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.(4 points)

Answer 1

To compare the functions \( f(x) = (x + 2)^2 \) and \( g(x) = 2x + 4 \), we can graph them. The function \( f(x) \) is a quadratic function, represented by a parabola that opens upwards with its vertex at the point (-2, 0). In contrast, \( g(x) \) is a linear function, represented by a straight line with a slope of 2 and a y-intercept at (0, 4).

Both functions have positive values for \( x \) greater than or equal to -2. However, they are fundamentally different in shape: \( f(x) \) curves smoothly, while \( g(x) \) has a constant rate of change. Additionally, \( f(x) \) can take on a minimum value but never changes direction, while \( g(x) \) continues infinitely without any curvature. Thus, the key features of the two functions highlight the differences between quadratic and linear relationships.