To sketch a polynomial function (a polynomial is a function involving powers of x, for example y = x3 + 2x2 - 13x + 10):
1. Start by differentiating the function to find its turning points (where the first differential is equal to zero). In this case, the first differential of the function is 3x2 + 4x -13. Solve for the roots of the differential (3x2 + 4x -13 = 0) by using the quadratic equation. The roots of the first differential are the points at which the graph has a maximum or minimum turning point (or, more unusually at A level, a point of inflection).
2.To check whether the turning point are maxima or minima, differentiate again for the second differential. Then put in the x-coodinates of each turning point. If the second differential is positive for that x-value, then the turning point is a minimum; if the second differential is negative then the point is a maximum.
3. Find the roots of the polynomial. Often, for a function with powers of x that are 3 or higher, an A-level question will give you a root; if not, try putting in values of x to the equation to find out if they are roots (a root is where y=0 on your graph). Once you have a root, use algebraic long division to factorise your polynomial. For example, with the polynomial y = x3 + 2x2 -13x +10, x=1 is a root, so you would divide (x3 + 2x2 -13x +10) by (x-1). This would give you (x2 + 3x - 10), which you can find the roots of using the quadratic equation. The graph crosses the x-axis at the roots - in this case, the graph crosses at x = -5, x = 1 and x = 2.
4. What does the graph do as x approaches infinity? Think about what happens as x gets very large and positive (going to the right on your graph). If the x term with the highest power is positive, then y will also be very large and positive. If the x term with the highest power is multiplied by a negative number, then y will be very large and negative. For example, for the funtion y = -x3 + 2x +1, as x becomes very large and positive, y becomes very large and negative, so the graph heads off downwards to the right after its last crossing point on the x-axis.
5. What happens as x approaches negative infinity? Again, think about what happens when x is very large and negative. This is a little more complicated because you also have to think about whether the highest power is odd or even. A negative value raised to a even power is positive, whilst a negative value raised to an odd power is negative. For example, when y = -x3 +2x + 1, as x becomes very large and negative, y becomes very large and positive. Here, x3 is negative when x is negative, but because x3 is multiplied by -1, y will be very large and positive when x approaches negative infinity.
6. When x is zero, what is y? It is always good to mark the y-intercept on your graph - just put x = 0 into the graph equation to find the value of the y-intercept.
Using all the information above, start your sketch. I'd usually start by marking the roots on the x-axis, then mark your turning points and join them up. Remember to only draw turning points you have found using the first differential - don't add any in by accident!