Target: On completion of this worksheet you should be able to sketch a rational
graph. To sketch the graph of a rational function we need to find where in cuts the
...
MATHEMATICS SUPPORT CENTRE
Title: Graphs of rational functions. Target: On completion of this worksheet you should be able to sketch a rational graph.
To sketch the graph of a rational function we need to find where in cuts the axes. It cuts the x-axis when y = 0 and it cuts the y-axis at x = 0. To find the point of intersection we substitute these values into the equation. If you have difficulty with this refer to the graph sheet “sketching graphs 2”.
Example. Find where the graph of the function y =
2x − 2 x−3
Exercise. Find the points at which the graphs of the following functions cut the axes. 4x − 6 1. y = . 2x + 3 x . 2. y = 3x + 6 3x + 6 . 3. y = 2 x + 4x + 2 1 4. y = − 2. x −3
cuts the axes. It cuts the y-axis when x = 0. Therefore: 2×0 − 2 2 y= = . 0−3 3 Hence it cuts at (0, 2 ). 3 It cuts the x-axis when y = 0. Therefore: 2x − 2 0= × [( x − 3)] x−3 0 = 2x − 2 ⇒ x = 1. Hence it cuts the x-axis at (1, 0). Remark. Finding the point where the graph cuts the x-axis is equivalent to finding when the numerator of the fraction equals 0.
C. Leech, Coventry University, July 2000.
(Answers: {(0, -2), (3/2, 0)}; {(0, 0)}; {(0,3), (-2, 0)} {(0, -7/3),(3.5, 0)}.) To sketch the graph of a rational function we should: • Find the points where the graph cuts the axes. • Plot these points. • Find the asymptotes. • Draw the asymptotes. • Draw a smooth curve that does not cross the asymptotes or the axes except at those points that are already plotted.
Examples. Sketch the graphs of 3x + 6 1. y = . x−2 2x − 6 . 2. y = 2 x − x−2 1. Cuts y-axis at (0, -3). Cuts x- axis at (-2, 0). Not defined when x - 2 = 0. Therefore has an asymptote at x = 2. As x gets large positively y gets closer to 3 As x gets large negatively y gets closer to 3. Therefore y = 3 is an asymptote. 6 4 2
-6
-4
Exercise. Sketch the graphs of the following functions. 3x + 4 1. y = . x−8 2x . 2. y = x+5 2x + 1 . 3. y = 2 x − x − 20 10 x + 2 . 4. y = 3 − 5x 2x . 5. y = 2 5 x − 3x + 2 (Answers: 1)
2)
3)
4)
y=3
-2
2
4
6
-2 -4 -6
x=2
2. Cuts y-axis at (0, 3). Cuts x-axis at (3, 0). Not defined when x 2 − x − 2 = 0. ⇒ ( x − 2)( x + 1) = 0 ⇒ x = 2, x = −1. Therefore asymptotes when x = 2 and x = -1. As x gets large both positively and negatively y gets closer to 0. Therefore y = 0 is an asymptote. 6 4 2
-6
-4
-2 -2
2
4
6
-4 -6
C. Leech, Coventry University, July 2000.
5)
).
C. Leech, Coventry University, July 2000.
