Given that the zeros of a quadratic function are –2 and 8, what is the standard form of the quadratic function?
A f(x) = x² + 6x + 16
B f(x) = x² + 6x – 16
C f(x) = x² – 6x + 16
D f(x) = x² – 6x – 16
Given that the zeros of a quadratic function are –2 and 8, what is the standard form of the quadratic function?
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To find the standard form of a quadratic function given its zeros, we can use the factored form of the quadratic function. If the zeros are ( r_1 ) and ( r_2 ), the quadratic function can be expressed as:
[ f(x) = a(x – r_1)(x – r_2) ]
Given the zeros are –2 and 8, we can plug these values in:
[ f(x) = a(x + 2)(x – 8) ]
Now let’s expand this:
1. First, expand the factors:
[
f(x) = a[(x + 2)(x – 8)] = a(x^2 – 8x + 2x – 16) = a(x^2 – 6x – 16)
]
2. By setting ( a = 1 ) (for the standard form), we get:
[
f(x) = x^2 – 6x – 16
]
So the standard form of the quadratic function is:
D ( f(x) = x^2 – 6x – 16 )
This choice matches the expanded expression, confirming it’s correct.