# Maths

How do you sketch curves like x to the power of 4 or x to the power of 6 ?
What are the steps involved ?

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1. it will depend what other terms are along with the x^4 term

in general for y = ax^4 + bx^3 +cx^2 + dx + e , the most dramatic graph would be in the shape of a "W", with a maximum possibility of 4 x-intercepts.
in its simplest form of y = x^4, it will resemble a very fast-rising 'parabola'.
I stress that it looks like a parabola, but does not have any of the parabolic qualities.

for y = ax^6 + bx^5 ....

the basic shape would be a double "W" with a maximum of 6 x-intercepts possible.

Your best bet is to make a table of values of x and y. You might want to scale the y axis different from the x axis because the y's tends to become large in a hurry.

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2. I don't really know how much you know about mathematics, but I'll give you some basic steps in sketching a function.

1) find the domain of the function. You can't draw the curves in places where the function can't go.
(For polynomials of x, the function reaches from-infinity untill +infinity)

2) Solve the function. These are the places where your function crosses the x-axis of your graph.
(if you should have any problem solving polynomials of higher degrees, please feel free to ask).

3) Check which value you get for x=0 (this is the place where your curve crosses the y-axis)

4) Next, take the first derivative of the function and solve it. When you solve the first derivative of your function you find the places where your function reaches a maximum or minimum (you can find out if it's a maximum or minimum by making a sign table of the first derivative, if the graph is rising, the first derivative is positive, if the graph is descending, the first derivative is negative)

5)Calculate the second derivative of the function and solve it. The second derivative describes the curvature of your graph. I'll refer to this picture:
Search Google for pictures of "second derivative" and look at the first picture that pops up titled rulers.gif.
In the points P and T, the graph has a negative curvature (negative second derivative) and in R it has a positive curvature (positive second derivative). When the graph flips from positive to negative curvature (or vice versa) it has a second derivative that equals 0.
So by solving the second derivative and making a sign table of it, you can find the points where the graph changes curvature.

6) When you have found all the points I previously discusses, you put them on a graph and connect them with a fluid line. You now should have a graph that is very close to the graph of the function.

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