Learn Implementing Identity-Quadratic Functions in Python

Identity Function

The identity function returns the input value unchanged, following the form $f(x) = x$ . In Python, we implement it as:

# Identity Function
def identity_function(x):
    return x

The identity function returns the input value unchanged, following the form $f(x)=x$ . To visualize it, we generate x-values from -10 to 10, plot the line, mark the origin $(0,0)$ , and include labeled axes and grid lines for clarity.


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import numpy as np 
import matplotlib.pyplot as plt 
# Identity Function 
def identity_function(x): 
    return x 

x = np.linspace(-10, 10, 100) 
y = identity_function(x) 

plt.plot(x, y, label="f(x) = x", color='blue', linewidth=2) 
plt.scatter(0, 0, color='red', zorder=5) # Mark the origin 

# Add axes 
plt.axhline(0, color='black', linewidth=1) 
plt.axvline(0, color='black', linewidth=1) 

# Add labels 
plt.xlabel("x") 
plt.ylabel("f(x)") 

# Add grid 
plt.grid(True, linestyle='--', alpha=0.6) 
plt.legend() 
plt.title("Identity Function: f(x) = x") 
plt.show()

Constant Function

A constant function always returns the same output, regardless of the input. It follows $f(x) = c$ .

# Constant Function
def constant_function(x, c):
    return np.full_like(x, c)

A constant function always returns the same output, regardless of the input, following the form $f(x) = c$ . To visualize it, we generate x-values from -10 to 10 and plot a horizontal line at $y = 5$ . The plot includes axes, labels, and a grid for clarity.


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import numpy as np
import matplotlib.pyplot as plt

def constant_function(x, c):
    return np.full_like(x, c)

x = np.linspace(-10, 10, 100)
y = constant_function(x, c=5)

plt.plot(x, y, label="f(x) = 5", color='blue', linewidth=2)
plt.axhline(0, color='black', linewidth=1)
plt.axvline(0, color='black', linewidth=1)
plt.xlabel("x")
plt.ylabel("f(x)")
plt.grid(True, linestyle='--', alpha=0.6)
plt.legend()
plt.title("Constant Function: f(x) = 5")
plt.show()

Linear Function

A linear function follows the form $f(x) = mx + b$ , where $m$ represents the slope and $b$ the y-intercept.

# Linear Function
def linear_function(x, m, b):
    return m * x + b

A linear function follows the form $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. We generate x-values from -20 to 20 and plot the function with both axes, a grid, and marked intercepts.


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import numpy as np
import matplotlib.pyplot as plt

def linear_function(x, m, b):
    return m * x + b

x = np.linspace(-20, 20, 400)
y = linear_function(x, m=2, b=-5)

plt.plot(x, y, color='blue', linewidth=2, label="f(x) = 2x - 5")
plt.scatter(0, -5, color='red', label="Y-Intercept")
plt.scatter(2.5, 0, color='green', label="X-Intercept")
plt.axhline(0, color='black', linewidth=1)
plt.axvline(0, color='black', linewidth=1)
plt.xlabel("x")
plt.ylabel("f(x)")
plt.grid(True, linestyle='--', alpha=0.6)
plt.legend()
plt.title("Linear Function: f(x) = 2x - 5")
plt.show()

Quadratic Function

A quadratic function follows $f(x) = ax^2 + bx + c$ , creating a parabolic curve. Key features include the vertex and x-intercepts.

# Quadratic Function
def quadratic_function(x):
    return x**2 - 4*x - 2

A quadratic function follows $f(x) = ax^2 + bx + c$ , forming a parabolic curve. We generate x-values from -2 to 6, plot the function, and mark the vertex and intercepts. The plot includes both axes, a grid, and labels.


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import numpy as np
import matplotlib.pyplot as plt

def quadratic_function(x):
    return x**2 - 4*x - 2

x = np.linspace(-2, 6, 200)
y = quadratic_function(x)

plt.plot(x, y, color='blue', linewidth=2, label="f(x) = x² - 4x - 2")
plt.scatter(2, quadratic_function(2), color='red', label="Vertex (2, -6)")
plt.scatter(0, quadratic_function(0), color='green', label="Y-Intercept (0, -2)")

# X-intercepts from quadratic formula
x1, x2 = (4 + np.sqrt(24)) / 2, (4 - np.sqrt(24)) / 2
plt.scatter([x1, x2], [0, 0], color='orange', label="X-Intercepts")

plt.axhline(0, color='black', linewidth=1)
plt.axvline(0, color='black', linewidth=1)
plt.xlabel("x")
plt.ylabel("f(x)")
plt.grid(True, linestyle='--', alpha=0.6)
plt.legend()
plt.title("Quadratic Function: f(x) = x² - 4x - 2")
plt.show()

Which code correctly defines a quadratic function in Python that calculates (f(x) = x^2 - 4x - 2)?

Select the correct answer

def quadratic_function(x):
    return x - 4 * x - 2

def quadratic_function(x):
    return x * 2 - 4 * x - 2

def quadratic_function(x):
    return x**2 - 4 * x - 2

def quadratic_function(x):
    return x**3 - 4 * x - 2

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Section 1. Chapter 5

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