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?

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.