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

Question

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

Answer 1

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!