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.
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.
