To find the function rule that represents the data in the table, we need to look for a pattern. We notice that as x increases by 1, y increases by a different amount each time. Specifically, y decreases by 1 when x increases from -1 to 0, then increases by 1 when x increases from 0 to 1, then increases by 3 when x increases from 1 to 2, and finally increases by 5 when x increases from 2 to 3. This means that the relationship between x and y is not linear, but rather it involves some kind of exponential growth or decay.
To determine the specific form of the function rule, we can use the method of finite differences. We start by calculating the first differences, which are the differences between consecutive y values:
5 - 6 = -1
6 - 5 = 1
9 - 6 = 3
14 - 9 = 5
Next, we calculate the second differences, which are the differences between consecutive first differences:
1 - (-1) = 2
3 - 1 = 2
5 - 3 = 2
We notice that all the second differences are equal to 2, which means that the function rule is quadratic. Specifically, it has the form y = ax^2 + bx + c, where a, b, and c are constants that we need to determine. To do this, we can use the three data points where x = 0, x = 1, and x = 2, since they form a unique quadratic curve.
When x = 0, we have y = 5, so we can substitute these values to get:
5 = a(0)^2 + b(0) + c
5 = c
When x = 1, we have y = 6, so we can substitute these values to get:
6 = a(1)^2 + b(1) + 5
6 = a + b + 5
a + b = 1
When x = 2, we have y = 9, so we can substitute these values to get:
9 = a(2)^2 + b(2) + 5
9 = 4a + 2b + 5
4a + 2b = 4
Now we have two equations in two variables (a and b), which we can solve by substitution or elimination. Let's use elimination:
a + b = 1
4a + 2b = 4
Multiply the first equation by -2:
-2a - 2b = -2
Add this equation to the second equation:
2a = 2
a = 1
Substitute a = 1 into the first equation:
1 + b = 1
b = 0
Therefore, the function rule that represents the data in the table is:
y = x^2 + 5
Checking our work, we can verify that this function produces the correct values for all five data points:
When x = -1, y = 6 (-1)^2 + 5 = 6
When x = 0, y = 5 (0)^2 + 5 = 5
When x = 1, y = 6 (1)^2 + 5 = 6
When x = 2, y = 9 (2)^2 + 5 = 9
When x = 3, y = 14 (3)^2 + 5 = 14
Therefore, the answer is:
y = x^2 + 5
Which function rule represents the data in the table?
X -1, 0, 1, 2, 3
Y 6, 5, 6, 9, 14
y = -2x>2 + 5
y = -x>2 + 5
y = x>2 - 5
y = x>2 + 5
Show your work.
1 answer