Q. Which is the graph of \( f(x) = 2(3)^x \)?

Problem

Graph the function \( f(x) = 2 \cdot 3^{x} \).

Step-by-step explanation

1. Recognize the type of function.

   The function is an exponential function of the form \( f(x) = a \cdot b^{x} \) with base \( b = 3 \) and vertical scale factor \( a = 2 \). This means it is exponential growth because \( b > 1 \).

2. Domain.

   Exponential functions are defined for all real inputs. Therefore the domain is
   \[ \text{Domain} = (-\infty, \infty). \]

3. Range.

   For any real \( x \), \( 3^{x} > 0 \). Multiplying by 2 keeps the values positive. Thus
   \[ \text{Range} = (0, \infty). \]

4. Horizontal asymptote.

   As \( x \) becomes very negative, \( 3^{x} \) approaches 0. Multiplying by 2 still approaches 0, so the horizontal asymptote is
   \[ y = 0. \]

5. Intercepts.

   Compute the y-intercept by evaluating at \( x = 0 \):
   \[ f(0) = 2 \cdot 3^{0} = 2 \cdot 1 = 2. \]
   So the graph passes through \( (0,2) \).

   There is no x-intercept because \( f(x) > 0 \) for all \( x \), so the graph never crosses the x-axis.

6. Monotonicity (increasing/decreasing).

   Differentiate to check the slope:
   \[ f'(x) = 2 \cdot 3^{x} \cdot \ln 3. \]
   Since \( 2 > 0 \), \( 3^{x} > 0 \), and \( \ln 3 > 0 \), we have \( f'(x) > 0 \) for all \( x \). Therefore the function is strictly increasing on its entire domain.

7. Concavity.

   Compute the second derivative:
   \[ f”(x) = 2 \cdot 3^{x} \cdot (\ln 3)^{2}. \]
   This is positive for all \( x \), so the graph is concave upward everywhere.

8. Key points to plot.

   Compute a small table of values to guide the sketch.

   x	f(x) = 2 · 3^x
   -2	\( f(-2) = 2 \cdot 3^{-2} = 2 \cdot \tfrac{1}{9} = \tfrac{2}{9} \approx 0.222\)
   -1	\( f(-1) = 2 \cdot 3^{-1} = 2 \cdot \tfrac{1}{3} = \tfrac{2}{3} \approx 0.667\)
   0	\( f(0) = 2 \)
   1	\( f(1) = 2 \cdot 3 = 6 \)
   2	\( f(2) = 2 \cdot 9 = 18 \)

   Plot these points and draw a smooth curve through them that is increasing and concave up, approaching the line \( y = 0 \) on the left.

9. Transformation description.

   Start with the basic graph of \( y = 3^{x} \), which has y-intercept 1 and horizontal asymptote \( y = 0 \). Multiplying by 2 produces a vertical stretch by factor 2. This doubles all y-values, so the y-intercept moves from \( (0,1) \) to \( (0,2) \), and other y-values are doubled accordingly.

10. Final sketch description.

    The graph of \( f(x) = 2 \cdot 3^{x} \) is an increasing exponential curve that:

    • has domain \( (-\infty,\infty) \),
    • has range \( (0,\infty) \),
    • has horizontal asymptote \( y = 0 \),
    • passes through \( (0,2) \), \( (1,6) \), \( (-1,\tfrac{2}{3}) \),
    • is strictly increasing and concave up.

    When sketched, the curve lies very close to the x-axis for large negative x, crosses the y-axis at 2, and then rises rapidly for positive x.