Find a quadratic model for the sequence.
-4, -4, -3, -1, 2
A.y = 0.5x^2 - 0.5x - 4
B.y = 0.5x^2 - 1.5x - 3
C.y = 4.5x^2 - 21.5x+21
D.y = -4.5x^2 + 21.4x - 21
Consider it as the points (1,-4), (2,-4), (3,-3) etc
let the function be ax^2 + bx + c = y
for:
(1,-4) ---> a + b + c = -4 *
(2,-4) ---> 4a + 2b + c = -4 **
(3,-3) ---> 9a + 3b + c = -3 ***
subtract * from **
3a + b = 0 #
subtract ** from ***
5a + b = 1 ##
subtract # from ##
2a = 1
a = 1/2
sub into #
3/2 + b = 0
b = -3/2
sub those two values into * to get
c = -3
so y = (1/2)x^2 - (3/2)x - 3
or
y = .5x^2 - 1.5x - 3 <------ B
or
y = (1/2)(x^2 - 3x - 6)
test it for the last two value of your data
To find a quadratic model for the given sequence, we need to look for a pattern in the data.
First, let's write down the input and output values in a table:
x | y
--------------
-2 | -4
-1 | -4
0 | -3
1 | -1
2 | 2
Now, let's analyze the pattern in the output values. Looking at the table, we can see that the values do not have a linear or exponential relationship. However, we can observe that the sequence is increasing and decreasing, and the change in the output values seems to be related to the square of the input values.
Let's test the options given:
A. y = 0.5x^2 - 0.5x - 4
B. y = 0.5x^2 - 1.5x - 3
C. y = 4.5x^2 - 21.5x + 21
D. y = -4.5x^2 + 21.4x - 21
By substituting the x-values into each equation, we can compare the result with the corresponding y-values in the table.
For option A:
x = -2, y = 0.5(-2)^2 - 0.5(-2) - 4 = 3 - (-1) - 4 = 4 - 4 = 0 not equal to -4
For option B:
x = -2, y = 0.5(-1)^2 - 1.5(-1) - 3 = 0.5 - (-1.5) - 3 = 0.5 + 1.5 - 3 = 1.5 - 3 = -1.5 not equal to -4
For option C:
x = -2, y = 4.5(-2)^2 - 21.5(-2) + 21 = 4.5 * 4 + 43 + 21 = 18 + 43 + 21 = 62 + 21 = 83 not equal to -4
For option D:
x = -2, y = -4.5(-2)^2 + 21.4(-2) - 21 = -4.5 * 4 - 42.8 - 21 = -18 - 42.8 - 21 = -71.8 - 21 = -92.8 not equal to -4
None of the options match the sequence. It appears that none of the given options provide the correct quadratic model for the given sequence. Therefore, none of the options A, B, C, or D is the correct answer.
It is possible that there was an error in recording the sequence or in providing the answer options. Double-checking the data or seeking clarification would be recommended to find the correct quadratic model.