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) = x^{3} – 3x^{2}

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

f(x) = y = x^{3} – 3x^{2} = 0

y = x^{2}(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) = x^{3} – 3x^{2}

f(-x) = (-x)^{3} – 3(-x)^{2} = -x^{3} – 3x^{2}

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 , , since = large value.

, , since = 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) = 3x^{2} – 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 = 2^{3} – 3×2^{2} = 8 – 12 = -4

(2, -4) is another stationary point

**Step 6**: Find the point of inflection

f”(x) = 6x – 6

f”(x) = 0 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 | ||

f”(x) | 6 x – 6 | 0 | 6 x – 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 (0, 0)
- minimum point (2, -4)
- point of inflection (1, -2)
- x-intercepts (0, 0), (3, 0)
- y-intercept (0, 0)

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

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