Question 2

1. functions

Figure 12

Figure 11

f(x)=

f(x)=

If to take a simple example of the y= function as y= where one positive value is even and its separate curve will be parabola shape, and another positive value is odd which has cubic function shape (Figure 11), composition of this values will change their curves (Figure 12).

f(x)=

y= function is a combination of quadratic and cubic functions. This function has local minimum and maximum its shape is a cubic function with point of inflection but which has not a gradient zero.

f(x)=

f(x)=

Figure 14

Figure 13

In the example when function y= has both numbers even shape of such function will depend on negative and positive values, but we all know that a separate curve of the even number instead of a will be parabola. For example function y= will have different shapes depending on negative and positive signs in the fraction (see figure 13 and 14)

f(x)=

f(x)=

f(x)=

f(x)=

In the case when both values are positive or both negative their curve is an ordinary parabola, but when one value is negative and one positive their curve has three stationary points, in case with f(x)=x²-x⁴ the curve has one maxima and two minima, and when f(x)=-x²+x⁴ the curve has one minima and two maxima.

Figure 16

Figure 15

If to put two odd values into the y= this values will have two possible curves which will depend on the positive and negative numbers as in the example with even numbers. So if to take the y= function and change cases with positive and negative signs – shapes of the curve will change as well:

f(x)=

f(x)=

f(x)=

f(x)=

Figures 15 and 16 show that in the case when both values are positive or both negative their curve is an ordinary shape of a cubic function, but when one value is negative and one positive their curve has three stationary points, in case with f(x)=x³-x⁵ the curve has one maxima, one minima and one point of inflection, and f(x)=-x³+x⁵ function has the same amount of stationary points but they are reflected along y-axis.

Figure 17

There is another pattern with values in the y= function: if values become bigger the curve becomes more sharp and if and values are far different (as 3 and 55) the curve’s gradient will be broader and if values are close to each other (as 51 and 57) the curve’s gradient will be slim (see figure 17)

f(x)=

f(x)=

f(x)=

2. Curves y= can have cusp the fraction instead of a has a fraction with even numerator and odd denominator

Figure 18

C urve with a shift along x and y-axis: (figure 18)

f(x)=

f(x)= function has a cusp due to even numerator over odd denominator, (x-4) in brackets represents a shift to the right along x-axis, +2 at the end represent a shift up along the y-axis

Figure 19

Inverse curve with shift along x and y-axis: (figure 19)

f(x)=

N egative sign in front of the f(x)= equation represents the inverse of the graph upside down, (x+7) in brackets represents a shift to the left along x-axis, -5 at the end represent a shift down along the y-axis, a fraction made a curves range very small that is why it looks almost as a straight line because 2 and 23 are far from each other, it looks like that the cusp is at (-7; -5.3) point, but table of values shows that the cusp as at (-7;-5), so it is reasonable to assume that slim graph is distorted.

Figure 20

C urves with cusp and straight line: (figure20)

f(x)=

Negative sign in front of the f(x)= equation represents the inverse on the graph upside down, (x-3) in brackets represents a shift to the right along x-axis, -3 at the end represent a shift down along the y-axis, a fraction change a curve from smooth to sharp straight line due to big values which are close to each other

f(x)=

Figure 21

A n ordinary function f(x)= which has a cusp at x=0 (Figure 21) can be modified by adding or subtracting values.

Features of the f(x)= function: (Figure 22)

f(x)= + x

f(x)=

Figure 22

B y adding x to the function with a cusp the shape of the graph changes: the pattern of the new graph is that it adds an x value of f(x)= to its y value

Eg.1 Point (1; 1) of the f(x)= will become:

1+1=2

Point (1; 2) of the f(x)=

Eg.2 Point (-3; 2) of the f(x)= will become:

-3+2=-1

Point (-3;-1) of the f(x)=

Figure 23

f(x)= + 2

F eatures of the f(x)= function: (Figure 23)

Some function + or – some number means that that function will shift up or down.

f(x)=

Function f(x)= +2 shifts up by two values along the y-axis

A curve with a cusp at x=0 and x=1:

Figure 24

A s it is already investigated that functions with a cusp should have a fraction as a power, a fraction needs to have even numerator and odd denominator. To get a cusp at x=0 the function should have x to the power of the investigated fraction, to get a function with a cusp at x=1 x needs to take 1 to make a shift along x-axis.

Figure 24 has two graphs:

f(x)=

f(x)= which has a cusp at x=0 and its shape is almost straight because 22 and 23 values are big and they are close to each other

f(x)=

f(x)= which has a cusp at x=1 and its shape is smooth because 2 and 3 values are small and it has a broad gradient because values are close.

f(x)=

Figure 25

T o get a curve with two cusps at x=0 and x=1 two investigated functions above need to be summed up, other functions with other values except of (x-1) will create required curve as well.

f(x)=

f(x)=

A new curve is created by adding value of one curve to another (Figure 25)

Eg.1 f(x)= has a point (0; 0)

f(x)= has a point (0; 1)

0+1=1

So the point at x=0 of the

f(x)= will be (0; 1)

Eg.2 f(x)= has a point (2; 1) f(x)= has a point (2; 2) 1+2=3

So the point at x=0 of the f(x)= will be (2; 3)