To graph the function [tex]\( g(x) = (0.5)^{x+3} - 4 \)[/tex], let's go through a step-by-step solution.
### Step-by-Step Solution:
1. Understand the Function:
- The function is [tex]\( g(x) = (0.5)^{x+3} - 4 \)[/tex].
- This is an exponential function with a base of 0.5 and shifted downward by 4 units.
2. Identify Key Features of the Function:
- Base: 0.5 indicates the function is decreasing because [tex]\( 0 < 0.5 < 1 \)[/tex].
- Exponent (x+3): The function is shifted horizontally.
- Vertical Shift: Subtracting 4 shifts the function down by 4 units.
3. Evaluate Key Points: Let's determine the values of [tex]\( g(x) \)[/tex] for some key points of [tex]\( x \)[/tex]:
- [tex]\( x = 0: \)[/tex]
[tex]\[
g(0) = (0.5)^{0+3} - 4 = (0.5)^3 - 4 = 0.125 - 4 = -3.875
\][/tex]
- [tex]\( x = -3: \)[/tex]
[tex]\[
g(-3) = (0.5)^{-3+3} - 4 = (0.5)^0 - 4 = 1 - 4 = -3
\][/tex]
- [tex]\( x = -5: \)[/tex]
[tex]\[
g(-5) = (0.5)^{-5+3} - 4 = (0.5)^{-2} - 4 = 4 - 4 = 0
\][/tex]
- [tex]\( x = 3: \)[/tex]
[tex]\[
g(3) = (0.5)^{3+3} - 4 = (0.5)^6 - 4 = 0.015625 - 4 = -3.984375
\][/tex]
4. Determine Asymptote:
- As [tex]\( x \to \infty \)[/tex], [tex]\( 0.5^{x+3} \)[/tex] approaches 0.
- Therefore, [tex]\( g(x) \to -4 \)[/tex]. The horizontal asymptote is [tex]\( y = -4 \)[/tex].
5. Plot the Graph:
- On a coordinate plane, start by plotting the key points we computed and draw the horizontal asymptote at [tex]\( y = -4 \)[/tex].
- The curve will be steeply decreasing on the left (as [tex]\( x \to -\infty \)[/tex]), and it approaches the asymptote as [tex]\( x \to \infty \)[/tex].
### Graphing the Function:
- Plot points we evaluated:
1. [tex]\( (0, -3.875) \)[/tex]
2. [tex]\( (-3, -3) \)[/tex]
3. [tex]\( (-5, 0) \)[/tex]
4. [tex]\( (3, -3.984375) \)[/tex]
- Draw the horizontal asymptote [tex]\( y = -4 \)[/tex].
- Sketch the curve through these points, ensuring it approaches the asymptote without touching it.
### Summary:
- The function [tex]\( g(x) = (0.5)^{x+3} - 4 \)[/tex] is an exponential decay function shifted downward by 4 units.
- Its horizontal asymptote is [tex]\( y = -4 \)[/tex].
- The graph passes through points like [tex]\( (-3, -3) \)[/tex], [tex]\( (0, -3.875) \)[/tex], [tex]\( (-5, 0) \)[/tex], and it continually decreases.
By using these steps, you can sketch the graph of [tex]\( g(x) = (0.5)^{x+3} - 4 \)[/tex]. The curve will be steeply decreasing on the left side and will gradually approach the line [tex]\( y = -4 \)[/tex] without crossing it as [tex]\( x \)[/tex] increases.
