Final answer:
To graph a piecewise-defined function and construct a histogram, identify intervals, plot relevant points, connect points with appropriate lines, and then determine the continuity or properties of data distribution. Use a ruler and pencil to create an accurate graph, and scale the axes properly.
Step-by-step explanation:
Constructing a Histogram and Analyzing Continuity
To graph the piecewise-defined function and construct a histogram, follow these steps:
- Identify intervals: Divide the dataset into five to six intervals of equal range if the data is continuous, or based on natural breaks in the dataset.
- Plot points for each piece: For the histogram, plot the frequency of data points within each interval. For the piecewise function, plot points that define each piece of the function on their respective intervals.
- Connect points: In a histogram, connect the top of the bars to represent the frequency of data within each interval. For the piecewise function, connect the points with straight lines or curves, depending on the function's formula within each interval.
- Determine continuity: Analyze the piecewise function to see if there is a smooth transition from one piece to another. A function is continuous if there are no breaks, jumps, or holes at the points where the pieces meet.
Use a ruler and pencil to ensure straight lines and clear demarcations of the intervals. Scale the axes so that all data points can be accurately plotted and the entire range of the dataset is represented. For the piecewise function, check whether the limit of the function as it approaches the boundary point from each side is the same. This will help in determining the continuity.
Finally, analyze the graph. If it's a histogram, note the skewness or symmetry of data distribution. For the piecewise function, observe if the graph is continuous throughout its domain or if there are discontinuities.