Question

Please Help! 30 POINTS! Given the following three points, find by hand the quadratic function they represent. (−1,−8), (0,−1),(1,2) A. f(x)=−3x2+4x−1 B. f(x)=−2x2+5x−1 C. f(x)=−3x2+10x−1 D. f(x)=−5x2+8x−1 Determine if the following set of ordered pairs represents a quadratic function. Explain. (5, 7), (7, 11), (9, 14), (11, 18) A. The y-values go up by the square of the x-value (22=4). Therefore, the ordered pairs represent a quadratic equation. B. The y-values go up by the square of the x-value (22=4). Therefore, the ordered pairs do not represent a quadratic equation. C. Since the differences between the x-values is 2 and the differences between the y-values is 4, that means that the differences between the differences of the y-values are all zero. Therefore, the ordered pairs represent a quadratic equation. D. Since the differences between the differences of the y-values is not consistent, the ordered pairs do not represent a quadratic equation.

Answer 1

(1) B

(2) D

Explanation:

(1)

Let the quadratic function be:

`y = ax^(2) + bx + c`

For the point, (0,-1),

`y = ax^(2) + bx + c`

`-1=(a*0)+(b*0}+c\\-1=c\\c=-1`

Then the equation is:

`y = ax^(2) + bx -1`

For the point (-1, -8) ,

`y = ax^(2) + bx -1`

`-8=(a* (-1)^(2))+(b* -1)-1\\-8=a-b-1\\a-b=-7...(i)`

For the point (1, 2) ,

`y = ax^(2) + bx -1`

`2=(a* (1)^(2))+(b* 1)-1\\2=a+b-1\\a+b=3...(ii)`

Add the two equations and solve for a as follows:

`a-b=-7\\a+b=3\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\2a = -4\\a = -2`

Substitute a = -2 in (i) and solve for b as follows:

`a-b=-7\\-2-b=-7\\b=5`

Thus, the quadratic function is:

`f(x)=-2x^(2)+5x-1`

The correct option is (b).

(2)

The ordered pairs are:

(5, 7), (7, 11), (9, 14), (11, 18)

Represent them in an XY table as follows:

X : 5 | 7 | 9 | 11

Y : 7 | 11 | 14 | 18

Compute the difference between the Y values as follows:

Diff = 11 - 7 = 4

Diff = 14 - 11 = 3

Diff = 18 - 14 = 4

Now compute the difference between the Diff values:

d = 3 - 4 = -1

d = 4 - 3 = 1

Since the differences between the differences of the y-values is not consistent, the ordered pairs do not represent a quadratic equation.

The correct option is D.