Aims: To be able to find equations of asymptotes to graphs of To be able to find equations of asymptotes to graphs of rational functions rational functions To be able to find points of intersection of these graphs To be able to find points of intersection of these graphs with the co-ordinate axes. with the co-ordinate axes. To be able to sketch these graphs. To be able to sketch these graphs. Graphs Lesson 1 1

2 Graphs of Rational Functions We are going to look at how to sketch rational functions of the form where the numerator N(x) and the denominator D(x) are polynomials, and D(x) is not the zero polynomial 2 Looking at the graph y = If you translate it 3 units to the right and 2 units up, you get the graph of y =

3 y = + 2 = = x cannot be 3 otherwise there would be division by zero This is shown on the graph by the vertical line x =, the curve cannot cross this If x gets numerically large the equation approximates to y =, which is the horizontal line on the graph.

4 The vertical & horizontal line are examples of asymptotes An asymptote is a straight line which a curve approaches tangentially as x and/or y approaches infinity. The line x = 3 is a vertical asymptote; the line y = 2 is a horizontal asymptote. It is usual for asymptotes to be shown as dashed lines Generally the line x = a is a vertical asymptote for the curve y = if this a value makes the bottom zero. What are the vertical asymptotes of y =

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6 1. Find the intercepts with the axes y = if x = 0, y =curve goes through ( ) if y = 0,curve goes through ( ) We are now going to go through the steps involved in drawing a sketch graph

7 2. Find the vertical asymptotes y = Asymptotes at x =, x =

8 We need to know if y is positive or negative just on each side of the asymptotes y = If x < -1, y is If x > -1, y is If x < 2, y is If x > 2, y is If you have a graphical calculator you would normally miss out the work on this slide!

9 3. Find the horizontal asymptote. y = As x gets numerically large, the equation approaches y  so the curve approaches the x- axis at the extremes To see which side of 0 it approaches at, enter x=100 and x= -100 into your calculator So +∞ tends to y=0 from -∞ tends to y= 0 from The standard approach is to remove any brackets in the fraction and divide each term by the highest power of x in the fraction found. Again you might miss out the part of finding which side the graph approaches the line y = 0, if you own a graphical calculator. y =

10 From this we can get a rough sketch of the shape y =

11 Summary of Steps 1. Find the intercepts with the axes 2. Find the vertical asymptotes 3. Find the horizontal asymptotes. (Examine the behaviour as x tends to   ) Properties of today’s types of graphs! ; reciprocal shape, 1 vertical and 1 horizontal asymptote. ; 2 vertical and 1 horizontal asymptote. ; 2 vertical and horizontal asymptote is always y = 0

12 Sketch the graph of y = 1. Find the intercepts with the axes When x = 0, y =When y = 0, x = or

13 2. Find the vertical asymptotes y = x cannot be or as that would lead to division by zero So asymptotes at x =, x = We need to know if y is positive or negative just on each side of the asymptotes. Enter graph into calculator to see! (Graph Mode) Press draw then to see this particular graph more clearly press F3 to change the scale. Select F1 (INIT) then EXE twice.

14 3. Find the horizontal asymptote. (Examine the behaviour as x tends to   ) y = Divide each term by x From this we can get a rough sketch of the shape

15 Sketch On w/b Do Ex 5A page 57 Ex 5B page 59 Ex 5C page 61