which quadratic rule represents the data in the table?
X |-1 | 0 | 1 | 2 | 3 |
Y | 4 | 5 | 4 | 1 | -4|
A: y= -2x^2+5
B: y= -x^2+5
C: y= x^2-5
D: y= x^2+5
The correct answer is B: y= -x^2+5.
To determine the quadratic rule, we need to fit a quadratic equation of the form y = ax^2 + bx + c to the given data points.
Substituting the x and y values from the table into the equation, we get the following system of equations:
4 = a(-1)^2 + b(-1) + c
5 = a(0)^2 + b(0) + c
4 = a(1)^2 + b(1) + c
1 = a(2)^2 + b(2) + c
-4 = a(3)^2 + b(3) + c
Simplifying these equations gives:
a - b + c = 4
c = 5
a + b + c = 4
4a + 2b + c = 1
9a + 3b + c = -4
Solving this system of equations (by substitution or elimination) gives a = -1, b = 0, and c = 5.
Therefore, the quadratic rule that represents the data in the table is y = -x^2 + 5.
To determine the quadratic rule that represents the data in the table, we can start by comparing the values of x and y.
Given:
X: -1, 0, 1, 2, 3
Y: 4, 5, 4, 1, -4
We can see that the values of y decrease and then go back up, indicating a parabolic shape.
Let's analyze the options one by one:
A: y = -2x^2 + 5
When we substitute the x values into this equation, we get:
For x = -1, y = -2(-1)^2 + 5 = -2 + 5 = 3
For x = 0, y = -2(0)^2 + 5 = 0 + 5 = 5
For x = 1, y = -2(1)^2 + 5 = -2 + 5 = 3
For x = 2, y = -2(2)^2 + 5 = -8 + 5 = -3
For x = 3, y = -2(3)^2 + 5 = -18 + 5 = -13
As we can see, this equation does not match the y values in the table.
B: y = -x^2 + 5
For x = -1, y = -(-1)^2 + 5 = -1 + 5 = 4
For x = 0, y = -(0)^2 + 5 = 0 + 5 = 5
For x = 1, y = -(1)^2 + 5 = -1 + 5 = 4
For x = 2, y = -(2)^2 + 5 = -4 + 5 = 1
For x = 3, y = -(3)^2 + 5 = -9 + 5 = -4
This equation matches the y values in the table.
Therefore, the quadratic rule that represents the data in the table is:
B: y = -x^2 + 5