Graph of a Function

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1. What is a Graph of a Function?

A graph of a function is the visual representation of all ordered pairs (x, f(x)). It gives us insight into the function’s behavior, such as:

• Where it increases or decreases

• Its symmetry

• Maximum or minimum points

• Intercepts and asymptotes

A function maps every input x to a unique output f(x). Plotting this for various x-values results in a curve or line.

2. Basic Example

Let’s consider:

$$f(x)=x^{2}$$

To graph this, we compute several (x, f(x)) values:

x	f(x) = x²
-3	9
-2	4
-1	1
0	0
1	1
2	4
3	9

Plotting these points gives a parabola opening upwards.

3. Key Features of a Graph

When analyzing graphs, observe:

Feature	Description
Domain	All x-values for which f(x) is defined
Range	All possible f(x) values
x-intercepts	Where the graph crosses the x-axis (f(x) = 0)
y-intercept	Where the graph crosses the y-axis (x = 0)
Asymptotes	Lines that the graph approaches but never touches
Symmetry	Even/Odd/None
Increasing/Decreasing Intervals	Where the graph goes up/down
Maximum/Minimum	Peaks and valleys of the graph

4. Python Example: Plotting a Function

Example 1: Plotting 𝑓(𝑥)=x²

Let’s use matplotlib and numpy to graph a function.

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def f(x):
    return x**2

# Generate x values
x = np.linspace(-10, 10, 400)
y = f(x)

# Plot
plt.figure(figsize=(8,5))
plt.plot(x, y, label="f(x) = x²", color='blue')
plt.axhline(0, color='black', linewidth=0.5)  # x-axis
plt.axvline(0, color='black', linewidth=0.5)  # y-axis
plt.title("Graph of f(x) = x²")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.grid(True)
plt.legend()
plt.show()

Output:

You can also plot using sympy.

import sympy as sp

# Define the symbolic variable and function
x = sp.symbols('x')
f = x**2

# Plot using sympy
sp.plotting.plot(f, (x, -10, 10),
                 title="Graph of f(x) = x²",
                 xlabel="x", ylabel="f(x)",
                 line_color='blue', legend=True)

Output:

Example 2: 𝑓(𝑥)=sin(x)

Let’s use matplotlib and numpy to graph a function.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-2*np.pi, 2*np.pi, 400)
y = np.sin(x)

plt.figure(figsize=(8,5))
plt.plot(x, y, label="f(x) = sin(x)", color='green')
plt.axhline(0, color='black', linewidth=0.5)
plt.axvline(0, color='black', linewidth=0.5)
plt.title("Graph of f(x) = sin(x)")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.grid(True)
plt.legend()
plt.show()

Output:

You can also plot using sympy.

import sympy as sp

# Define the symbolic variable and function
x = sp.symbols('x')
f = sp.sin(x)

# Plot using sympy
sp.plotting.plot(f, (x, -10, 10),
                 title="Graph of f(x) = sin(x)",
                 xlabel="x", ylabel="f(x)",
                 line_color='blue', legend=True)h

Output:

5. Try Multiple Graphs Together

Let’s use matplotlib and numpy to graph a function.

x = np.linspace(-10, 10, 400)

plt.figure(figsize=(10,6))

# Define functions
plt.plot(x, x, label="f(x) = x")
plt.plot(x, x**2, label="f(x) = x²")
plt.plot(x, x**3, label="f(x) = x³")
plt.plot(x, np.sin(x), label="f(x) = sin(x)")
plt.plot(x[x!=0], 1/x[x!=0], label="f(x) = 1/x")

plt.axhline(0, color='black', linewidth=0.5)
plt.axvline(0, color='black', linewidth=0.5)

plt.title("Multiple Function Graphs")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.grid(True)
plt.legend()
plt.show()

Output:

You can also plot using sympy.

import sympy as sp

# Define symbol
x = sp.symbols('x')

# Define functions
f1 = x
f2 = x**2
f3 = x**3
f4 = sp.sin(x)
f5 = 1/x

# Plot all functions
sp.plot(
    f1, f2, f3, f4, f5,
    (x, -10, 10),
    title="Multiple Function Graphs",
    xlabel="x",
    ylabel="f(x)",
    legend=True,
    show=True
)

Output:

Keywords

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