Question

without graphing, complete the following for the function ( k(x) = 3(2^x) ).
(a) describe the shape of the graph of the function.
(b) find the second coordinates of the points with first coordinates 0 and 1.
(a) the graph is entirely above the x - axis and rises from left to right, more steeply than the graph of ( y = 2^x ).
(b) the second coordinate of the point with first coordinate 0 is (square). (type an integer or a simplified fraction.)

Explanation:

Step1: Analyze the exponential function form

The function $k(x)=3(2^x)$ is a vertical stretch of the parent exponential function $y=2^x$ by a factor of 3. Exponential functions with base $b>1$ are always positive, rise left to right, and stretch vertically when multiplied by a positive constant greater than 1.

Step2: Calculate $k(0)$

Substitute $x=0$ into $k(x)$:
$k(0)=3(2^0)=3(1)=3$

Step3: Calculate $k(1)$

Substitute $x=1$ into $k(x)$:
$k(1)=3(2^1)=3(2)=6$

Answer:

(a)

The graph is entirely above the x-axis, rises from left to right, and increases more steeply than the graph of $y=2^x$.

(b)

• For first coordinate 0: 3
• For first coordinate 1: 6