Kwangmin Kim - 함수의 변환

1 Transformations of Functions

1.1 Translations

Thanslations are about vertical and horizontal sifts. To be more sepecific, if \(c\) is a positive number, then

\(y=f(x)+c\), shift the graph of \(y=f(x)\) a distance \(c\) units upward
\(y=f(x)-c\), shift the graph of \(y=f(x)\) a distance \(c\) units downward
\(y=f(x-c)\), shift the graph of \(y=f(x)\) a distance \(c\) units to the right
\(y=f(x+c)\), shift the graph of \(y=f(x)\) a distance \(c\) units to the left

1.1.1 Example

draw \(y=x\), \(y=(x-3)\), \(y=x-3\), \(y=(x+3)\), \(y=x+3\)

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-10, 10, 1000)
y = x
y2 = x-3
y3 = x+3


plt.xlabel('X')
plt.ylabel('Y')
plt.axhline(y=0, color='k')
plt.axvline(x=0, color='k')
plt.grid(True, which='both')

plt.plot(x,y,color='black',label='y=x')
plt.plot(x,y2,color='red',label='y=(x-3) or (y+3)=x')
plt.plot(x,y3,color='blue',label='y=(x+3) or (y-3)=x')

plt.title('Traslation of Functions')
plt.legend()
plt.show()

1.2 Transformations

There are largely two types of transofrmations of functions: stretching and reflecting. To be more sepecific, if \(c\) is larger than 1, then

\(y=cf(x)\), stretch the graph of \(y=f(x)\) vertically by a factor of \(c\)
\(y=\frac{1}{c}f(x)\), shrink the graph of \(y=f(x)\) vertically by a factor of \(c\)
\(y=f(cx)\), shrink the graph of \(y=f(x)\) horizontally by a factor of \(c\)
\(y=f(\frac{x}{c})\), stretch the graph of \(y=f(x)\) horizontally by a factor of \(c\)

1.2.1 Example

draw \(y=\sin x\), \(y=\sin 2x\), \(y=\frac{1}{2} \sin x\)


x = np.linspace(-10, 10, 1000)
y = np.sin(x)
y2 = np.sin(2*x)
y3 = np.sin(x/2)
y4 = 2*np.sin(x)
y5 = np.sin(x)/2

plt.xlabel('X')
plt.ylabel('Y')
plt.axhline(y=0, color='k')
plt.axvline(x=0, color='k')
plt.grid(True, which='both')

plt.plot(x,y,color='black',label='y=sin(x)')
plt.plot(x,y2,color='red',label='y=sin(2x)')
plt.plot(x,y3,color='blue',label=r'y=sin($\frac{x}{2}$)')
plt.plot(x,y4,color='green',label=r'y=2sin(x)')
plt.plot(x,y5,color='orange',label=r'y=$\frac{1}{2}$sin(x) or (2y)=sin(x)')

plt.title('Trasformation of Functions')
plt.legend(shadow=True, loc=(-0.2, 1.05), handlelength=1.5, fontsize=8)
plt.show()

1.2.2 Example

draw \(x^2+y^2=1\), \(\frac{x^2}{4}+\frac{y^2}{9}=1\), \((x-3)^2+(y-3)^2=1\), \(\frac{(x-3)^2}{4}+\frac{(y-3)^2}{9}=1\)


# initialize x and y using radian
# theta = np.linspace(0, 2*np.pi, 1000)
# x = np.cos(theta)
# y = np.sin(theta)

# initialize x and y without using radian
x = np.linspace(-1, 1, 1000)
y1 = np.sqrt(1 - x*x)
y2 = -np.sqrt(1 - x*x)

fig, ax = plt.subplots()

# unit circle
ax.plot(x, y1, color='black', label=r'$x^2+y^2=1$')
ax.plot(x, y2, color='black')
ax.plot(0,0,'o',color='black', label='(0,0)')

# unit circle translated by 2
ax.plot(x+3, y1+3, color='red', label=r'$(x-3)^2+(y-3)^2=1$')
ax.plot(x+3, y2+3, color='red')
ax.plot(3,3,'o',color='red', label='(3,3)')

# eplipse: a unit circle transformed by 2,3 in x, y
ax.plot(2*x, 3*y1, color='blue', label=r'$\frac{x^2}{4}+\frac{y^2}{9}=1$')
ax.plot(2*x, 3*y2, color='blue')
ax.plot(0,-np.sqrt(9-4), 'o', color='blue', label=r'$F_1=(0,\sqrt{5}),F_2=(0,-\sqrt{5})$')
ax.plot(0,np.sqrt(9-4), 'o', color='blue')

# translated eplipse: a unit circle translated by 2 and transformed by 2,3 in x, y
ax.plot(2*x+3, 3*y1+3, color='green', label=r'$\frac{(x-3)^2}{4}+\frac{(y-3)^2}{9}=1$')
ax.plot(2*x+3, 3*y2+3, color='green')
ax.plot(0+3,-np.sqrt(9-4)+3, 'o', color='green', label=r'$F_1=(3,\sqrt{5}+3),F_2=(3,-\sqrt{5}+3)$')
ax.plot(0+3,np.sqrt(9-4)+3, 'o', color='green')

# height/width=1
ax.set_aspect(1)

ax.set_xlim([-4, 7])
ax.set_ylim([-4, 7])

ax.grid(True)
ax.set_title("Transformation of Functions")
ax.set_xlabel("x-axis")
ax.set_ylabel("y-axis")

plt.legend(shadow=True, loc=(-0.5, 1.05), handlelength=1.5, fontsize=8)

# show the plot
plt.show()

Try using \(y=\sin (\theta)\), \(x=\cos (\theta)\), by yourself, to draw \(x^2+y^2=1\), \(\frac{x^2}{4}+\frac{y^2}{9}=1\), \((x-3)^2+(y-3)^2=1\), \(\frac{(x-3)^2}{4}+\frac{(y-3)^2}{9}=1\)

1.3 Reflections

\(y=-f(x)\), reflect the graph of \(y=f(x)\) about the x-axis
\(y=f(-x)\), reflect the graph of \(y=f(x)\) about the y-axis
\(-y=f(-x)\), reflect the graph of \(y=f(x)\) about the origin on the 2D plain
\(x=f(y)\), reflect the graph of \(y=f(x)\) about the \(y=x\)

1.3.1 Example

draw \(y=\sin x\), \(y=\sin (-x)\), \(y=-\sin x\), \(-y=-\sin x\), \(x=-\sin y\)


x = np.linspace(0, 2*np.pi, 1000)
x1 = np.linspace(-2*np.pi, 2*np.pi, 1000)
y = np.sin(x)

fig, ax = plt.subplots()

# unit circle
ax.plot(x, y, color='black', label='$y=\sin (x)$')
ax.plot(-x, y, color='blue', label='$y=\sin (-x)$')
ax.plot(x, -y, color='green', label='$y=-\sin (x)$')
ax.plot(-x, -y, color='orange', label='$-y=-\sin (x)$')
ax.plot(x1,x1, color='red', label='$y=x$')
ax.plot(np.sin(y),x, color='black', label='$y=sin^{-1}(x)$',linestyle='dashed')

# height/width=1
ax.set_aspect(1)

ax.set_xlim([-7, 7])
ax.set_ylim([-7, 7])

ax.grid(True)
ax.set_title("Reflection of Functions")
ax.set_xlabel("x-axis")
ax.set_ylabel("y-axis")
plt.legend(shadow=True, loc=(-0.25, 1.05), handlelength=1.5, fontsize=8)
# show the plot
plt.show()

Draw \(S(x)=\frac{1}{1+e^{-x}}\)


x = np.linspace(-10, 10, 1000)
y = 1/(1+np.exp(-x))

fig, ax = plt.subplots()

# unit circle
ax.plot(x, y, color='black', label='$y=S(x)$')
ax.plot(-x, y, color='blue', label='$y=S(-x)$')
ax.plot(x, -y, color='green', label='$y=-S(x)$')
ax.plot(-x, -y, color='orange', label='$-y=-S(x)$')
ax.plot(x,x, color='red', label='$y=x$')
ax.plot(np.sin(y),x, color='black', label='$y=S^{-1}(x)$',linestyle='dashed')

# height/width=1
ax.set_aspect(1)

ax.set_xlim([-3, 3])
ax.set_ylim([-3, 3])

ax.grid(True)
ax.set_title("Reflection of Functions")
ax.set_xlabel("x-axis")
ax.set_ylabel("y-axis")
plt.legend(shadow=True, loc=(-0.25, 1.05), handlelength=1.5, fontsize=8)
# show the plot
plt.show()

1.3.2 Application: Standardization

Distribution of Raw Data


data1 = np.random.normal(loc=5,scale=6.0,size=1000) #loc=mean, scale=sd
data2 = np.random.normal(loc=-4,scale=2.0,size=1000)
data3 = np.random.normal(loc=-7,scale=2.5,size=1000)
data4 = np.random.normal(loc=0,scale=1.0,size=1000)

# the range of values to evaluate the PDF
x = np.linspace(data2.min(), data1.max(), 10000)

bins_number=100

# Plot the data and PDF
plt.hist(data1, density=True, color="red", alpha=0.5, bins=bins_number,label=r'$X \sim N(5,6^2)$')
plt.hist(data2, density=True, color="green", alpha=0.5, bins=bins_number,label=r'$X \sim N(-4,2^2)$')
plt.hist(data3, density=True, color="blue", alpha=0.5, bins=bins_number,label=r'$X \sim N(-7,2.5^2)$')
plt.hist(data4, density=True, color="black", alpha=0.5, bins=bins_number,label=r'$X \sim N(0,1)$')

plt.legend(shadow=True, loc=(-0.25, 1.05), handlelength=1.5, fontsize=8)
plt.title("Distribution of Raw Data")
plt.show()

Distribution of Translated Raw Data


# Plot the data and PDF
plt.hist(data1-data1.mean(), density=True, color="red", alpha=0.5, bins=bins_number,label=r'$X \sim N(5-5,6^2)$')
plt.hist(data2-data2.mean(), density=True, color="green", alpha=0.5, bins=bins_number,label=r'$X \sim N(-4-(-4),2^2)$')
plt.hist(data3-data3.mean(), density=True, color="blue", alpha=0.5, bins=bins_number,label=r'$X \sim N(-7-(-7),2.5^2)$')
plt.hist(data4-data4.mean(), density=True, color="black", alpha=0.5, bins=bins_number,label=r'$X \sim N(0-0,1)$')

plt.legend(shadow=True, loc=(-0.25, 1.05), handlelength=1.5, fontsize=8)
plt.title("Distribution of Translated Raw Data")
plt.show()

Distribution of Transformed Raw Data


# Plot the data and PDF
plt.hist(data1/data1.std(), density=True, color="red", alpha=0.5, bins=bins_number,label=r'$X \sim N(5,6^2/6^2)$')
plt.hist(data2/data2.std(), density=True, color="green", alpha=0.5, bins=bins_number,label=r'$X \sim N(-4,2^2/2^2)$')
plt.hist(data3/data3.std(), density=True, color="blue", alpha=0.5, bins=bins_number,label=r'$X \sim N(-7,2.5^2/5^2)$')
plt.hist(data4/data4.std(), density=True, color="black", alpha=0.5, bins=bins_number,label=r'$X \sim N(0,1/1)$')

plt.legend(shadow=True, loc=(-0.25, 1.05), handlelength=1.5, fontsize=8)
plt.title("Distribution of Transformed Raw Data")
plt.show()

Distribution of Standardized Raw Data


# Plot the data and PDF
plt.hist((data1-data1.mean())/data1.std(), density=True, color="red", alpha=0.5, bins=bins_number,label=r'$X \sim N(0,1)$')
plt.hist((data2-data2.mean())/data2.std(), density=True, color="green", alpha=0.5, bins=bins_number,label=r'$X \sim N(0,1)$')
plt.hist((data3-data3.mean())/data3.std(), density=True, color="blue", alpha=0.5, bins=bins_number,label=r'$X \sim N(0,1)$')
plt.hist((data4-data4.mean())/data4.std(), density=True, color="black", alpha=0.5, bins=bins_number,label=r'$X \sim N(0,1)$')

plt.legend(shadow=True, loc=(-0.25, 1.05), handlelength=1.5, fontsize=8)
plt.title("Distribution of Standardized Raw Data")
plt.show()

2 Thinking About …

Transformation in Statistics? ex) transformation of random variable
Transformation in Linear Algebra? ex) linear transformation or linear mapping (transformation matrix)
Transformation in Machine Learning? ex) scaling (min-max normalization or standardization)

3 Transformations of Functions

3.1 Translations