Curve Sketching

 
       As we have mentioned before, the graph of a function is a geometric representation of the function. Graphs have their usefulness. We can study certain properties to solve problems, for example, in inequalities. Graphs are common in physics, where data collected in an experiment are plotted and a graph drawn to reduce errors. Relations between variables can also be derived from graphs.

In this section, we start by introducing techniques used to sketch curves. These techniques can be applied to any graph on the cartesian plane. We then proceed to explain specific properties of various types of graphs, which are listed in our contents. This section serves as a basic introduction to the various shapes of graphs, though it is not exhaustive. In the next section, we will discuss transformations of graphs. Graphs of exponential and logarithmic, as well as trigonometric graphs, are treated separately in detail.

Certain mathematical foundations must be present before tackling this topic. Knowledge of algebraic concepts as well as the cartesian plane is required. Also, differentiation and its application to stationary points is also used.

Are you ready to go forth and learn curve sketching?

 

Important Points

1. X-intercepts. These can be found by putting y=0 and solve for x.

2. Y-intercepts. These can be found by putting x=0 and solve for y.

3. Stationary points. Found by calculating the derivative of the function, set it to zero and solve for x.

4. Asymptotes. Lines on the plane that the curve tends towards but does not cut. Can be vertical, horizontal or oblique. Found by letting x or y tend to infinity and check if the function approaches a certain value.

5. Gradient. Is the gradient positive or negative for the relevant portions of the graphs. It can be deduced from the derivative.

 These techniques can be used for any graph, especially if you do not know the general shape of the graph.

 

 

Example: Sketch the graph of x² - 2x - 3.

 

Solution:

 

1. Find y-intercept.

    When x = 0, y = -3

 

2. Find x-intercepts.

    When y = 0, x² - 2x - 3 = 0 --> x = 3 or x = -1

 

3. Find stationary points.

    dy/dx = 2x - 2 = 0

    x = 1, y = 4

 

4. Asymptotes: none

 

5. Gradient.

    When x is slightly less than 1, dy/dx < 0

    When x is slightly greater than 1, dy/dx > 0

 

Mark these points on a set of axes and draw a smooth curve through tem.

 

 

More examples in individual sections.

 

 

Curve Sketching | Quadratic Equation | Rectangular Hyperbolas | Conic sections | Modulus Functions