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2. Graph and compare the shapes and key features of the functions f(x) = (x+2)2

and g(x) = 2x+4
. In 3-5 sentences, explain how the two functions are similar and how they are different.
Show the math

1 answer

To compare the functions \( f(x) = (x + 2)^2 \) and \( g(x) = 2x + 4 \), let's first determine their shapes and key features.

  1. Function \( f(x) = (x + 2)^2 \):

    • This is a quadratic function (a parabola) with the standard form \( f(x) = a(x - h)^2 + k \).
    • It opens upwards because the coefficient of the \( x^2 \) term is positive.
    • The vertex of this parabola can be found at \( (-2, 0) \).
    • The axis of symmetry is \( x = -2 \).
    • The y-intercept can be found by setting \( x = 0 \): \( f(0) = (0 + 2)^2 = 4 \).

  2. Function \( g(x) = 2x + 4 \):

    • This is a linear function with a slope of 2 and a y-intercept of 4.
    • The equation can be rewritten in slope-intercept form as \( y = mx + b \) (where \( m = 2 \) and \( b = 4 \)).
    • The graph of this function will be a straight line.

Comparison:

  • Similarities: Both functions can be evaluated at any real number \( x \) and have y-intercepts, though they are different. Both functions can be transformed into their respective graph forms.
  • Differences: The function \( f(x) \) is nonlinear (quadratic) and has a unique minimum point (the vertex), while \( g(x) \) is linear with no curvature, continually increasing without bound. Additionally, \( f(x) \) is symmetric around its vertex, whereas \( g(x) \) has no symmetry.

Graphs: To visualize the functions, you can plot them:

  • For \( f(x) = (x + 2)^2 \):

    • The vertex at (-2, 0)
    • The y-intercept at (0, 4)

  • For \( g(x) = 2x + 4 \):

    • The y-intercept at (0, 4)
    • A slope of 2 indicates a steep increase.

Plotting the Functions

import matplotlib.pyplot as plt
import numpy as np

# Define the range of x values
x = np.linspace(-6, 2, 400)

# Define the functions
f = (x + 2)**2
g = 2*x + 4

# Create the plot
plt.figure(figsize=(10, 6))
plt.plot(x, f, label='f(x) = (x + 2)^2', color='blue')
plt.plot(x, g, label='g(x) = 2x + 4', color='red')

# Add key features
plt.scatter([-2, 0], [0, 4], color='blue')  # f(x) features
plt.scatter([0], [4], color='red')          # g(x) feature

# Annotations
plt.text(-2, 0, 'Vertex (-2, 0)', fontsize=10, verticalalignment='bottom')
plt.text(0, 4, 'Y-Intercept (0, 4)', fontsize=10, verticalalignment='bottom')
plt.axhline(0, color='black',linewidth=0.5, ls='--')
plt.axvline(0, color='black',linewidth=0.5, ls='--')

# Set title and labels
plt.title('Comparison of f(x) and g(x)')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid()
plt.ylim(-5, 10)
plt.xlim(-6, 2)

# Show the plot
plt.show()

This graph illustrates the differences and similarities effectively!