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To graph the function \( f(x) = 2 \left( \frac{1}{4} \right)^x - 5 \) as requested, let’s analyze its characteristics.

1. **Function Structure**: The function \( f(x) = 2 \left( \frac{1}{4} \right)^x - 5 \) is an exponential decay function due to the base \( \frac{1}{4} \) (which is less than 1).

2. **Horizontal Asymptote**: For exponential functions of the form \( a \cdot b^x + c \), the horizontal asymptote is \( y = c \). Therefore, in this function, the asymptote is:
   \[
   y = -5
   \]

3. **Points with Integer Coordinates**:
   - To find points, we can calculate \( f(x) \) for a few integer values of \( x \).
   
   Let's find values at \( x = 0 \) and \( x = 1 \):
   
   - **For \( x = 0 \)**:
     \[
     f(0) = 2 \left( \frac{1}{4} \right)^0 - 5 = 2 \cdot 1 - 5 = -3
     \]
     So, one point is \( (0, -3) \).

   - **For \( x = 1 \)**:
     \[
     f(1) = 2 \left( \frac{1}{4} \right)^1 - 5 = 2 \cdot \frac{1}{4} - 5 = \frac{1}{2} - 5 = -4.5
     \]
     However, this does not give an integer coordinate.

   - **For \( x = -1 \)**:
     \[
     f(-1) = 2 \left( \frac{1}{4} \right)^{-1} - 5 = 2 \cdot 4 - 5 = 8 - 5 = 3
     \]
     So, another point with integer coordinates is \( (-1, 3) \).

### Summary:
- **Asymptote**: \( y = -5 \)
- **Integer Coordinate Points**: \( (0, -3) \) and \( (-1, 3) \)

### Graphing Steps:
1. Plot the horizontal asymptote \( y = -5 \).
2. Mark the points \( (0, -3) \) and \( (-1, 3) \) on the graph.
3. Draw a smooth curve passing through these points and approaching the asymptote as \( x \) increases.

Would you like additional help with plotting, or any further clarification on these steps?

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Here are some questions to extend your understanding:

1. How does the base \( \frac{1}{4} \) affect the shape of the graph compared to a base greater than 1?
2. What would happen to the graph if the constant \( -5 \) were changed?
3. Why does the function approach \( y = -5 \) but never reach it?
4. How would this graph differ if the base were negative?
5. What other points could we choose to make the graph more accurate?

**Tip**: When working with transformations, focus on the asymptote and key points to guide your sketch efficiently.

Graph the function \( f(x) = 2 \left( \frac{1}{4} \right)^x - 5 \) on the axes below. You must plot the asymptote and any two points with integer coordinates.

Learn how to graph the exponential decay function f(x) = 2(1/4)^x - 5, including plotting the horizontal asymptote at y = -5 and identifying points with integer coordinates. This problem covers concepts like asymptotes and exponential functions, suitable for Grades 9-11.

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