Sure, let's first discuss the function [tex]\( f(x) = -5^{x+2} \)[/tex], determine whether it is increasing or decreasing, and then find two points on the graph.
### Step 1: Understanding the Function
The function given is [tex]\( f(x) = -5^{x+2} \)[/tex]. Note that [tex]\(5^{x+2}\)[/tex] is an exponential function with base 5, and the negative sign in front of it flips the graph vertically.
### Step 2: Determine if the Function is Increasing or Decreasing
To determine whether the function is increasing or decreasing, let's analyze the behavior of the exponential function [tex]\(5^{x+2}\)[/tex]:
1. The base 5 is greater than 1, so [tex]\(5^{x+2}\)[/tex] is an increasing function.
2. Since we're multiplying by -1, [tex]\( -5^{x+2} \)[/tex] will be a decreasing function. This indicates that as [tex]\( x \)[/tex] increases, the value of [tex]\( -5^{x+2} \)[/tex] will decrease.
Therefore, the graph of [tex]\( f(x) = -5^{x+2} \)[/tex] is a decreasing function.
### Step 3: Find Points on the Graph
Let's choose two "easy" points for [tex]\( x \)[/tex] such as 0 and 1.
Point 1: When [tex]\( x = 0 \)[/tex]
[tex]\[ f(0) = -5^{0+2} = -5^2 = -25 \][/tex]
So, the first point is [tex]\( (0, -25) \)[/tex].
Point 2: When [tex]\( x = 1 \)[/tex]
[tex]\[ f(1) = -5^{1+2} = -5^3 = -125 \][/tex]
So, the second point is [tex]\( (1, -125) \)[/tex].
### Conclusion:
- The graph of [tex]\( f(x) = -5^{x+2} \)[/tex] is decreasing.
- Two points on the graph are [tex]\( (0, -25) \)[/tex] and [tex]\( (1, -125) \)[/tex].
