Question

Using the included equations, compare linear and quadratic graph shapes

f(x)=(x+3)^2 and g(x) =2x+6

Responses:

Linear is an L-shaped curve and Quadratic is an s-shaped curve, where one side goes up and one side goes down.

Linear is a straight, slanted line and Quadratic is a U shaped curve.

Quadratic is a U that is turned on it's side. Linear is a U that is turned on the opposite side.

Answer 1

To compare the shapes of the graphs of the functions \( f(x) = (x + 3)^2 \) (quadratic) and \( g(x) = 2x + 6 \) (linear), let's analyze each function.

1. **Linear Function**:
- The function \( g(x) = 2x + 6 \) is a linear function. It represents a straight line with a slope of 2. This means that for every unit increase in \( x \), \( g(x) \) increases by 2. The graph of a linear function is always a straight line, which can be slanted but does not curve.

2. **Quadratic Function**:
- The function \( f(x) = (x + 3)^2 \) is a quadratic function. Its graph is a parabola that opens upwards. The vertex of this parabola is located at the point (-3, 0). The U-shape of the parabola means that both ends of the graph extend upwards as \( x \) moves away from -3.

Based on this analysis, the most accurate comparison is:

**Linear is a straight, slanted line and Quadratic is a U shaped curve.**

This description correctly captures the essence of how each graph behaves and their respective shapes.