Question

question 18 of 25
on a piece of paper, graph ( f(x) = 2 cdot (3)^x ). then determine which answer choice matches the graph you drew.
four graphs labeled a, b, c, d
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a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

Step1: Analyze the function type

The function \( f(x) = 2\cdot(3)^x \) is an exponential function. For an exponential function \( y = a\cdot b^x \), if \( b>1 \), it is an increasing exponential function; if \( 0 < b < 1 \), it is a decreasing exponential function. Here, \( b = 3>1 \), so it is an increasing exponential function. This eliminates options C and D (which are decreasing or negative - valued for positive \( x \) in a wrong way).

Step2: Find the y - intercept

To find the y - intercept, we set \( x = 0 \). Substitute \( x = 0 \) into the function \( f(x)=2\cdot(3)^x \). We know that any non - zero number to the power of 0 is 1, so \( f(0)=2\cdot(3)^0=2\times1 = 2 \). So the graph should pass through the point \( (0,2) \).

Now let's check the graphs:

• Graph A: When \( x = 0 \), the y - value is 1 (from the graph), which does not match \( f(0) = 2 \).
• Graph B: When \( x = 0 \), the y - value is 2 (from the graph), and it is an increasing exponential function (since \( b = 3>1 \)), which matches our function \( f(x)=2\cdot(3)^x \).

Answer:

B. Graph B