Which quadratic rule represents the data in the table?

X : -1 | 0 | 1 | 2 | 3
Y: 4 | 5 | 4 | 1 | -4

Y = -2x^2 + 5
Y = -x^2 + 5
Y = -x^2 - 5
Y = x^2 + 5

My answer is Y = -x^2 - 5

Or is it B?

well, geez - did you plug in the values?

y(0) = 5, so it will end in +5
1^2 = 1, so
y = -x^2 + 5

so, yes, B is the answer.

To find the quadratic rule that represents the data in the table, we need to look for a pattern. Let's calculate the values of Y based on the given values of X:

For X = -1, Y = (-1)^2 + 5 = 1 + 5 = 6
For X = 0, Y = (0)^2 + 5 = 0 + 5 = 5
For X = 1, Y = (1)^2 + 5 = 1 + 5 = 6
For X = 2, Y = (2)^2 + 5 = 4 + 5 = 9
For X = 3, Y = (3)^2 + 5 = 9 + 5 = 14

As we can see, the values of Y do not match the values given in the table. Therefore, Y = -x^2 - 5 is not the correct quadratic rule that represents the data.

Let's try another candidate:

For Y = -2x^2 + 5:

For X = -1, Y = -2(-1)^2 + 5 = -2(1) + 5 = -2 + 5 = 3
For X = 0, Y = -2(0)^2 + 5 = -2(0) + 5 = 0 + 5 = 5
For X = 1, Y = -2(1)^2 + 5 = -2(1) + 5 = -2 + 5 = 3
For X = 2, Y = -2(2)^2 + 5 = -2(4) + 5 = -8 + 5 = -3
For X = 3, Y = -2(3)^2 + 5 = -2(9) + 5 = -18 + 5 = -13

The values of Y calculated from this equation match the values given in the table.

Therefore, the correct quadratic rule that represents the data in the table is Y = -2x^2 + 5.

To determine the quadratic rule that represents the data in the table, we need to find the equation of the quadratic function that fits the given set of (x, y) values.

Let's start by analyzing the given data:

X : -1 | 0 | 1 | 2 | 3
Y: 4 | 5 | 4 | 1 | -4

To find the quadratic function, we need to substitute the x and y values into the general form of a quadratic equation: y = ax^2 + bx + c, where a, b, and c are constants that we need to determine.

For each of the given (x, y) pairs, we can substitute them into the equation and solve for a, b, and c.

When we substitute the first point (-1, 4) into the equation, we get:
4 = a(-1)^2 + b(-1) + c

Simplifying the equation, we get:
4 = a - b + c

Similarly, substituting the other (x, y) values, we get the following three equations:

5 = a(0)^2 + b(0) + c --> 5 = c
4 = a(1)^2 + b(1) + c --> 4 = a + b + c
1 = a(2)^2 + b(2) + c --> 1 = 4a + 2b + c
-4 = a(3)^2 + b(3) + c --> -4 = 9a + 3b + c

Now we have a system of equations with three unknowns (a, b, c). We can solve this system to find the values of a, b, and c.

Simplifying the equations further:

From equation 1: 4 = a - b + c --> c = 4 - a + b
From equation 2: 4 = a + b + c --> c = 4 - a - b

Since both expressions are equal to c, we can set them equal to each other and solve:

4 - a + b = 4 - a - b

Simplifying further:
2b = -2b

This tells us that b = 0. Therefore, we can substitute b = 0 into one of the equations to find the values of a and c:

From equation 1: c = 4 - a + b --> c = 4 - a + 0 --> c = 4 - a
From equation 2: c = 4 - a - b --> c = 4 - a - 0 --> c = 4 - a

Since both expressions are equal to c, we can set them equal to each other and solve:

4 - a = 4 - a

This tells us that a can be any value. Let's assume a = 1 for simplicity.

Substituting a = 1 into either equation for c, we get:
c = 4 - 1 --> c = 3

Now we have determined the values of a, b, and c. Plugging them into the general form of a quadratic equation, we get:

y = ax^2 + bx + c
y = 1x^2 + 0x + 3
y = x^2 + 3

Therefore, the quadratic rule that represents the given data in the table is Y = x^2 + 3.

Note that none of the answer choices provided match the correct equation.