The graphs of g(x) = 3x² and f(x) = 3x represent different types of functions that exhibit distinct behaviors and shapes.
Firstly, g(x) = 3x² is a quadratic function. Its graph is a parabola that opens upwards. The vertex of this parabola is at the origin (0, 0), and as x moves away from 0 in either direction, g(x) increases rapidly because of the squaring effect. The shape of this graph implies that for negative values of x, g(x) is still positive, and it has a minimum point at (0, 0).
On the other hand, f(x) = 3x is a linear function. The graph of f(x) is a straight line that passes through the origin and has a slope of 3. This means that for every unit increase in x, f(x) increases by 3 units. The line goes upward indefinitely as x increases, and it crosses the x-axis at (0, 0) as well.
In summary, here’s how the two graphs compare:
- Shape: g(x) is a parabola, while f(x) is a straight line.
- Behavior: g(x) increases rapidly outside the vertex, whereas f(x) increases linearly.
- Vertex/Intercept: Both graphs intersect at the origin, but g(x) has a minimum point there, where it changes from decreasing to increasing.
The differences in their shapes and behaviors provide valuable insights into how these functions behave as their inputs vary.