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Curve Sketching

We can sketch a curve by finding all the important parameters, such as the x-intercepts, y-intercepts, stationary points, and the points of inflection. Here are the steps to follow to do that:

  • Step 1: Find the x-intercepts, by turning y = 0 and solving the equation
  • Step 2: Find the y-intercepts, by turning x = 0
  • Step 3: Check if the curve is symmetric, i.e. is the function odd or even. Use symmetry properties of a curve for this.
  • Step 4: Check for any discontinuities, and find the asymptotes, if any, or the limits
  • Step 5: Find stationary points (put y’ = f'(x) = 0)
  • Step 6: Find the point of inflection
  • Step 7: Draw a table of values to help sketch the curve. This will help you find the in-between points.

Note: Sometimes, for simplicity, we can start with steps 5 and 6, then work with steps 1 to 4 and then finally step 7.

 

Let’s now apply these steps to the following examples.

Example 1: Sketch a curve for f(x) = x3 – 3x2

Step 1: Find the x-intercepts, by putting y = 0 and solving the equation

f(x) = y = x3 – 3x2 = 0

y = x2(x – 3)  = 0

Therefore x = 0 or x = 3

So x-intercept points are (0, 0) and (3, 0)

 

Step 2: Find the y-intercepts, by putting x = 0

y = 0 – 0  =  0

So y-intercept point is (0, 0)

 

Step 3: Check if the curve is symmetric, i.e. is the function odd or even. Use symmetry properties of a curve for this.

f(x) = x3 – 3x2

f(-x) = (-x)3 – 3(-x)2  =   -x3 – 3x2 

This is neither an odd nor an even function. So there is no symmetry.

 

Step 4: Check for any discontinuities, and find the asymptotes, if any, or the limits

When {x}~right~infty{y}~right~infty, since {infty}^3 = large value.

{x}~right~-infty{y}~right~-infty, since {infty}^-3 = very small value.

There are no discontinuities or asymptotes to find for this function.

 

Step 5: Find stationary points (put y’ = f'(x) = 0)

f'(x) = 3x2 – 6x = 0

3x(x – 2) = 0

So x = 0 or x = 2

When x = 0, y = 0    (0, 0) is a stationary point

When x = 2, y = 23 – 3×22 = 8 – 12  = -4

(2, -4) is another stationary point

 

Step 6: Find the point of inflection

f”(x) = 6x – 6

f”(x) =  0    doubleright 6x – 6 = 0, or 6x = 6 or x = 1

When x = 1, y = (1)3 – 3(1)2 = -2

(1, -2) is a possible point of inflection.

 

Step 7: Draw a table of values to help sketch the curve. This will help you find the in-between points.

x 1/2 1 3/2
f”(x) 6 x 1/2 – 6 0 6 x 3/2 – 6
f”(x) = -3 < 0 0 = 3 > 0

Note that here we choose points as close as possible to 1, since x = 0 is a stationary point, so we pick a point that is past 0, but before x = 1.

The concavity of the curve changes, hence (1, -2) is a point of inflection.

To find the type of stationary point –

f”(0) = 6×0 – 6  =  -6 < 0  (concave downwards)

So (0, 0) is a maximum point

f”(2) = 6×2 – 6  =  6 > 0  (concave upwards)

So (2, -4) is a minimum point

 

Now we’ve got all the parameters required to sketch the curve

  • maximum point right (0, 0)
  • minimum point  right (2, -4)
  • point of inflection right  (1, -2)
  • x-intercepts right (0, 0), (3, 0)
  • y-intercept right (0, 0)

Now using these points, we sketch a graph with suitable scale.

curve sketching for a cubic function

 

We’ll now take a look at how to sketch a curve for a polynomial function.

See also examples of sketching curves with asymptotes –

Curve Sketching with asymptotes – Example 1

Curve Sketching with asymptotes – Example 2

Curve Sketching with asymptotes – Example 3

Curve Sketching with asymptotes – Example 4