The graphs of the equations y = 8x² + 1 and y = 8x² are both parabolas that open upwards because their leading coefficient, 8, is positive. However, there are key differences between them that affect their positioning on the Cartesian plane.
1. Vertical Shift: The main difference between the two graphs is the vertical shift. The equation y = 8x² + 1 is shifted upwards by 1 unit. This means that every point on the graph of y = 8x² will be moved up by 1 unit to create the graph of y = 8x² + 1. Consequently, the vertex of the first parabola is at (0, 0), while the vertex of the second parabola is at (0, 1).
2. Y-Intercepts: The y-intercept of the graph of y = 8x² is at (0, 0), meaning that it crosses the y-axis at the origin. In contrast, the y-intercept of y = 8x² + 1 is at (0, 1), where the graph crosses the y-axis one unit higher than the first graph.
3. Overall Appearance: Visually, while both parabolas share the same width and general shape due to their identical leading coefficient, the graph of y = 8x² + 1 appears to be floating above the graph of y = 8x² throughout the plane. They never intersect and maintain a consistent vertical distance from each other.
Overall, while the shapes of the two graphs are the same, the vertical shift of the second graph causes it to be repositioned one unit higher on the Cartesian plane compared to the first graph.