y = f(-x) y = -28(x+ 2) y = f(x-1)Example 2: Factor the following polynomials completely. Sketch each graph. a) P(x) = x'+6x2 - x-30 b) P(x) = x3 -5x2 - 2x+241. Domain: All possible values of The domain of a rational function consists of all real number except where the denominator Set and solve for X. 2. x-intercepts/zeros: These occur when 3. y-intercepts: These occur when 4. Vertical, Horizontal and Oblique Asymptotes a) Vertical Asymptote: can be found where the function is If x = a is a vertical asymptote then h(a) = 0 and g(a) # 0(i.e. lim f(x) D.N.E.). If a vertical asymptote occurs at x = a then consider the behaviour x-> a on each side of the asymptote. Important Note: When both g(a) = 0 and h(a) = 0 then the function is indeterminate and a may occur in the graph of the function. Cancel the common factors from the numerator and denominator and then sub in x = a to calculate the y- coordinate of the hole. If the answer is undefined for y, then there is not a hole, but a vertical asymptote at x = a .b) Horizontal Asymptote: indicates the general behaviour of f(x) far off to both sides of the graph (x>i) called . You will have a horizontal asymptote if the degree in the denominator is the degree in the numerator. You can nd the horizontal asymptote by determining lim f (x) . X)00 c) Oblique Asymptote: occurs when there is no horizontal asymptote. In the case the degree of the is larger than the degree in the denominator. If this is the case, f(x) must be re-expressed by carrying out long division. Then take the limit as x > ioo. 5. Specific Points: Use a small table of values to help place the curve in the approximate position. 6. Symmetry about the axis: It is sometimes benecial to know if a function is even or odd when sketching its graph. Even if f (x) = f (x) and odd when f (x) = f (x). Putting it all together... Try graphing y = x- +5x -4 x+3