Consider the table of values given below. The quadratic equation that describe the relationship between y and x is?

To find the quadratic equation that describes the relationship between y and x, we can use the method of interpolation.

Let's first write down the given table of values:

x | y
--|--
1 | 3
2 | 8
3 | 15
4 | 24
5 | 35

Now, let's try to find the pattern in the values of y as x increases:

From x=1 to x=2, y increases by 5.
From x=2 to x=3, y increases by 7.
From x=3 to x=4, y increases by 9.
From x=4 to x=5, y increases by 11.

The pattern in the increase of y seems to be increasing by consecutive odd numbers. This suggests that the equation may be of the form y = ax^2 + bx + c, where a, b, and c are constants to be determined.

Let's substitute the values of x and y from the table into the equation:

For x=1, y=3:
3 = a(1)^2 + b(1) + c
3 = a + b + c

For x=2, y=8:
8 = a(2)^2 + b(2) + c
8 = 4a + 2b + c

For x=3, y=15:
15 = a(3)^2 + b(3) + c
15 = 9a + 3b + c

Now we have a system of three equations with three unknowns (a, b, c). By solving this system of equations, we can find the values of a, b, and c, and thus determine the quadratic equation that describes the relationship between y and x.

| x | y |
|---|---|
| -2 | 12 |
| -1 | 5 |
| 0 | 0 |
| 1 | 5 |
| 2 | 12 |

The quadratic equation that describe the relationship between \(y\) and \(x\) is:

$$y=2x^2-2x$$

Explanation:

1. Find the differences between the consecutive \(y\)-values:

$$12-5=7$$

$$5-0=5$$

$$0-5=-5$$

$$5-12=-7$$

2. Find the differences between the consecutive differences:

$$7-5=2$$

$$5-(-5)=10$$

$$-5- (-7)=-2$$

3. We can observe that the second differences are constant and equal to 10. This indicates that the equation is a quadratic equation.

4. Since the second differences are constant, the equation can be written in the form:

$$y=ax^2+bx+c$$

5. We can use the given table of values to solve for \(a\), \(b\), and \(c\):

When \(x=-2\), \(y=12\):

$$12=a(-2)^2+b(-2)+c$$

$$12=4a-2b+c$$

When \(x=-1\), \(y=5\):

$$5=a(-1)^2+b(-1)+c$$

$$5=a-b+c$$

When \(x=0\), \(y=0\):

$$0=a(0)^2+b(0)+c$$

$$0=c$$

6. Substituting \(c=0\) into the first two equations, we get:

$$12=4a-2b$$

$$5=a-b$$

7. Solving these equations simultaneously, we get:

$$a=2$$

$$b=2$$

8. Therefore, the quadratic equation that describes the relationship between \(y\) and \(x\) is:

$$y=2x^2+2x+0$$

$$y=2x^2+2x$$