Question

1. which of the following is not a feature of the graph of $f(x) = 2 \cdot 3^x$?

a. the y - intercept is 2.

b. the function increases as $x$ increases.

c. it has a minimum point.

d. it has a horizontal asymptote at $y = 0$.

Explanation:

Step1: Analyze the y - intercept

To find the y - intercept, we set \(x = 0\) in the function \(f(x)=2\cdot3^{x}\). Then \(f(0)=2\cdot3^{0}=2\times1 = 2\). So the y - intercept is 2, and option a is a feature of the graph.

Step2: Analyze the monotonicity

The function \(y = a\cdot b^{x}\), when \(a>0\) and \(b > 1\), the function is an increasing function. In \(f(x)=2\cdot3^{x}\), \(a = 2>0\) and \(b=3>1\), so as \(x\) increases, \(f(x)\) increases. Option b is a feature of the graph.

Step3: Analyze the existence of a minimum point

The function \(f(x)=2\cdot3^{x}\) is an exponential function with a base \(b = 3>1\) and a coefficient \(a = 2>0\). The graph of an exponential function \(y=a\cdot b^{x}(a>0,b > 1)\) is a curve that is always increasing (since the derivative \(f^\prime(x)=2\cdot3^{x}\ln3>0\) for all real \(x\)) and it has no minimum point (it approaches \(y = 0\) as \(x
ightarrow-\infty\) but never reaches it, and it goes to \(+\infty\) as \(x
ightarrow+\infty\)).

Step4: Analyze the horizontal asymptote

For the exponential function \(y = a\cdot b^{x}\), as \(x
ightarrow-\infty\), \(b^{x}
ightarrow0\) (since \(b>1\)), so \(\lim_{x
ightarrow-\infty}2\cdot3^{x}=0\). So the horizontal asymptote is \(y = 0\), and option d is a feature of the graph.

Answer:

c. It has a minimum point.