02Haddords

toc = Addition of Ordinates =

Aim: To sketch a non-standard graph by sketching each part separately then adding the ordinates together.

For a demonstration of this process in Powerpoint, download the following file (3.75 Mbytes):

** Example **
math y=\dfrac{1}{(x+1)^2}+x-1 math

By examining this equation, we see there will be a vertical asymptote with the equation x = –1 and an oblique asymptote with the equation y = x – 1. We can divide the function into two sections math \\ y_1=\dfrac{1}{(x+1)^2} \quad \text{and} \\ \\ y_2 = x-1 math

Use a pencil or a light colour to sketch these two simple functions on the one set of axes. (y 1 is in blue, y 2 is in green)

Notice the vertical asymptote of the truncus at x = –1. The domain of the truncus will therefore be x Î R\ {–1}

Now work across the graph, comparing and adding the y-coordinates of the two simple graphs for a variety of different x-coordinates. (marked in red)

Note 1: The final value is always greater than the y ``=`` x – 1 value. At each end of the graph, the final value is very close to y ``=`` x – 1. The graph will be approaching (from above) the line y ``=`` x – 1 as an oblique asymptote as x approaches ¥ and as x approaches – ¥.

Note 2: To the left of x ``=`` 1 our final line is below the original truncus. To the right of x ``=`` 1, it is above the original truncus. The point x ``=`` 1 is important because it is the x-intercept for y ``=`` x – 1.

Using all of this information, we can draw our final graph (in red).

Domain: x Î R\ {–1} Range: y Î R

Asymptotes: x = –1 y = x – 1

y-intercept: y = 0 x-intercepts: x = –1.62, 0, 0.62

Stationary points: Local minimum at (0.26, –0.11)

Summary
To draw a graph by addition of ordinates > * values close to any vertical asymptote(s) > * each end of the x-axis > * y-intercept > * x-intercepts of either original graph > * placement of x-intercepts and turning points
 * 1) Sketch (using pencil or a light colour) the two original graphs
 * 2) Draw and label the asymptotes
 * 3) For a variety of x-values, add the y-coordinates and mark points.
 * 4) Pay particular attention to:
 * 1) Sketch the resultant graph by drawing a smooth curve through the points.
 * 2) Take care to draw the graph approaching the asymptotes (not touching or curling away)
 * 3) Label graphs, axes, asymptotes, intercepts, turning points,
 * 4) State the domain, range and the equations of asymptotes

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