To graph the function $f(x) = (0.05)^x$, we can follow these steps:

1. Make a Table of Coordinates: Select a few $x$-values (positive, negative, and 0) and compute $f(x)$ for each. Let’s calculate the values for $x = -2, -1, 0, 1, 2$.

   • When $x = -2$: $f(-2) = (0.05)^{-2} = \frac{1}{(0.05)^2} = 400$
   • When $x = -1$: $f(-1) = (0.05)^{-1} = \frac{1}{0.05} = 20$
   • When $x = 0$: $f(0) = (0.05)^0 = 1$
   • When $x = 1$: $f(1) = (0.05)^1 = 0.05$
   • When $x = 2$: $f(2) = (0.05)^2 = 0.0025$

   So, the table of coordinates becomes:

   $x$	$f(x)$
   -2	400
   -1	20
   0	1
   1	0.05
   2	0.0025

2. Graph the Coordinates: Plot these points on a graph. Note that the function $f(x) = (0.05)^x$ is an exponential function with a base less than 1, so it decays quickly for positive $x$ and grows rapidly for negative $x$.

3. Graphing Utility: Use a graphing tool or calculator to confirm the graph. I will plot this graph for you now.import matplotlib.pyplot as plt import numpy as np

Define the function

def f(x): return (0.05)**x

Generate x values and corresponding f(x) values

x_values = np.linspace(-3, 3, 100) # For a smooth curve y_values = f(x_values)

Points from the table

x_points = [-2, -1, 0, 1, 2] y_points = [f(x) for x in x_points]

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x_values, y_values, label=r"$f(x) = (0.05)^x$", color="blue") plt.scatter(x_points, y_points, color="red", label="Points from Table") plt.axhline(0, color="black", linewidth=0.5, linestyle="--") plt.axvline(0, color="black", linewidth=0.5, linestyle="--")

Add labels, title, and legend

plt.title("Graph of $f(x) = (0.05)^x$") plt.xlabel("x") plt.ylabel("f(x)") plt.yscale("log") # To show exponential behavior clearly plt.grid(True, which="both", linestyle="--", linewidth=0.5) plt.legend()

Show the plot

plt.show() <<ImageDisplayed>>successHere is the graph of $f(x) = (0.05)^x$. The red points represent the coordinates derived from the table, and the blue curve is the smooth graph of the function. You can see how the function decreases exponentially for positive $x$ and grows rapidly for negative $x$.

Do you have any questions or would you like further clarification?

5 Related Questions:

1. How does the base of an exponential function affect its graph?
2. What happens to the graph of $f(x) = (a)^x$ when $0 < a < 1$?
3. Can we find the asymptotes of $f(x) = (0.05)^x$? If so, where are they?
4. How would the graph change if the base were $0.1$ instead of $0.05$?
5. How do we calculate the inverse of $f(x) = (0.05)^x$?

Tip:

For exponential functions, $f(x) = a^x$, if $a > 1$, the graph increases as $x$ increases; if $0 < a < 1$, the graph decreases as $x$ increases. Always check the base to understand its behavior!