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.(4 points)

Answer 1

To graph the functions \( f(x) = (x + 2)^2 \) and \( g(x) = 2x + 4 \):

1. Shape: The function \( f(x) = (x + 2)^2 \) is a parabola that opens upwards, with its vertex at the point (-2, 0). The function \( g(x) = 2x + 4 \) is a linear function represented by a straight line with a slope of 2 and a y-intercept at (0, 4).

2. Key Features: The parabola \( f(x) \) has a minimum point at (-2, 0) and is symmetric around the vertical line \( x = -2 \). The linear function \( g(x) \) has no curvature and continues infinitely in both directions with a constant increase, crossing the y-axis at (0, 4).

Comparison: Both functions can take on positive and negative values depending on the input \( x \), but they differ significantly in shape and behavior. \( f(x) \) is quadratic and exhibits curvature with a minimum point, while \( g(x) \) is linear with a constant slope and no minimum or maximum. Overall, they showcase different mathematical properties, with \( f(x) \) representing a polynomial function and \( g(x) \) representing a linear function.