Identify the function rule from the following xy table:
x y
-3 -5
0 1
5 11
8 16
a. y = x + 2
b. y = 2x + 1
c. y = x - 6
d. y = x + 3
To identify the function rule from the xy table, we need to analyze the relationship between the values of x and y.
Let's start by examining the change in y as x increases.
From x = -3 to x = 0, y increases from -5 to 1. The change in y is 1 - (-5) = 6. So, when x increases by 3 (from -3 to 0), y increases by 6.
From x = 0 to x = 5, y increases from 1 to 11. The change in y is 11 - 1 = 10. So, when x increases by 5 (from 0 to 5), y increases by 10.
From x = 5 to x = 8, y increases from 11 to 16. The change in y is 16 - 11 = 5. So, when x increases by 3 (from 5 to 8), y increases by 5.
By analyzing these changes, we can see that there's a constant rate of change between x and y. For every change of 3 in x, y increases by a constant amount.
Comparing the given function rules:
a. y = x + 2
b. y = 2x + 1
c. y = x - 6
d. y = x + 3
We can eliminate option (a) because the rate of change between x and y is not 1.
Next, let's evaluate option (b), y = 2x + 1.
Using the given xy table:
For x = -3: y = 2(-3) + 1 = -6 + 1 = -5 (matches the table)
For x = 0: y = 2(0) + 1 = 0 + 1 = 1 (matches the table)
For x = 5: y = 2(5) + 1 = 10 + 1 = 11 (matches the table)
For x = 8: y = 2(8) + 1 = 16 + 1 = 17 (doesn't match the table)
Option (b) does not provide the correct output for x = 8, so it is not the correct function rule.
Next, let's evaluate (c), y = x - 6.
Using the given xy table:
For x = -3: y = (-3) - 6 = -9 (doesn't match the table)
For x = 0: y = 0 - 6 = -6 (doesn't match the table)
For x = 5: y = 5 - 6 = -1 (doesn't match the table)
For x = 8: y = 8 - 6 = 2 (doesn't match the table)
None of the values match the table, so option (c) is not the correct function rule.
Lastly, let's evaluate (d), y = x + 3.
Using the given xy table:
For x = -3: y = (-3) + 3 = 0 (doesn't match the table)
For x = 0: y = 0 + 3 = 3 (doesn't match the table)
For x = 5: y = 5 + 3 = 8 (doesn't match the table)
For x = 8: y = 8 + 3 = 11 (matches the table)
Option (d) correctly provides the output for x = 8, which matches the table. Therefore, the correct function rule for the given xy table is:
d. y = x + 3
To identify the function rule from the given xy table, we can examine the relationship between the x and y values. Let's calculate the differences between the pairs:
x y difference (y - x)
-3 -5 -5 - (-3) = -2
0 1 1 - 0 = 1
5 11 11 - 5 = 6
8 16 16 - 8 = 8
Looking at the differences, we can see that the value of y is always greater than the value of x by a constant amount. So, the function rule can be expressed as y = x + (constant).
Comparing this pattern with the given options, the option that matches the pattern is:
a. y = x + 2
Therefore, the function rule for the given xy table is y = x + 2.