To graph the exponential function [tex]\( g(x) = 4^{x+3} \)[/tex], follow these steps:
1. Plotting Points:
- Choose [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 4^{0+3} = 4^3 = 64
\][/tex]
So, one point on the graph is [tex]\((0, 64)\)[/tex].
- Choose [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 4^{1+3} = 4^4 = 256
\][/tex]
Another point on the graph is [tex]\((1, 256)\)[/tex].
Using these two points, [tex]\((0, 64)\)[/tex] and [tex]\((1, 256)\)[/tex], you can start plotting the function on the graph.
2. Identifying the Asymptote:
- An exponential function of the form [tex]\( g(x) = 4^{x+3} \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex]. This is because as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 4^{x+3} \)[/tex] approaches zero but never actually reaches it.
3. Graphing the Function:
- On a coordinate plane, plot the points [tex]\((0, 64)\)[/tex] and [tex]\((1, 256)\)[/tex].
- Draw a smooth curve passing through these points that approaches the asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] decreases and increases very steeply as [tex]\( x \)[/tex] increases.
4. Domain and Range:
- The domain of [tex]\( g(x) = 4^{x+3} \)[/tex] includes all real numbers since there are no restrictions on the values that [tex]\( x \)[/tex] can take.
[tex]\[
\text{Domain: } (-\infty, \infty)
\][/tex]
- The range of [tex]\( g(x) = 4^{x+3} \)[/tex] includes all positive real numbers, because [tex]\( 4^{x+3} \)[/tex] is always positive for all real numbers [tex]\( x \)[/tex].
[tex]\[
\text{Range: } (0, \infty)
\][/tex]
In summary:
- Points to plot: [tex]\((0, 64)\)[/tex] and [tex]\((1, 256)\)[/tex]
- Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( (0, \infty) \)[/tex]
With these details, you can graph the function accurately.
