## Curve Sketching

When sketching curves of a polynomial function 1. try to factorise it first, if it is not already factorised. Writing a polynomial as a product of factors - - for example - makes it easier to identify the roots – the points where The roots will be points on the x – axis, since each root is a solution of the equation 2. Find the – intercept by substituting into 3. Decide whether the curve tends to or as x tends to or If the coefficient of the highest power of is positive when the expression is expanded, then as tends to so does and if the coefficient is negative, then as tends to  tends to 4. Each distinct root - - with no power - will give rise to a point on the – axis where the curve CROSSES the – axis, and each repeated root - given by a factor – will give rise to a point on the curve which touches the – axis but does not cross it if is even, or which forms a tangent to the – axis and crosses it if is odd. Examples are shown below. For example, to sketch The roots are the solutions to These are   Substituting into the expression gives The highest power of is and the coefficient of is 2 (consider so as Each root is distinct, so the graph crosses the – axis at each root. The graph is sketch below. To sketch The roots are given by  Substituting into the expression gives The highest power of is and the coefficient of is -1 (consider so as The root at is a double root, so the curve touches the axis at but does not cross it, and the root at is a single root so the graph crosses the – axis there.

The curve is sketched below.  