Graph y=x+3, y=x squared +3 and [x]+3

What are the similarities and differences.

To analyze the similarities and differences between the equations y = x+3, y = x^2+3, and [x]+3, we can consider their graphs and their algebraic properties.

1. y = x + 3:
This equation represents a straight line with a slope of 1 and a y-intercept of 3. Its graph is a diagonal line that intersects the y-axis at the point (0, 3).

2. y = x^2 + 3:
This equation represents a parabola that opens upward. It has a vertex at the point (-3, 3) and is symmetric with respect to the vertical line x = -3. The vertex is the minimum point of the parabola, which means the y-value is greater than or equal to 3 for all x-values.

3. [x] + 3:
The notation [x] represents the greatest integer function, also known as the floor function. It rounds down any decimal value to the nearest integer. The graph of this equation consists of staircase-like segments. It starts at y = 3 when x is an integer, and increases by 1 when x crosses an integer value.

Similarities:
- All three equations have a constant term of +3, meaning they all have a y-intercept at (0, 3).
- They all have the same y-value when x is an integer.

Differences:
- The equation y = x+3 is a straight line, while y = x^2+3 is a parabola, and [x]+3 is a staircase-like graph.
- The graph of y = x^2+3 has a vertex and opens upward, while the other two graphs do not have a vertex or specific direction.
- The graph of y = x^2+3 is continuous, while the graph of [x]+3 is composed of discrete segments.

In summary, the similarities between the equations are the y-intercept and the y-values when x is an integer. The differences lie in the shapes of their graphs and their algebraic properties.