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f(x)=(x-6)^(2)-9
Which of the following is an equivalent form of the function
f in which the zeros of
f appear as constants or coefficients?
Choose 1 answer:
(A)
f(x)=(x-9)(x-6)
(B)
f(x)=(x-9)(x-3)
(c)
f(x)=(x+3)(x+9)
(D)
f(x)=x^(2)-12 x+27

$f(x)=(x-6)^{2}-9$ $\newline$ Which of the following is an equivalent form of the function $f$ in which the zeros of $f$ appear as constants or coefficients? $\newline$ Choose $1$ answer: $\newline$ (A) $f(x)=(x-9)(x-6)$ $\newline$ (B) $f(x)=(x-9)(x-3)$ $\newline$ (C) $f(x)=(x+3)(x+9)$ $\newline$ (D) $f(x)=x^{2}-12 x+27$

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Compare linear and exponential growth

Full solution

Q.

f(x)=(x-6)^{2}-9

\newline

Which of the following is an equivalent form of the function

f

in which the zeros of

f

appear as constants or coefficients?

\newline

Choose

1

answer:

\newline

(A)

f(x)=(x-9)(x-6)

\newline

(B)

f(x)=(x-9)(x-3)

\newline

(C)

f(x)=(x+3)(x+9)

\newline

(D)

f(x)=x^{2}-12 x+27

Expand and Simplify: We are given the function $f(x)=(x-6)^{2}-9$ . To find an equivalent form where the zeros are visible, we need to expand the squared term and then simplify the expression. $\newline$ Let's expand $(x-6)^{2}$ first: $\newline$ $(x-6)^{2} = x^{2} - 2\times(6)\times x + 6^{2} = x^{2} - 12x + 36$
Subtract Constant: Now, we subtract $9$ from the expanded form to get the complete function: $\newline$ $f(x) = x^{2} - 12x + 36 - 9$ $\newline$ $f(x) = x^{2} - 12x + 27$
Factor Quadratic: We now have the function in the form $f(x) = x^{2} - 12x + 27$ . To find the zeros, we can factor this quadratic equation. The factors of $27$ that add up to $-12$ are $-3$ and $-9$ . Therefore, we can write the function as: $\newline$ $f(x) = (x - 3)(x - 9)$
Match Factored Form: We check the options given to see which one matches our factored form of the function: $\newline$ (A) $f(x)=(x-9)(x-6)$ does not match. $\newline$ (B) $f(x)=(x-9)(x-3)$ matches our factored form. $\newline$ (C) $f(x)=(x+3)(x+9)$ does not match. $\newline$ (D) $f(x)=x^{2}-12x+27$ is the expanded form but does not show the zeros as constants or coefficients.

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