To determine the characteristics and shape of the graph of the function [tex]\( g(x) = (0.5)^{x+3} - 4 \)[/tex], let's analyze it step-by-step.
### Step 1: Understanding the Form of the Function
First, recognize that we are dealing with an exponential function. The general form of an exponential function can be expressed as [tex]\( f(x) = a^x \)[/tex]. Here, our base [tex]\( a \)[/tex] is [tex]\( 0.5 \)[/tex], and the exponent is [tex]\( x + 3 \)[/tex]. The function also includes a vertical shift of 4 units downward.
### Step 2: Basic Shape of the Exponential Function
The base of our exponent is [tex]\( 0.5 \)[/tex]. An exponential function with a base between 0 and 1 is a decreasing function. Therefore, [tex]\( (0.5)^x \)[/tex] starts high when [tex]\( x \)[/tex] is very negative and decreases as [tex]\( x \)[/tex] becomes positive.
### Step 3: Adjusting for the Horizontal Shift
In the function [tex]\( g(x) = (0.5)^{x+3} - 4 \)[/tex], the term [tex]\( x+3 \)[/tex] implies a horizontal shift. Specifically, the graph of [tex]\( (0.5)^x \)[/tex] is shifted 3 units to the left.
### Step 4: Vertical Shift
The function further involves a vertical shift down by 4 units, as indicated by the [tex]\(-4\)[/tex] at the end of the function. This means every point on the graph of [tex]\( (0.5)^{x+3} \)[/tex] is lowered by 4 units.
### Step 5: Domain and Range
- Domain: The domain of [tex]\( g(x) \)[/tex] is all real numbers, [tex]\( \mathbb{R} \)[/tex], since exponential functions are defined for all real numbers.
- Range: The range of an exponential function with vertical shift [tex]\( -4 \)[/tex] is [tex]\( (-4, +\infty) \)[/tex]. This is because while [tex]\( (0.5)^{x+3} \)[/tex] asymptotically approaches 0, [tex]\( -4 \)[/tex] is subtracted from it. Hence, [tex]\( g(x) \)[/tex] will asymptotically approach [tex]\(-4\)[/tex] from above.
### Step 6: Calculating Key Points
You will calculate a few key points to get a more precise understanding of the graph:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = (0.5)^{0+3} - 4 = (0.5)^3 - 4 = 0.125 - 4 = -3.875
\][/tex]
- When [tex]\( x = -3 \)[/tex]:
[tex]\[
g(-3) = (0.5)^{-3+3} - 4 = (0.5)^0 - 4 = 1 - 4 = -3
\][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = (0.5)^{1+3} - 4 = (0.5)^4 - 4 = 0.0625 - 4 = -3.9375
\][/tex]
- As [tex]\( x \to \infty \)[/tex]:
[tex]\[
g(x) \to -4
\][/tex]
### Step 7: Asymptotes
- The horizontal asymptote of the function is [tex]\( y = -4 \)[/tex].
### Conclusion
Given this detailed analysis, the graph of the function [tex]\( g(x) = (0.5)^{x+3} - 4 \)[/tex] is a decreasing exponential curve due to the base [tex]\( 0.5 \)[/tex], shifted 3 units to the left, and 4 units downward. The graph approaches but never reaches the horizontal line [tex]\( y = -4 \)[/tex].
Based on the key values and shape determined, you can sketch the graph accurately.
