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.

Answer 1

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

1. Shape: The function \( f(x) \) is a quadratic function, which means its graph is a parabola that opens upwards, with its vertex at the point (-2, 0). On the other hand, \( g(x) \) is a linear function, producing a straight line with a slope of 2 and a y-intercept of 4.

2. Key Features: The parabola of \( f(x) \) has a minimum point and is symmetric around the vertical line \( x = -2 \). Conversely, the linear function \( g(x) \) continuously increases and has no maximum or minimum points.

Comparison: Both functions pass through the plane but behave differently: \( f(x) \) curves with a vertex, while \( g(x) \) is a straight line. They intersect at specific points; however, one represents a quadratic relationship and the other a linear relationship, showcasing their differing rates of change.