Properties of Functions

First derivative test. 1.Differentiate 2.Set derivative equal to zero 3.Use nature table to determine the behaviour of the graph Second derivative test 1.Differentiate 2.Set derivative equal to zero 3.Find second derivative 4.Substitute x values in to second derivative 5.If second derivative is positive, minimum 6.If second derivative is negative, maximum 7.If second derivative is zero or does not exist, use nature table

To find any vertical asymptotes, we set the denominator equal to zero

Non vertical Asymptotes What happens to the y value if x tends to infinity For the degree of the denominator is greater than the numerator, hence the function tends to zero.

To find any vertical asymptotes, we set the denominator equal to zero

Non vertical Asymptotes What happens to the y value if x tends to infinity For the degree of the denominator is greater than the numerator, hence the function tends to zero.

Find the coordinates off the points where the graph crosses the coordinate axes Find the equation of all vertical and non vertical asymptotes Find the coordinates of any stationary points, and, if they exist determine their nature. Sketch the graph of State the range of values of the constant k such that the equation has no real solution