Question

the radioisotope iron - 55 has a half - life of about 2.736 years. suppose a scientist has a 15 gram sample of iron - 55. the function (y = 15(0.5)^{\frac{x}{2.736}}) represents the amount of iron - 55 remaining in grams, (y), after (x) days.
a. determine whether the function represents exponential growth or exponential decay.
b. graph the function.
c. find and interpret the key features of the graph.

Explanation:

Step1: Analyze the form of the exponential - function

The general form of an exponential function is $y = a(b)^x$. In the given function $y = 15(0.5)^{\frac{x}{2.756}}$, where $a = 15$ and $b=0.5$. Since $0 < b=0.5<1$, the function represents exponential decay.

Step2: Prepare for graphing

To graph the function $y = 15(0.5)^{\frac{x}{2.756}}$, we can create a table of values. Choose some values of $x$ (e.g., $x = 0,2.756,5.512,8.268$). When $x = 0$, $y=15(0.5)^0=15$; when $x = 2.756$, $y = 15(0.5)^1 = 7.5$; when $x=5.512$, $y=15(0.5)^2=3.75$; when $x = 8.268$, $y=15(0.5)^3 = 1.875$. Plot these points $(0,15),(2.756,7.5),(5.512,3.75),(8.268,1.875)$ and connect them with a smooth curve. The $y$ - intercept is the point where $x = 0$, which is $(0,15)$.

Step3: Identify key features

1. $y$ - intercept:
The $y$ - intercept is $(0,15)$. This represents the initial amount of iron - 55, which is 15 grams.
2. Asymptote:
The horizontal asymptote is $y = 0$. As $x$ approaches positive infinity, the value of $y = 15(0.5)^{\frac{x}{2.756}}$ approaches 0, meaning that over time, the amount of iron - 55 will get closer and closer to 0 grams but will never actually reach 0.
3. Domain:
The domain of the function is all real non - negative numbers $x\geq0$ (since $x$ represents time in days and time cannot be negative in this context).
4. Range:
The range of the function is $0 < y\leq15$. The function starts at $y = 15$ (when $x = 0$) and decreases towards 0 as $x$ increases.

Answer:

a. The function represents exponential decay.
b. Graph by creating a table of values with points such as $(0,15),(2.756,7.5),(5.512,3.75),(8.268,1.875)$ and connecting them with a smooth curve. The $y$ - intercept is $(0,15)$.
c. Key features: $y$ - intercept at $(0,15)$ (initial amount), horizontal asymptote $y = 0$ (amount approaches 0 over time), domain $x\geq0$, range $0 < y\leq15$.