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# Sketching Curves with Asymptotes – Example 1

We earlier saw how to sketch the curve of a function and a polynomial function (with and without solving the polynomial function).

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

y =

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) =

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+

=

=  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

 x 0.4 = 0.447 0.5 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

 x -0.5 = -0.447 -0.4 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.