Point of inflexion and concavity

The concept of increasing function, point of inflexion and concavity can be understood with an illustration.

 

Concavity

a. Strictly increasing. f(x) is strictly increasing when f'(x) > 0. ( x<1.423 or x> 2.577) . These points also coincide with the maximum and minimum point.

b. Strictly decreasing. f(x) is strictly decreasing when f'(x) < 0. (1.423  < x <  2.577)

c. Point of inflexion is the point where the curve changes curvature or concavity. f'(x) does not change sign, and  f” (x) = 0 and changes sign. In the above graph, the curve changes from convex to concave at x = 2.

d. Concave upwards. f(x) concave upwards when f”(x) > 0 or f'(x) is strictly increasing. f(x) concave upwards at x> 2, after f'(x) reaches a minimum, at the point of inflexion.

e. Concave downwards or convex. f(x) concave downwards when f”(x) < 0 or f'(x) is strictly decreasing. f(x) concave downwards for x < 2.

Transformation of graphs: Order of Transformation

In the transformation of graphs, knowing the order of transformation is important. Knowing whether to scale or translate first is crucial to getting the correct transformation.

Let’s look at this example to illustrate the difference:

Example 1

Original point on y=f(x) is x=8

For f(2x+4), we do translation first, then scaling. ie. Move left by 4 units, then scale parallel to the X axis by a factor of 1/2. Hence, the original point becomes x= (8-4)/2 = 2

If we want to do scaling first, we need to factorise into f 2(x+2). So scale parallel to the X axis by a factor of 1/2, then move left by 2 units. Hence, the original point becomes x= (8/2)-2 = 2

Example 2

Describe the transformation of 3f(2x-4) + 5.

Translate 4 units in the positive X direction

Scale by a factor of 1/2 parallel to the X axis

Scale by a factor of 3 parallel to the Y axis

Translate 5 units in the positive Y direction

In summary,

cf(bx+a)+d = Translate by a units in the negative X direction, then scale by a factor of 1/b parallel to the X-axis, then scale by a factor of c parallel to the Y-axis, then translate by d units in the positive Y direction.

c[f a(x+b)]+d = Scale by a factor of 1/a parallel to the X-axis, then translate by b units in the negative X direction, then scale by a factor of c parallel to the Y axis, then translate by d units in the positive Y direction.

 

Sketching graphs of polynomial functions

Sketching graphs of polynomial functions are useful in graphing techniques, and solving inequalities.

Steps

1. First mark down the roots of the polynomial function.

2. Decide how the tail ends behave, whether it is above or below the X axis. This can be determined from the coef of the highest power. For instance, if the polynomial is of degree 6 and the coef of the highest power is positive, then when X approaches either positive or negative infinity, the function approaches positive infinity, so the tail ends are above the X axis.

3. Determine how the function behaves at the roots. When there are 3 or more odd number of roots at the same point, there is a point of inflexion at the root. When there are 2 or more even number of roots at the same  point, there is a minimum or maximum point at the root.

Example 1

graph 1

Example 2

 

2

Using GC to determine the nature of stationary points

Suppose we are required to determine the nature of stationary points for the following:

y

Using TI-84 Plus (OS 2.55) det

Observation of stationary points:

Left stationary point: First derivative changes from +ve to -ve. Therefore, it is a maximum point

Middle stationary point: No change in sign of deriative. Therefore, it is a point of inflexion.

Right stationary point. First derivative changes from -ve to +ve. Therefore, it is a minimum point