You have a third degree equation. If you graph this you see that it starts when x large negative the function is also large negative and when x is large positive the function is also large positive. That is because at large values only the x^3 counts (take for instance x = 1000000) so the sign of the x^3 is an indication whether it is is x large neg, then f large neg and x large pos. f large pos. It the coefficient of x^3 is negative the function is mirrored with regard to the x -axis.
A third grade function like this when you go form x neg. large to x pos large starts negative, rises until it reaches a local maximum, then descends until it reaches a local minimum, the it rises again. The local extremes are easily to compute: the derivative must be zero: 3.x^2 - 12.x + 8 = 0 is a simple 2-grade function.
As the function starts negative and ends positive the must be at least 1 point where it crosses the x-axis, so where x = 0. It can also have 3 zero points when the bumps (local extremes) are above (the p maximum) and below (the minimum) of the x-axis.
So you start with a "fishbone" a horizontal line about 10 cm with at every 1 cm a vertical stroke of about 2 cm. Then you fill in some values of x. The best way to start is starting with x equal to the biggest factor in the function: here 8, and go from -8 to + 8. For some simple values of x you see: x = 0, f = -2 and x = 1, f = 1 - 6 + 8 - 2 = +1.
After you have calculated som 10 values pairs (x,f) you can sketch the function.
The best way to get a good estimate of the zero points is by trail and error. Take two values of x one with f positive and one with f negative. Calculate f at the point half way between those 2 values. Then you know whether the zero point is left or right form this new point. Then take this new point and from the two old points the one that gives an opposite value of f ( so when f at the new point is negative, chose the old point where x is positive etc.) . Then again chose a point half way between the two (new) points. etc.
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You have a third degree equation. If you graph this you see that it starts when x large negative the function is also large negative and when x is large positive the function is also large positive. That is because at large values only the x^3 counts (take for instance x = 1000000) so the sign of the x^3 is an indication whether it is is x large neg, then f large neg and x large pos. f large pos. It the coefficient of x^3 is negative the function is mirrored with regard to the x -axis.
A third grade function like this when you go form x neg. large to x pos large starts negative, rises until it reaches a local maximum, then descends until it reaches a local minimum, the it rises again. The local extremes are easily to compute: the derivative must be zero: 3.x^2 - 12.x + 8 = 0 is a simple 2-grade function.
As the function starts negative and ends positive the must be at least 1 point where it crosses the x-axis, so where x = 0. It can also have 3 zero points when the bumps (local extremes) are above (the p maximum) and below (the minimum) of the x-axis.
So you start with a "fishbone" a horizontal line about 10 cm with at every 1 cm a vertical stroke of about 2 cm. Then you fill in some values of x. The best way to start is starting with x equal to the biggest factor in the function: here 8, and go from -8 to + 8. For some simple values of x you see: x = 0, f = -2 and x = 1, f = 1 - 6 + 8 - 2 = +1.
After you have calculated som 10 values pairs (x,f) you can sketch the function.
The best way to get a good estimate of the zero points is by trail and error. Take two values of x one with f positive and one with f negative. Calculate f at the point half way between those 2 values. Then you know whether the zero point is left or right form this new point. Then take this new point and from the two old points the one that gives an opposite value of f ( so when f at the new point is negative, chose the old point where x is positive etc.) . Then again chose a point half way between the two (new) points. etc.