The function \( y = 3^x + 4 \) represents an exponential equation where the base of the exponent is 3, and it has been vertically shifted by 4 units.

### Key characteristics of the equation:

1. **Exponential Growth:** The term \( 3^x \) grows exponentially as \( x \) increases, so the graph will rise steeply as \( x \to \infty \).
2. **Vertical Shift:** The "+4" causes the entire graph to be shifted upwards by 4 units compared to the basic function \( y = 3^x \).
3. **Intercept:** When \( x = 0 \), \( y = 3^0 + 4 = 1 + 4 = 5 \). So the y-intercept of the graph is 5.
4. **Asymptote:** As \( x \to -\infty \), \( y \to 4 \), meaning the horizontal asymptote of the graph is \( y = 4 \).
5. **Behavior for Negative \(x\):** For negative \( x \), the value of \( 3^x \) approaches 0, so \( y \) will approach 4 but never reach it.

Would you like a detailed graph of this function or any further explanation on its behavior?

### 5 Questions to Explore:
1. How does the behavior of \( y = 3^x \) change if the base is different (e.g., 2 or 4)?
2. What would the graph of \( y = 3^{-x} + 4 \) look like?
3. How would adding a coefficient to \( 3^x \), like \( y = 2 \cdot 3^x + 4 \), affect the graph?
4. Can you find the inverse of \( y = 3^x + 4 \)?
5. What happens if you replace the base 3 with \( e \), resulting in \( y = e^x + 4 \)?

### Tip:
When graphing exponential functions, always look for the y-intercept and asymptote to get an idea of the overall shape.

Explore the exponential function y = 3^x + 4, focusing on key characteristics like exponential growth, vertical shift, and asymptote behavior. The solution includes step-by-step insights into the graph's intercept, asymptote, and behavior for positive and negative x-values. Ideal for Grades 9-11 students.

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