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f(x)=-(x+1)(x+7)
The function represents a parabola in the
xy-plane. Which of the following is an equivalent form of
f in which the
y-intercept of the graph of
f appears as a constant or coefficient?
Choose 1 answer:
(A)
f(x)=-(x+4)^(2)+9
(B)
f(x)=-(x+4)^(2)-9
(C)
f(x)=-x^(2)+8x+7
(D)
f(x)=-x^(2)-8x-7

$f(x) = -(x+1)(x+7)$ $\newline$ The function represents a parabola in the $xy$ -plane. Which of the following is an equivalent form of $f$ in which the $y$ -intercept of the graph of $f$ appears as a constant or coefficient? $\newline$ Choose $1$ answer: $\newline$ (A) $f(x) = -(x+4)^{2}+9$ $\newline$ (B) $f(x) = -(x+4)^{2}-9$ $\newline$ (C) $f(x) = -x^{2}+8x+7$ $\newline$ (D) $f(x) = -x^{2}-8x-7$

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Reflections of functions

Full solution

Q.

f(x) = -(x+1)(x+7)

\newline

The function represents a parabola in the

xy

-plane. Which of the following is an equivalent form of

f

in which the

y

-intercept of the graph of

f

appears as a constant or coefficient?

\newline

Choose

1

answer:

\newline

(A)

f(x) = -(x+4)^{2}+9

\newline

(B)

f(x) = -(x+4)^{2}-9

\newline

(C)

f(x) = -x^{2}+8x+7

\newline

(D)

f(x) = -x^{2}-8x-7

Expand and Simplify: To find the y-intercept of the graph of f, we need to express $f(x)$ in a form where $x = 0$ yields the y-intercept directly. This means we need to expand the given product and simplify. $\newline$ Calculation: $\newline$ $f(x) = -(x+1)(x+7)$ $\newline$ $f(x) = -(x^2 + 7x + x + 7)$ $\newline$ $f(x) = -(x^2 + 8x + 7)$ $\newline$ $f(x) = -x^2 - 8x - 7$
Check Options: Now, we check the options to see which one matches the expanded form of $f(x)$ and has the $y$ -intercept as a constant or coefficient. $\newline$ Option (A) $f(x) = -(x+4)^2 + 9$ does not match because when expanded it would not yield $-x^2 - 8x - 7$ . $\newline$ Option (B) $f(x) = -(x+4)^2 - 9$ does not match for the same reason as (A). $\newline$ Option (C) $f(x) = -x^2 + 8x + 7$ is incorrect because the signs of the $x$ terms are positive, not negative. $\newline$ Option (D) $f(x) = -x^2 - 8x - 7$ matches our expanded form and has the $y$ -intercept $(-7)$ as a constant.

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