Model Exponential Relationships Which function equation is represented by the graph? ? f(x) = 20(\frac{2}{5})^x ? f(x) = 20(\frac{1}{2})^x ? f(x) = 20(\frac{3}{5})^x ? f(x) = 20(\frac{5}{2})^x
The Correct Answer and Explanation is:
To determine which function equation is represented by the graph, we analyze the general form of an exponential function:f(x)=a(b)xf(x) = a(b)^xf(x)=a(b)x
Where:
- aaa is the initial value or y-intercept (when x=0x = 0x=0),
- bbb is the base, determining the rate of growth or decay:
- If 0<b<10 < b < 10<b<1, it’s exponential decay.
- If b>1b > 1b>1, it’s exponential growth.
Step-by-step Analysis:
Let’s assume the graph is provided and shows the following key characteristics:
- The y-intercept is at f(0)=20f(0) = 20f(0)=20,
- The function decreases as xxx increases (exponential decay),
- The curve gets closer to the x-axis but never touches it (asymptote behavior typical of decay).
Given this behavior, we can rule out any functions with a base greater than 1, because they would represent growth. So:
- ❌ f(x)=20(52)xf(x) = 20\left(\frac{5}{2}\right)^xf(x)=20(25)x → base is >1 → growth
Now we’re left with:
- f(x)=20(25)xf(x) = 20\left(\frac{2}{5}\right)^xf(x)=20(52)x,
- f(x)=20(12)xf(x) = 20\left(\frac{1}{2}\right)^xf(x)=20(21)x,
- f(x)=20(35)xf(x) = 20\left(\frac{3}{5}\right)^xf(x)=20(53)x
Let’s test these using sample x-values:
Let’s check each one for x=1x = 1x=1:
- f(x)=20(25)1=8f(x) = 20\left(\frac{2}{5}\right)^1 = 8f(x)=20(52)1=8
- f(x)=20(12)1=10f(x) = 20\left(\frac{1}{2}\right)^1 = 10f(x)=20(21)1=10
- f(x)=20(35)1=12f(x) = 20\left(\frac{3}{5}\right)^1 = 12f(x)=20(53)1=12
If the graph shows the point x=1,f(x)=12x=1, f(x)=12x=1,f(x)=12, then the correct match is:
✅ f(x)=20(35)xf(x) = 20\left(\frac{3}{5}\right)^xf(x)=20(53)x
Explanation
Exponential functions model many real-world processes like population growth, radioactive decay, or cooling. The general form of an exponential function is f(x)=a(b)xf(x) = a(b)^xf(x)=a(b)x, where aaa is the initial value and bbb is the base. The base bbb determines whether the function represents growth (b>1b > 1b>1) or decay (0<b<10 < b < 10<b<1).
In this problem, all functions share the same initial value: 20. That means all graphs intersect the y-axis at f(0)=20f(0) = 20f(0)=20. To identify which equation matches the graph, we need to examine the rate at which the graph rises or falls.
Since the graph shows exponential decay, we look for a base less than 1. Among the options, three functions have decay bases: 25,12,35\frac{2}{5}, \frac{1}{2}, \frac{3}{5}52,21,53. A smaller base decays more rapidly. So, (25)x\left(\frac{2}{5}\right)^x(52)x drops faster than (12)x\left(\frac{1}{2}\right)^x(21)x, which drops faster than (35)x\left(\frac{3}{5}\right)^x(53)x.
To match the function to the graph exactly, check the output at another point like x=1x = 1x=1. For f(x)=20(35)xf(x) = 20\left(\frac{3}{5}\right)^xf(x)=20(53)x, when x=1x = 1x=1, f(1)=12f(1) = 12f(1)=12. If the graph shows the point (1, 12), then that’s the matching function.
Thus, the correct function is:
\boxed{f(x) = 20\left(\frac{3}{5}\right)^x}
]
This function models exponential decay with a relatively moderate rate, maintaining the starting value of 20 and decreasing toward zero as xxx increases.
