We now look at an example of sketching curves with asymptotes, i.e. how to sketch a curve that has asymptotes.
Example 1: Sketch a curve for f(x) =
Step 1: Find the y-intercepts, when x = 0
Therefore (0, ) is the y-intercept
Step 2: We cannot find the x-intercepts, since
Step 3: Check if the curve is symmetric, i.e. is the function odd or even.
f(x) = f(-x)
So this is an even function, and is symmetric about y-axis.
Step 4: Check for any discontinuities, and find the asymptotes, if any, or the limits
= = 0+
Step 5: Find stationary points (put y’ = f'(x) = 0)
= 0 So = 0
x = 0
is a stationary point.
Step 6: Find the point of inflection
For point of inflection, = 0.
Therefore 15x2 – 3 = 0
x2 = , or x =
When x = 0, = = = < 0
Hence is a maximum point.
Test for point of inflection at x = = 0.447
|f”(x)||-0.07 < 0||0||0.087 > 0|
When x = 0.447, y = 0.625
So (0.447, 0.625) is a point of inflection.
Test for point of inflection at x = = -0.447
|f”(x)||0.087 > 0||0||-0.07 < 0|
When x = -0.447, y = 0.625
So (-0.447, 0.625) is a point of inflection.
See also other examples of sketching curves with asymptotes –