Question

analyzing exponential decay graphs
analyze the graph of the exponential decay function.
the initial value is
the base, or rate o is
the domain is
graph has points (-1, 3), (0, 1), (1, 1/3), (2, 1/9)

Explanation:

Step1: Find Initial Value

The initial value of an exponential function \( y = ab^x \) is the value when \( x = 0 \). From the graph, when \( x = 0 \), \( y = 1 \)? Wait, no, wait. Wait the point \( (0,1) \) is on the graph? Wait the graph has points \( (-1, 3) \), \( (0,1) \), \( (1, \frac{1}{3}) \), \( (2, \frac{1}{9}) \). So the general form of exponential decay is \( y = ab^x \). Let's use \( x = 0 \): \( y = a \cdot b^0 = a \). So when \( x = 0 \), \( y = 1 \)? Wait no, the point \( (0,1) \) is on the graph? Wait the graph shows \( (0,1) \)? Wait the grid: at \( x = 0 \), the y - coordinate is 1? Wait the red dot at \( (0,1) \)? Wait the point \( (-1, 3) \), \( (0,1) \), \( (1, 1/3) \), \( (2, 1/9) \). So for \( x = 0 \), \( y = 1 \). Wait but the dropdown has options 0, 1/3, 1, 3. Wait maybe I made a mistake. Wait let's check the function. Let's take two points. Let's take \( x=-1 \), \( y = 3 \); \( x = 0 \), \( y =? \); \( x = 1 \), \( y = 1/3 \); \( x = 2 \), \( y = 1/9 \). Let's assume the function is \( y = ab^x \). When \( x = 0 \), \( y = a \cdot b^0 = a \). If \( x = 1 \), \( y = ab^1 = ab \); \( x = 2 \), \( y = ab^2 \). From \( x = 1 \) to \( x = 2 \), \( y \) goes from \( 1/3 \) to \( 1/9 \), so the ratio is \( (1/9)/(1/3)=1/3 \), so \( b = 1/3 \). Then when \( x = 1 \), \( y = ab = 1/3 \), and \( b = 1/3 \), so \( a \cdot (1/3)=1/3 \), so \( a = 1 \). Wait but when \( x=-1 \), \( y = ab^{-1}=a/b \). If \( a = 1 \) and \( b = 1/3 \), then \( y = 1/(1/3)=3 \), which matches the point \( (-1, 3) \). So the function is \( y = 1 \cdot (1/3)^x \). So the initial value (when \( x = 0 \)) is \( a = 1 \)? Wait no, the initial value is \( a \), which is 1? But the dropdown has 0, 1/3, 1, 3. Wait maybe the initial value is the value when \( x = 0 \), which is 1. Wait but let's check the graph again. The point \( (0,1) \) is on the graph. So the initial value (the y - intercept) is 1. Wait but the dropdown options include 1. Wait maybe I was wrong. Wait let's re - evaluate. Wait the function is \( y = 3 \cdot (1/3)^{x + 1}\)? Let's check \( x=-1 \): \( 3\cdot(1/3)^{0}=3\cdot1 = 3 \), correct. \( x = 0 \): \( 3\cdot(1/3)^{1}=3\cdot(1/3)=1 \), correct. \( x = 1 \): \( 3\cdot(1/3)^{2}=3\cdot(1/9)=1/3 \), correct. \( x = 2 \): \( 3\cdot(1/3)^{3}=3\cdot(1/27)=1/9 \), correct. So the function can also be written as \( y = 3\cdot(1/3)^{x + 1}=3\cdot(1/3)^x\cdot(1/3)^{-1}=3\cdot3\cdot(1/3)^x = 9\cdot(1/3)^x \)? No, that's not right. Wait \( (1/3)^{-1}=3 \), so \( 3\cdot(1/3)^{x + 1}=3\cdot(1/3)^x\cdot(1/3)= (1/3)^x \cdot1 \). Wait no, \( 3\times(1/3)=1 \), so \( 3\cdot(1/3)^{x + 1}=(1/3)^x \). Wait that's the same as \( y=(1/3)^x \). Wait when \( x=-1 \), \( (1/3)^{-1}=3 \), correct. \( x = 0 \), \( (1/3)^0 = 1 \), correct. \( x = 1 \), \( (1/3)^1=1/3 \), correct. So the function is \( y=(1/3)^x \)? Wait no, \( (1/3)^{-1}=3 \), which matches \( (-1,3) \). So the initial value (when \( x = 0 \)) is \( y=(1/3)^0 = 1 \). So the initial value is 1.

Step2: Find the Base

The base of the exponential function. Looking at the points: from \( x = 0 \) to \( x = 1 \), \( y \) goes from 1 to \( 1/3 \), so the ratio is \( (1/3)/1 = 1/3 \). From \( x = 1 \) to \( x = 2 \), \( y \) goes from \( 1/3 \) to \( 1/9 \), ratio is \( (1/9)/(1/3)=1/3 \). So the base \( b = 1/3 \).

Step3: Find the Domain

The domain of an exponential function \( y = b^x \) (or any exponential function) is all real numbers, because we can plug in any real number for \( x \). So the domain is all real numbers (\( (-\infty, \infty) \)).

Wait but the first dropdown is for initial value. Let's re - check the initial value. The initial value of an exponential function is the value when \( x = 0 \). From the graph, when \( x = 0 \), the point is \( (0,1) \), so the initial value is 1. The base: since the function is decaying, the base is between 0 and 1. From the points, the ratio between consecutive \( y \) - values (when \( x \) increases by 1) is \( 1/3 \), so the base is \( 1/3 \)? Wait no, wait the function is \( y = ab^x \). If \( x = 0 \), \( y = a \). If \( x = 1 \), \( y = ab \). If \( x=-1 \), \( y = a/b \). We have \( x=-1 \), \( y = 3 \); \( x = 0 \), \( y = 1 \); \( x = 1 \), \( y = 1/3 \); \( x = 2 \), \( y = 1/9 \). So from \( x=-1 \) to \( x = 0 \), \( x \) increases by 1, \( y \) goes from 3 to 1, which is a ratio of \( 1/3 \). So the base \( b = 1/3 \). And the initial value (when \( x = 0 \)) is 1.

So for the initial value dropdown, the correct option is 1. For the base, the correct option is \( 1/3 \). For the domain, it's all real numbers.

Answer:

Initial value: 1; Base: \( \frac{1}{3} \); Domain: All real numbers (\( (-\infty, \infty) \))

(Assuming the first dropdown is for initial value, the second for base, and the third for domain. So the initial value is 1, the base is \( 1/3 \), and the domain is all real numbers.)