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Let's solve the questions step-by-step.

### Given Information
1. The functions are \( f(x) = a^x \) and \( g(x) = -x + 5 \).
2. \( R \) is the \( y \)-intercept of \( g \), and \( S \) is the \( y \)-intercept of \( f \).
3. The graphs intersect at the point \( (1, 4) \).
4. The dotted line \( ST \) is parallel to the \( x \)-axis (i.e., it's a horizontal line at \( y = 0 \)).

### Question 4.1: Show that \( a = 4 \)

Since the two graphs intersect at \( (1, 4) \), both functions must have the same value when \( x = 1 \).

1. **Using \( f(x) = a^x \):**
   \[
   f(1) = a^1 = a
   \]
   Since the point is \( (1, 4) \), we have \( f(1) = 4 \).
   \[
   a = 4
   \]

2. **Conclusion:**
   This shows that \( a = 4 \), as required.

### Question 4.2: Calculate the coordinates of \( R \), \( S \), and \( T \)

Now that we know \( a = 4 \), let's find the coordinates of points \( R \), \( S \), and \( T \).

1. **Point \( R \)** - the \( y \)-intercept of \( g(x) \):
   - The \( y \)-intercept occurs when \( x = 0 \).
   \[
   g(0) = -0 + 5 = 5
   \]
   So, \( R = (0, 5) \).

2. **Point \( S \)** - the \( y \)-intercept of \( f(x) \):
   - Again, the \( y \)-intercept occurs when \( x = 0 \).
   \[
   f(0) = 4^0 = 1
   \]
   So, \( S = (0, 1) \).

3. **Point \( T \)** - where the line \( g(x) \) intersects the \( x \)-axis (i.e., where \( y = 0 \) for \( g(x) \)):
   - Set \( g(x) = 0 \):
   \[
   -x + 5 = 0
   \]
   \[
   x = 5
   \]
   So, \( T = (5, 0) \).

### Summary of Answers
- **4.1:** \( a = 4 \).
- **4.2:** The coordinates are:
  - \( R = (0, 5) \)
  - \( S = (0, 1) \)
  - \( T = (5, 0) \)

Would you like further details or have any questions?

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Here are 5 related questions for further exploration:
1. What is the general shape of the graph of an exponential function, and how does the base affect it?
2. How would the intersection points change if the slope of \( g(x) \) were different?
3. What are the transformations involved in shifting the line \( g(x) = -x + 5 \)?
4. How would we determine the points of intersection if \( f(x) = b^x \) for an arbitrary \( b \)?
5. How would you find the area between the two curves between \( x = 0 \) and \( x = 1 \)?

**Tip:** When two functions intersect, their values and slopes at that point can reveal more about their relative behaviors around the intersection.

The graphs of f(x) = a^x and g(x) = -x + 5 are drawn below. R is the y-intercept of g and S is the y-intercept of f. Both graphs intersect at point (1, 4). The dotted line ST is parallel to the line y = 0. Show that the value of a = 4 and calculate the coordinates of R, S, and T.

Learn how to find the value of a for an exponential function f(x) = a^x that intersects with a linear function g(x) = -x + 5. Calculate the y-intercepts of both functions and the intersection points with the x-axis. This solution covers core concepts of exponential and linear functions, suitable for Grades 10-12.

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