells you that x will tend toward positive infinity for values of x above "b". The function will be less than (or equal to) "c" for values of x below "b".

You are usually not terribly interested in values of the expression when those values are very large, so your graph will probably want to extend from x=-2 or so to about x=b+2 or so. Computation is probably easiest if you use mostly integers for values of x.

For the first function, we can make a short table of values. We see that a=1, b=4, c=2.
For x=-2, (-2)^2(-2-4)+2 = -22, so (-2, -22) is a point on the curve
For x=-1, (-1)^2(-1-4)+2 = -3, so (-1, -3) is another point on the curve
From above, we know that points (0, 2) and (4, 2) are on the curve
For x=1, 1^2(1-4)+2 = -1, so (1, -1) is on the curve
For x=2, 2^2(2-4)+2 = -6, so (2, -6) is on the curve
For x=3, 3^2(3-4)+2 = -7, so (3, -7) is on the curve
For x=5, 5^2(5-4)+2 = 27, so (5, 27) is on the curve
A graph of the first expression can be seen here.

A graph of the second expression can be seen here.
thanked the writer.
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Anonymous commented
So I can choose any reasonable x-incercept and plug that in, such as -2?
Oddman commented
Yes, you can choose any x value you like. -2 is one I used in the example because it is close to where the curve changes direction. Understand that an "x-intercept" is a value of x where the value of the expression is zero. The "x-intercepts" of the first expression are approximately x=-.655, x=.789, and x=3.866. They are called this because these points are where the curve intercepts the x-axis on the graph.
Anonymous commented
Ok! Thank you so much I really owe you one :).
S.K. SINGH Profile
S.K. SINGH answered
You can draw the graphs of any function through "skeching of curves studied in caculas".Be the function of any type trigonomeric, exponential, logarithimic, algrebric or combined all. Explaining here every thing is a difficult task YOU CAN REFFER TO THE BOOK 'PLAY WITH GRAPH'.

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