Give an example of a quadratic function that has zeroes at x=2 and x=4, and that takes the value 6 when x=3.

An example of the quadratic function that has zeros at x = 2 and x = 4, and that takes the value 6 when x = 3 is  f(x) = -6·x² + 36·x  48

Step-by-step explanation:

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

Which gives;

When x = 3, f(x) = 6 which gives;

6 = a·3² + b·3 + c  = a·9 + b·3 + c(1)

The zeros of the function are x = 2 and x = 4 which gives

0 = a·2² + b·2 + c = a·4 + b·2 + c(2)

0 = a·4² + b·4 + c = a·16 + b·4 + c(3)

Subtracting equation (2) from (3) gives;

a·16 + b·4 + c - (a·4 + b·2 + c)  = a·12 + b·2 = 0

b·2 = -a·12

b = -a·12/2 = -6·a

Substituting b = -6·a in equation (1) we have;

a·9 + b·3 + c = a·9 + (-6·a)·3 + c = 6

a·9 -18·a + c = 6

-9·a + c = 6

c = 6 + 9·a

Substituting b = -6·a and c = 6 + 9·a in equation (2), we get

0 = a·4 + b·2 + c = a·4 + (-6·a)·2 + 6 + 9·a = a·4 -12·a + 6 + 9·a

a + 6 = 0

a = - 6

b = -6·a = -6*(-6) = 36

c = 6 + 9·a = 6 + 9*(-6) = 6 - 54 = -48

The equation of the function is therefore, f(x) = -6·x² + 36·x  48.

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