Let's go through the problem step by step to determine the correct statement about the data that Sarah has included in her lab report.
1. Understanding the [tex]$y$[/tex]-Intercept:
- The [tex]$y$[/tex]-intercept of a graph is the value of the function when [tex]$x$[/tex] is equal to [tex]$0$[/tex].
- From the table given, when [tex]$x = 0$[/tex] years, the amount remaining [tex]\( f(0) \)[/tex] is [tex]\( 900 \)[/tex].
- Therefore, the [tex]$y$[/tex]-intercept is [tex]\( 900 \)[/tex].
2. Behavior of [tex]$f(x)$[/tex] as [tex]$x$[/tex] Increases:
- We need to analyze the function's behavior to understand how the amount remaining changes as the number of years increases.
- Observing the values in the table, as [tex]$x$[/tex] increases from [tex]$0$[/tex] to [tex]\( 10 \)[/tex] years, the amount remaining [tex]\( f(x) \)[/tex] is gradually decreasing.
- Specifically, the remaining amount starts at [tex]\( 900 \)[/tex] and reduces to just over [tex]\( 100 \)[/tex] after 10 years. It seems to be approaching a value close to [tex]\( 100 \)[/tex] but not dipping below it.
3. Determining the Asymptote:
- An asymptote of a graph is a line that the graph approaches but never really touches.
- When examining the reductions in the remaining amount, it is evident that [tex]\( f(x) \)[/tex] is approaching [tex]\( 100 \)[/tex] as [tex]\( x \)[/tex] increases.
Based on these observations:
- The [tex]$y$[/tex]-intercept of the graph is indeed [tex]\( 900 \)[/tex].
- As [tex]\( x \)[/tex] increases (i.e., as the number of years increases), [tex]\( f(x) \)[/tex] is approaching [tex]\( 100 \)[/tex].
4. Evaluating the Statements:
- Statement A: "The [tex]$y$[/tex]-intercept of the graph is 900, and as [tex]$x$[/tex] increases, [tex]$f(x)$[/tex] approaches infinity."
- This is incorrect because [tex]\( f(x) \)[/tex] approaches [tex]\( 100 \)[/tex] rather than infinity.
- Statement B: "The x-intercept of the graph is 900, and as [tex]$x$[/tex] increases, [tex]$f(x)$[/tex] approaches 100."
- This is incorrect because the [tex]$x$[/tex]-intercept is the point where [tex]\( f(x) \)[/tex] is [tex]\( 0 \)[/tex], not [tex]\( 900 \)[/tex].
- Statement C: "The amount of radioactive substance will continue to decay at a constant rate over time until it reaches the [tex]$x$[/tex]-axis."
- This is incorrect because the decay is not at a constant rate and the function does not reach the [tex]$x$[/tex]-axis (since it approaches [tex]\( 100 \)[/tex], not [tex]\( 0 \)[/tex]).
- Statement D: "The [tex]$y$[/tex]-intercept of the graph is 900, and as [tex]$x$[/tex] increases, [tex]$f(x)$[/tex] approaches 100."
- This is correct because the [tex]$y$[/tex]-intercept is [tex]\( 900 \)[/tex] and [tex]\( f(x) \)[/tex] does indeed approach [tex]\( 100 \)[/tex] as [tex]\( x \)[/tex] increases.
Thus, the correct statement is [tex]\( \boxed{D} \)[/tex].
