Question

consider the table representing an exponential function. the equation for this function is f(x) = dropdown dropdown^x. the table has x values -2, -1, 0, 1, 2, 3 with corresponding f(x) values 32, 16, 8, 4, 2, 1. the dropdown options for the first box are 0.5, 1, 2, 8.

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x) = a \cdot b^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base.

Step2: Find the initial value \( a \)

When \( x = 0 \), \( f(0) = 8 \). Substituting into the general form: \( 8 = a \cdot b^0 \). Since \( b^0 = 1 \), we get \( a = 8 \). Wait, no, wait—wait, looking at the table, when \( x = 0 \), \( f(x) = 8 \). But let's check the ratio between consecutive terms. Let's take \( x = -1 \) and \( x = 0 \): \( f(-1) = 16 \), \( f(0) = 8 \). The ratio \( \frac{f(0)}{f(-1)} = \frac{8}{16} = 0.5 \). Similarly, \( f(1) = 4 \), \( \frac{f(1)}{f(0)} = \frac{4}{8} = 0.5 \). \( f(2) = 2 \), \( \frac{2}{4} = 0.5 \). \( f(3) = 1 \), \( \frac{1}{2} = 0.5 \). So the common ratio (base \( b \)) is \( 0.5 \)? Wait, no—wait, when \( x \) increases by 1, \( f(x) \) is multiplied by \( 0.5 \). But let's check the general form again. Wait, maybe I mixed up. Wait, the function is \( f(x) = a \cdot b^x \). Let's use \( x = 0 \): \( f(0) = a \cdot b^0 = a \), so \( a = 8 \). Then, for \( x = 1 \), \( f(1) = 4 = 8 \cdot b^1 \), so \( 8b = 4 \), so \( b = \frac{4}{8} = 0.5 \). Wait, but the first dropdown is for the coefficient (a) and the second for the base (b)? Wait, the equation is \( f(x) = [\text{first box}] \cdot [\text{second box}]^x \). Wait, looking at the dropdown options for the first box: 0.5, 1, 2, 8. Wait, maybe I made a mistake. Wait, let's re-express. Wait, when \( x = 0 \), \( f(0) = 8 \). Let's check the values:

- When \( x = -2 \), \( f(-2) = 32 \)
- \( x = -1 \), \( f(-1) = 16 \)
- \( x = 0 \), \( f(0) = 8 \)
- \( x = 1 \), \( f(1) = 4 \)
- \( x = 2 \), \( f(2) = 2 \)
- \( x = 3 \), \( f(3) = 1 \)

So each time \( x \) increases by 1, \( f(x) \) is divided by 2, which is multiplying by \( 0.5 \). So the base \( b = 0.5 \)? Wait, no—wait, if \( f(x) = a \cdot b^x \), then let's take \( x = 1 \): \( 4 = a \cdot b^1 \), \( x = 0 \): \( 8 = a \cdot b^0 \). So from \( x = 0 \), \( a = 8 \). Then \( 4 = 8 \cdot b \implies b = 0.5 \). But the first dropdown has options 0.5, 1, 2, 8. Wait, maybe the equation is \( f(x) = 8 \cdot (0.5)^x \)? Wait, but let's check \( x = -2 \): \( 8 \cdot (0.5)^{-2} = 8 \cdot 4 = 32 \), which matches. \( x = -1 \): \( 8 \cdot (0.5)^{-1} = 8 \cdot 2 = 16 \), matches. \( x = 0 \): \( 8 \cdot 1 = 8 \), matches. \( x = 1 \): \( 8 \cdot 0.5 = 4 \), matches. \( x = 2 \): \( 8 \cdot 0.25 = 2 \), matches. \( x = 3 \): \( 8 \cdot 0.125 = 1 \), matches. Wait, but the first dropdown is the coefficient (a) and the second is the base (b). Wait, the first dropdown options are 0.5, 1, 2, 8. So a is 8? Wait, no—wait, maybe I misread the equation. The equation is \( f(x) = [\text{first box}] \cdot [\text{second box}]^x \). Wait, looking at the dropdown for the first box, the options are 0.5, 1, 2, 8. Wait, when \( x = 0 \), \( f(0) = 8 \), so \( a = 8 \)? But 8 is an option in the first dropdown. Wait, but let's check the base. The base b: when x increases by 1, f(x) is multiplied by 0.5? Wait, no—wait, from x=0 to x=1, f(x) goes from 8 to 4, which is multiplying by 0.5. So b=0.5? But 0.5 is an option in the first dropdown? Wait, no, the first dropdown is the coefficient (a) and the second is the base (b). Wait, maybe the equation is \( f(x) = 8 \cdot (0.5)^x \), but the first dropdown is 8? Wait, the first dropdown has 8 as an option. Wait, let's check the options again. The first dropdown (the first box) has options 0.5, 1, 2, 8. The second box (base) would be... Wait, maybe I made a mistake. Wait, let's re-express the function. Let's take two points. Let's take (0, 8) and (1, 4). So \( f(0) = 8 = a \cdot b^0 \implies a = 8 \). Then \( f(1) = 4 = 8 \cdot b^1 \implies b = 4/8 = 0.5 \). So the function is \( f(x) = 8 \cdot (0.5)^x \). Wait, but the first box is the coefficient (a) which is 8, and the second box is the base (b) which is 0.5? But the second dropdown isn't shown, but the first dropdown has 8 as an option. Wait, the problem shows the first dropdown with options 0.5, 1, 2, 8. So the first box is the coefficient (a) which is 8, and the second box (base) would be 0.5? But the first dropdown's options include 8, so the first box should be 8? Wait, no—wait, maybe I messed up the form. Wait, maybe the function is \( f(x) = a \cdot (b)^x \), and when x=0, f(0)=8, so a=8. Then the base b: from x=0 to x=1, f(x) is 8 to 4, so b=4/8=0.5. So the first box (a) is 8, and the second box (b) is 0.5. But the first dropdown has 8 as an option. So the first box should be 8? Wait, the first dropdown's options are 0.5, 1, 2, 8. So the first box (the coefficient) is 8, and the second box (base) is 0.5. But the problem's first dropdown is the first box (a) and the second is (b). Wait, the question is "The equation for this function is \( f(x) = \) [first box] [second box] \( ^x \)". So first box: 8, second box: 0.5? But the first dropdown has 8 as an option. Wait, let's check with x=3: f(3)=1. 8*(0.5)^3 = 8*(1/8)=1, which matches. So the first box (the coefficient) is 8, and the base is 0.5. So the first dropdown's correct option is 8.

Step3: Confirm the coefficient

Since when \( x = 0 \), \( f(0) = 8 \), and in the exponential function \( f(x) = a \cdot b^x \), \( a = f(0) = 8 \). So the first box (the coefficient) is 8.

Answer:

The first box (coefficient) should be 8, and the base (second box) would be 0.5, but from the first dropdown options, the correct choice for the first box is 8. So the first box is 8.