Sketch a graph of y=1/2int(x-2)+1. State the domain range and intercepts. Hint: Lightly graph y=int(x) and then apply three transformations.

To sketch the graph of y = 1/2int(x-2) + 1 and determine its domain, range, and intercepts, we will follow these steps:

1. Start with the graph of y = int(x) (the greatest integer function). This graph consists of a series of "steps" where the output of the function jumps to the nearest integer as the input increases. The y-values of this graph are all integers.

2. Apply the first transformation by evaluating y = int(x-2). This shifts the graph 2 units to the right. For example, when x = 2, we now look at int(0) = 0, so the point (2,0) is on the graph.

3. Apply the second transformation by multiplying the entire expression by 1/2. This vertically compresses the graph. For example, with the previous point (2,0), we now have (2,0 * 1/2) = (2,0).

4. Apply the final transformation by shifting the graph up 1 unit. For example, if we previously had the point (2,0), we now have (2,0 + 1) = (2,1).

5. Now, we have the graph of y = 1/2int(x-2) + 1, which is a transformed version of the original greatest integer function.

The domain of the function is all real numbers since there are no restrictions on the x-values.

The range of the function is all real numbers between the indefinite step points. In this case, the range will be all real numbers between 1 and 2 (inclusive), since we have shifted the graph up by 1 unit.

To find the x-intercepts, we need to determine where the graph crosses the x-axis. In the case of the greatest integer function, it only crosses the x-axis at integer x-values. With the transformations, the x-intercepts will occur where int(x-2) = 0. Solving this equation, we get x-2 = 0, which results in x = 2. Therefore, the x-intercept is at x = 2.

To find the y-intercept, substitute x = 0 into the equation: y = 1/2int((0)-2) + 1. Simplifying further, we have y = 1/2int(-2) + 1. Since int(-2) = -2, the y-intercept is y = 1/2(-2) + 1 = -1 + 1 = 0. Hence, the y-intercept is at y = 0.

Combining all this information, we can sketch the graph of y = 1/2int(x-2) + 1 based on the transformations applied to the original greatest integer function. Remember to label the x and y-intercepts and indicate the range and domain of the function.