Given the function [tex]\( f(x) = a \sqrt{x + b} \)[/tex] and the points through which the graph passes, we can determine the nature of the constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
First, let's use the information that the graph passes through the point [tex]\((-24, 0)\)[/tex]. This means that when [tex]\( x = -24 \)[/tex], [tex]\( f(x) = 0 \)[/tex].
[tex]\[
f(-24) = a \sqrt{-24 + b} = 0
\][/tex]
Since [tex]\( \sqrt{-24 + b} = 0 \)[/tex], we have:
[tex]\[
-24 + b = 0 \implies b = 24
\][/tex]
So, we have determined that [tex]\( b = 24 \)[/tex].
Next, we were given that [tex]\( f(24) < 0 \)[/tex]. Let's use this condition to understand the nature of [tex]\( a \)[/tex].
[tex]\[
f(24) = a \sqrt{24 + b} = a \sqrt{24 + 24} = a \sqrt{48}
\][/tex]
Since [tex]\( \sqrt{48} \)[/tex] is positive, for [tex]\( f(24) < 0 \)[/tex], it must be that [tex]\( a \)[/tex] is negative. Thus,
[tex]\[
a < 0
\][/tex]
With [tex]\( b = 24 \)[/tex] and [tex]\( a < 0 \)[/tex], let's now evaluate the options provided:
- Option (A) [tex]\( f(0) = 24 \)[/tex]:
[tex]\[
f(0) = a \sqrt{0 + b} = a \sqrt{24}
\][/tex]
Since [tex]\( a < 0 \)[/tex], [tex]\( a \sqrt{24} \)[/tex] cannot be 24 because it would be a positive number if [tex]\( a \)[/tex] were positive, which is not the case here. So, this option is incorrect.
- Option (B) [tex]\( f(0) = -24 \)[/tex]:
[tex]\[
f(0) = a \sqrt{0 + b} = a \sqrt{24}
\][/tex]
We need to check if this equals -24:
[tex]\[
a \sqrt{24} = -24 \implies a = -\frac{24}{\sqrt{24}} = -\sqrt{24}
\][/tex]
Given [tex]\( \sqrt{24} \)[/tex] is positive and [tex]\( a = -\sqrt{24} \)[/tex] is indeed negative, so [tex]\( f(0) = -24 \)[/tex] is a valid option.
- Option (C) [tex]\( a > b \)[/tex]:
We know [tex]\( a < 0 \)[/tex] and [tex]\( b = 24 \)[/tex]. Clearly [tex]\( a \)[/tex] is not greater than [tex]\( b \)[/tex] because [tex]\( a < 0 \)[/tex] and [tex]\( b = 24 \)[/tex]. So this option is incorrect.
- Option (D) [tex]\( a < b \)[/tex]:
As [tex]\( a < 0 \)[/tex] and [tex]\( b = 24 \)[/tex], it is true that [tex]\( a < b \)[/tex].
Thus, the options that are true based on the given conditions are:
- [tex]\( f(0) = -24 \)[/tex]
- [tex]\( a < b \)[/tex]
Thus the correct answers must include:
(B) [tex]\( f(0) = -24 \)[/tex]
(D) [tex]\( a < b \)[/tex]
