20 POINTS! Please help.! 1) Given the following three points, find by hand the quadratic function they represent. (0,6), (2,16), (3, 33) A. f(x)=4x2−3x+6 B. f(x)=4x2+3x+6 C. f(x)=−4x2−3x+6 D. f(x)=−4x2+21x+6 2) Given the following three points, find by hand the quadratic function they represent. (−1,−8), (0,−1),(1,2) A. f(x)=−3x2+10x−1 B. f(x)=−3x2+4x−1 C. f(x)=−2x2+5x−1 D. f(x)=−5x2+8x−1 3) Find the equation of a parabola that has a vertex (3,5) and passes through the point (1,13). A. y=−3(x−3)2+5 B. y=2(x−3)2+5 C. y=−2(x−3)2+5 D. y=2(x+3)2−5

1) f(x) = 4·x² - 3·x + 6

2) f(x) = -2·x² + 5·x - 1

3) y = 2·(x - 3)² + 5

Step-by-step explanation:

1) The quadratic function that is represented by the points (0, 6), (2, 16), (3, 33) is found as follows

The general form of a quadratic function is f(x) = a·x² + b·x + c

Where, in (x, y), f(x) = y, and x = x

Therefore for the point (0, 6), we have;

6 = 0·x² + 0·x + c

c = 6

We have c = 6

For the point (2, 16), we have;

16 = a·2² + b·2 + 6

10 = 4·a + 2·b(1)

For the point (3, 33), we have;

33 = a·3² + b·3 + 6

27 = 9·a + 3·b(2)

Multiply equation (1) by 1.5 and subtract it from equation (2), we have;

1.5 × (10 = 4·a + 2·b)

15 = 6·a + 3·b

27 = 9·a + 3·b - (15 = 6·a + 3·b) gives;

27 - 15 = 9·a - 6·a+ 3·b - 3·b

12 = 3·a

a = 12/3 = 4

a = 4

From equation (1), we have;

10 = 4·a + 2·b = 4×4 + 2·b

10 - 4×4 = 2·b

10 - 16 = 2·b

-6 = 2·b

b = -3

The function, f(x) = 4·x² - 3·x + 6

2) Where the points are (-1, -8), (0, -1), (1, 2), we have;

For point (-1, -8), we have -8 = a·(-1)² - b·(-1) + c = a - b + c(1)

For point (0, 1), we have -1 = a×0² + b×0 + c = c(2)

For point (1, 2), we have 2 = a×1²+ b×1 + c = a + b + c(3)

Adding equation (1) to equation (3) gives

-8 + 2 = a - b + c +  a + b + c = 2·a + 2·c  where, c = -1, we have

-8 + 2 = -6 = 2·a + 2

2·a = -6 + 2 = - 4

a = -8/2 = -2

From equation (3), we have;

2 = a + b + c

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

f(x) = -2·x² + 5·x - 1

3) The equation of a parabola that has vertex (3, 5) and passing through the point (1, 13) is given by the vertex equation of a parabola

The vertex equation of a parabola is y = a(x - h)² + k

Where;

(h, k) = Vertex (3, 5)

(x, y) = (1, 13)

We have

13 = a·(1 - 3)² + 5

13 = a·(-2)² + 5

13 - 5 = a·(-2)² = 4·a

4·a = 8

a = 8/4 = 2

The equation is y = 2·(x - 3)² + 5.

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step-by-step explanation: