The graphs of y = 6x² + 4 and y = 6x² represent two parabolas that share a common general shape, but they differ significantly in their vertical positioning and vertex location.
1. Shape and Orientation: Both equations describe parabolas that open upwards since the coefficient of x² is positive (6 in this case). Therefore, the overall shape of the graphs remains the same, which is a U-shape.
2. Vertex Position: The vertex of a parabola described by the equation y = ax² + b is located at the point (0, k) where k is the constant term (b). For the function y = 6x², the vertex is at the origin (0, 0). However, for y = 6x² + 4, the vertex shifts vertically to (0, 4). This means the entire graph of y = 6x² + 4 is positioned 4 units higher than that of y = 6x², reflecting the vertical translation along the y-axis.
3. Y-Intercept: The y-intercept of a graph is found by substituting x with 0 in the equation. For the graph of y = 6x², when x = 0, y equals 0, giving a y-intercept at (0, 0). Meanwhile, in the case of the graph of y = 6x² + 4, substituting x with 0 yields y = 4, resulting in a y-intercept at (0, 4). This further emphasizes how the graph of y = 6x² + 4 is elevated compared to that of y = 6x².
In summary, the main difference between the graphs of y = 6x² + 4 and y = 6x² lies in the vertical shift, where the former graph is shifted upwards by 4 units, while they maintain the same shape and orientation as upward-opening parabolas.