The problem asks us to analyze the graph of the exponential function $y = 3(2^x)$ and determine the following:

1. The y-intercept of the graph.
2. The x-intercept of the graph.
3. Whether the graph is increasing or decreasing.

Step 1: Finding the y-intercept

The y-intercept is the point where the graph crosses the $y$-axis, which happens when $x = 0$.

Substitute $x = 0$ into the equation: $y = 3(2^0) = 3 \cdot 1 = 3$ Thus, the y-intercept is $y = 3$.

Step 2: Finding the x-intercept

The x-intercept is the point where the graph crosses the $x$-axis, which happens when $y = 0$.

For $y = 3(2^x)$, set $y = 0$: $0 = 3(2^x)$ Divide both sides by 3: $0 = 2^x$ An exponential term $2^x$ can never equal 0 for any real $x$. Therefore, the graph has no x-intercept.

Step 3: Analyzing the graph

The base of the exponential function is 2, which is greater than 1. Exponential functions with a base greater than 1 are increasing.

Final Answers:

• The y-intercept of the graph is $3$.
• The graph has no x-intercept.
• The graph is increasing.

Let me know if you'd like further clarifications or have any specific questions!

Related Questions:

1. How would the graph change if the equation were $y = 3(0.5^x)$?
2. What happens to the graph if the coefficient 3 is replaced by another constant?
3. What is the horizontal asymptote of the graph $y = 3(2^x)$?
4. Can an exponential function ever have an x-intercept? Why or why not?
5. How does the rate of increase for $y = 3(2^x)$ compare to $y = 3(3^x)$?

Tip:

Always remember that exponential functions of the form $y = a(b^x)$ have a horizontal asymptote at $y = 0$ when $b > 0$.