Question

The difference between differences of the y-values in a quadratic function is 2. If two ordered pairs in that quadratic function were (2,10) and (3,16), what is the y-value of the ordered pair (4,y)?

Answer 1

To determine the y-value for the ordered pair (4,y) in a quadratic function with given points (2,10) and (3,16), we consider that the second difference between y-values is 2. This information allows us to solve for the coefficients of the quadratic equation, which can then be used to calculate the y-value when x=4.

Step-by-step explanation:

To find the y-value of the ordered pair (4,y) for a quadratic function where the second differences are constant, we can use the given ordered pairs (2,10) and (3,16). The constant second difference implies that the function is in the form y = ax² + bx + c. The difference in y-values between consecutive x-values (difference of differences) would be represented by 2a, which is given as 2. Hence, a = 1.

Now, we already have two points that give us the linear equations:

• 10 = 4a + 2b + c
• 16 = 9a + 3b + c

Substituting a = 1 into these equations, we can solve for b and c. Once we have the complete function, we can then substitute x = 4 to find the corresponding y-value.

Step-by-Step Explanation:

1. The difference between differences (second difference) of 2 implies a = 1.
2. Use the point (2,10) to create the equation 10 = 4(1) + 2b + c.
3. Use the point (3,16) to create the equation 16 = 9(1) + 3b + c.
4. Solve the two simultaneous equations to find b and c.
5. Substitute x = 4 into the derived quadratic equation to find y.