Question

What are the important aspects to look for in a graph in order to accurately identify the function family the graph is from? What are the important aspects to look for in a table of values? Which is easier: determining a function family based on a graph, or based on a table of values? Why do you think so?

Answer 1

When looking at a graph, important aspects to consider are the shape, key points, and behavior. When looking at a table, important aspects to consider are patterns, consistent rate of change, and special values. Identifying the function family based on a graph can be easier because it provides visual insights, but both methods require careful analysis.

Step-by-step explanation:

When looking at a graph to identify the function family it is representing, there are a few important aspects to consider:

1. Shape: The overall shape of the graph can give clues about the type of function. For example, a linear function will have a straight line, while a quadratic function will have a curve.
2. Key Points: Look for any specific points on the graph that align with known function families. For example, a graph with a single point of intersection with the y-axis may indicate an exponential function.
3. Behavior: Pay attention to how the graph behaves as it extends beyond the visible area. This can provide insight into whether the function is increasing or decreasing, and whether it has any asymptotes or end behavior.

When looking at a table of values, the following aspects are important:

1. Patterns: Look for any patterns or relationships between the input and output values in the table. This can help identify the function family. For example, if the output values are all doubles of the input values, it may indicate a linear function.
2. Consistent Rate of Change: Determine if the rate of change between values is consistent. For example, if the difference between consecutive output values is always the same, it may indicate a linear function.
3. Special Values: Check for any specific input or output values that align with known function families. For example, if the input value is always 0 and the output values are all positive, it may be an absolute value function.

Determining a function family based on a graph or a table of values can both be challenging. However, some may find it easier to identify the function family based on a graph, as it provides a visual representation that can offer more immediate insights, such as the overall shape and behavior. However, both methods require careful analysis and consideration of the different aspects mentioned above.