Graph and compare the shapes and key features of the functions

Question

f (x) = (x + 2)? and g (x) = 2x + 4. In 3 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. Graph of \( f(x) = (x + 2)^2 \): This is a quadratic function that opens upwards, with its vertex at the point (-2, 0). As \( x \) moves away from -2 in either direction, the values of \( f(x) \) rise steeply.

2. Graph of \( g(x) = 2x + 4 \): This is a linear function with a slope of 2, indicating that it rises steadily as \( x \) increases. The y-intercept is at the point (0, 4), meaning the line crosses the y-axis there.

In summary, both functions exhibit increasing behavior, but they have different shapes: \( f(x) \) is a parabola (curved) whereas \( g(x) \) is a straight line. The key features are that \( f(x) \) has a minimum point at its vertex and is symmetric, while \( g(x) \) has no turning points and is linear. Additionally, the growth rates differ; \( f(x) \) accelerates as \( x \) moves away from -2, while \( g(x) \) increases at a constant rate.