Consider the linear function with equation y = 3x + 5.
Sketch the graph of the linear function on the grid.
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Here is a picture of the graph, and this is how I did it.
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Then it says.. Restrict the domain to the set of natural numbers. Mark the dots points on the graph which represent the function on the restricted domain.
Write the first 5 elements of the range in numerical order.
Show that the elements of the range form an arithmetic sequence and state the common difference.
I don't understand this at all.
To restrict the domain to the set of natural numbers, we need to consider only the x-values that are natural numbers (positive integers). In the given linear function y = 3x + 5, the x-values for the graph would be all the natural numbers.
To mark the points on the graph that represent the function on the restricted domain, you would need to find the corresponding y-values for each x-value. Start with x = 1 and substitute it into the equation: y = 3(1) + 5 = 8. Thus, the first point on the graph would be (1, 8).
Continue this process for the next four natural numbers:
For x = 2: y = 3(2) + 5 = 11. So the second point on the graph would be (2, 11).
For x = 3: y = 3(3) + 5 = 14. The third point would be (3, 14).
For x = 4: y = 3(4) + 5 = 17. The fourth point would be (4, 17).
For x = 5: y = 3(5) + 5 = 20. The fifth point would be (5, 20).
Now that we have the points on the graph, we can determine the first 5 elements of the range. The range refers to the set of all y-values corresponding to the given x-values. In this case, the first 5 y-values are 8, 11, 14, 17, and 20.
To show that these values form an arithmetic sequence, we need to check if the difference between consecutive terms is constant. Taking the differences between consecutive terms:
Second term - First term: 11 - 8 = 3
Third term - Second term: 14 - 11 = 3
Fourth term - Third term: 17 - 14 = 3
Fifth term - Fourth term: 20 - 17 = 3
As we can see, the difference between each pair of consecutive terms is 3, indicating that the elements of the range form an arithmetic sequence with a common difference of 3.