Certainly! Let's explore the function [tex]\( y = 2^x + 5 \)[/tex] in a detailed manner.
1. Understanding the Form of the Equation:
The given equation is [tex]\( y = 2^x + 5 \)[/tex]. This is an exponential function, where the base is 2, and it represents exponential growth.
2. Evaluating the Function for Different Values of [tex]\( x \)[/tex]:
To better understand the nature of this equation, let's compute [tex]\( y \)[/tex] for various values of [tex]\( x \)[/tex].
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 2^0 + 5 = 1 + 5 = 6
\][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 2^1 + 5 = 2 + 5 = 7
\][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 2^2 + 5 = 4 + 5 = 9
\][/tex]
- When [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 2^3 + 5 = 8 + 5 = 13
\][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 2^{-1} + 5 = \frac{1}{2} + 5 = 0.5 + 5 = 5.5
\][/tex]
- When [tex]\( x = -2 \)[/tex]:
[tex]\[
y = 2^{-2} + 5 = \frac{1}{4} + 5 = 0.25 + 5 = 5.25
\][/tex]
3. Graphical Representation:
Plotting this function would show an exponential curve. For [tex]\( x \ge 0 \)[/tex], the graph rises rapidly. For [tex]\( x < 0 \)[/tex], the graph approaches [tex]\( y = 5 \)[/tex] asymptotically but never actually reaches it.
4. Transformations:
The base function here is [tex]\( 2^x \)[/tex]. By adding 5 (i.e., [tex]\( +5 \)[/tex]), we are shifting the entire graph of [tex]\( 2^x \)[/tex] vertically upwards by 5 units.
5. Behavior of the Function:
- As [tex]\( x \)[/tex] increases, [tex]\( 2^x \)[/tex] increases exponentially, hence [tex]\( y \)[/tex] increases exponentially.
- As [tex]\( x \)[/tex] decreases, [tex]\( 2^x \)[/tex] approaches zero, and so [tex]\( y \)[/tex] approaches 5 but is always greater than 5.
6. Asymptotic Behavior:
As [tex]\( x \to -\infty \)[/tex], [tex]\( 2^x \to 0 \)[/tex]. Therefore, [tex]\( y \)[/tex] approaches 5 from above but never actually equals 5.
This outlines the detailed analysis of the function [tex]\( y = 2^x + 5 \)[/tex].
