To identify the graph that represents the function [tex]\(y = 5 \cdot\left(\frac{1}{3}\right)^x\)[/tex], let's carefully analyze the behavior of this exponential function through a detailed step-by-step explanation:
1. Function Analysis:
- The given function is [tex]\(y = 5 \cdot\left(\frac{1}{3}\right)^x\)[/tex].
- This is an exponential decay function with a base of [tex]\(\frac{1}{3}\)[/tex] and a coefficient of 5.
2. Understanding Exponential Decay:
- Since the base of the exponent [tex]\(\left(\frac{1}{3}\right)\)[/tex] is less than 1 but positive, the function represents exponential decay.
- As [tex]\(x\)[/tex] increases, [tex]\(\left(\frac{1}{3}\right)^x\)[/tex] decreases towards 0.
- For negative [tex]\(x\)[/tex], [tex]\(\left(\frac{1}{3}\right)^x\)[/tex] increases rapidly.
3. Key Points on the Graph:
- When [tex]\(x = 0\)[/tex], [tex]\(y = 5\cdot\left(\frac{1}{3}\right)^0 = 5\cdot 1 = 5\)[/tex].
- For [tex]\(x > 0\)[/tex], the value of [tex]\(y\)[/tex] decreases.
- For [tex]\(x < 0\)[/tex], the value of [tex]\(y\)[/tex] increases.
4. Detailed Values:
- The specific numerical results for a range of [tex]\(x\)[/tex] values can help pinpoint the graph accurately. Here are some key data points derived from the function:
- For [tex]\(x = -10\)[/tex], [tex]\(y \approx 295245.00\)[/tex]
- For [tex]\(x = -5\)[/tex], [tex]\(y \approx 231.00\)[/tex]
- For [tex]\(x = 0\)[/tex], [tex]\(y = 5.00\)[/tex]
- For [tex]\(x = 5\)[/tex], [tex]\(y \approx 0.01\)[/tex]
- For [tex]\(x = 10\)[/tex], [tex]\(y \approx 0.00005\)[/tex]
5. Behavior at Specific Intervals:
- For large negative [tex]\(x\)[/tex], [tex]\(y \)[/tex] becomes extremely large.
- For large positive [tex]\(x\)[/tex], [tex]\(y\)[/tex] approaches zero but never touches it.
- At [tex]\(x = 0\)[/tex], [tex]\(y\)[/tex] consistently hits the value 5.
Putting it all together:
- The graph should show a rapid increase for negative [tex]\(x\)[/tex]-values, crossing 5 at [tex]\( x = 0 \)[/tex], and a rapid decrease for positive [tex]\(x\)[/tex]-values.
- The graph should have a distinct shape where for [tex]\(x < 0\)[/tex], the values are large, and for [tex]\(x > 0\)[/tex], they approach zero.
By matching this behavior with the described data points, you should be able to identify the correct graph that represents [tex]\(y=5 \cdot\left(\frac{1}{3}\right)^x\)[/tex].
