Question

which of the following is not a feature of the graph of $f(x) = 3 \cdot 4^x$?
\\(\bigcirc\\) a. it has a horizontal asymptote at $y = 0$.
\\(\bigcirc\\) b. the y-intercept is 3.
\\(\bigcirc\\) c. it has a minimum point.
\\(\bigcirc\\) d. the function increases as $x$ increases.

Explanation:

Step1: Analyze the function \( f(x) = 3\cdot4^x \)

This is an exponential function of the form \( y = ab^x \), where \( a = 3 \) and \( b = 4 \) (and \( b>1 \)).

Step2: Check option a

For exponential functions \( y = ab^x \) with \( b>0, b
eq1 \), as \( x\to -\infty \), \( b^x\to 0 \) (since \( b > 1 \)), so \( y = ab^x\to 0 \). Thus, the horizontal asymptote is \( y = 0 \). So option a is a feature.

Step3: Check option b

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \): \( f(0)=3\cdot4^0=3\cdot1 = 3 \). So the y - intercept is 3. Option b is a feature.

Step4: Check option c

For the exponential function \( f(x)=3\cdot4^x \), since the base \( 4>1 \) and the coefficient \( 3>0 \), the function is always increasing (as \( x \) increases, \( 4^x \) increases, so \( 3\cdot4^x \) increases). An increasing exponential function (with \( b > 1 \) and \( a>0 \)) does not have a minimum point (it has a horizontal asymptote at \( y = 0 \) and the function values increase from near 0 to \(+\infty\) as \( x \) goes from \( -\infty \) to \(+\infty\)).

Step5: Check option d

Since \( b = 4>1 \) and \( a = 3>0 \), for an exponential function \( y=ab^x \) with \( b > 1 \) and \( a>0 \), as \( x \) increases, \( b^x \) increases, so \( y=ab^x \) increases. So the function increases as \( x \) increases. Option d is a feature.

Answer:

c. It has a minimum point.