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avanturin [10]
2 years ago
14

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.
Mathematics
1 answer:
ycow [4]2 years ago
8 0

Answer:

(1) B

(2) D

Step-by-step explanation:

(1)

Let the quadratic function be:

y = ax^{2} + bx + c

For the point, (0,-1),

y = ax^{2} + bx + c

-1=(a\times0)+(b\times0}+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\times (-1)^{2})+(b\times -1)-1\\-8=a-b-1\\a-b=-7...(i)

For the point (1, 2) ,

y = ax^{2} + bx -1

2=(a\times (1)^{2})+(b\times 1)-1\\2=a+b-1\\a+b=3...(ii)

Add the two equations and solve for <em>a</em> as follows:

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

Substitute <em>a</em> = -2 in (i) and solve for <em>b</em> 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 <em>Y</em> 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.

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