X | Y
-1| -4
0| -5
1| -4
2 | -1
3 | -4
A. Y= -2x^2+5
B. Y= -x^2+5
C. Y= x^2-5
D. Y= x^2+5
B. Y= -x^2+5
To determine the equation that represents the given data table, we need to find a pattern or relationship between the values of x and y.
Looking at the table, we can notice that the value of y changes as the value of x changes. However, the values of x and y do not have a linear relationship since the change in y is not constant for each unit change in x.
The values of y seem to be decreasing as x increases, and when x=0, y has its largest value. This suggests a quadratic relationship between x and y.
To find the equation, we need to substitute the x and y values from one of the data points into the general equation of a quadratic function and solve for the unknown coefficients.
Let's choose the point (0, -5) as it is the readily evident and lies on the x-axis.
Substituting the values into the general equation:
-5 = a(0)^2 + b(0) + c
Simplifying this equation:
-5 = 0 + 0 + c
-5 = c
Therefore, we have found the coefficient c, which is -5.
Now, we need to find the values of coefficients a and b by substituting the points (-1, -4) and (2, -1) into the equation.
Using the point (-1, -4):
-4 = a(-1)^2 + b(-1) + (-5)
-4 = a + (-b) - 5
-4 = a - b - 5
Using the point (2, -1):
-1 = a(2)^2 + b(2) + (-5)
-1 = 4a + 2b - 5
4 = 4a + 2b
We now have a system of two equations:
a - b - 5 = -4
4a + 2b = 4
To solve this system, we can eliminate one of the variables by multiplying one of the equations by a suitable factor so that the coefficients of one of the variables will cancel out.
Multiplying the first equation by 2:
2a - 2b - 10 = -8
Adding this new equation to the second equation:
2a - 2b - 10 + 4a + 2b = -8 + 4
6a - 10 = -4
Simplifying the equation:
6a = -4 + 10
6a = 6
a = 1
Substituting the value of a into the first equation:
1 - b - 5 = -4
-4 - b = -4
-b = 0
b = 0
Therefore, the values of the coefficients are:
a = 1
b = 0
c = -5
We can now substitute these values into the general equation for a quadratic function to obtain the final equation.
Y = ax^2 + bx + c
Y = 1x^2 + 0x - 5
Y = x^2 - 5
Thus, the equation that represents the given data table is C. Y = x^2 - 5.