The function f(x) = x² represents a parabolic graph that opens upwards. When comparing the graphs of f(x) = x³ and f(x) = x², we notice distinct characteristics due to their different degrees and forms.
1. Shape: The graph of f(x) = x² is a smooth, U-shaped curve, known as a parabola. This contrasts sharply with f(x) = x³, which has an S-like shape, appearing to rise steeply in both positive and negative directions.
2. Vertex: The vertex of the parabola f(x) = x² is at the origin (0,0), which is also the minimum point on the graph. In contrast, the graph of f(x) = x³ has no minimum or maximum points, as it continues to rise or fall indefinitely.
3. Intercepts: Both functions intersect the x-axis at the origin. However, f(x) = x² only touches the x-axis at (0,0) without crossing it, while f(x) = x³ crosses the axis at the origin.
4. Symmetry: The graph of f(x) = x² is symmetric about the y-axis, meaning that for every point (x,y), there is a matching point (-x,y). In contrast, f(x) = x³ is symmetric about the origin, leading to the corresponding point (-x,-y) for every point (x,y).
5. Domain and Range: The domain of both functions is all real numbers. However, the range of f(x) = x² is y ≥ 0, as the parabola never goes below the x-axis. Conversely, the range of f(x) = x³ is all real numbers, reflecting its ability to take on both positive and negative values.
Overall, while both functions have their unique properties, the parabolic nature of f(x) = x² stands in contrast to the cubic behavior of f(x) = x³. Understanding these differences helps appreciate the variety of shapes and behaviors that polynomial functions can exhibit.