To identify the graph that represents the function [tex]\( f(x) = 2^{x-3} + 3 \)[/tex], let's carefully analyze its shape and properties based on the calculated values:
1. Understanding the Function:
- The function [tex]\( f(x) = 2^{x-3} + 3 \)[/tex] involves an exponential term [tex]\( 2^{x-3} \)[/tex] which is shifted horizontally by 3 units to the right and vertically by 3 units upwards.
2. Behavior of the Function:
- As [tex]\( x \to -\infty \)[/tex]: The term [tex]\( 2^{x-3} \)[/tex] approaches 0, so [tex]\( f(x) \)[/tex] approaches 3.
- As [tex]\( x \to +\infty \)[/tex]: The term [tex]\( 2^{x-3} \)[/tex] grows exponentially, so [tex]\( f(x) \)[/tex] increases rapidly.
- At [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2^{3-3} + 3 = 1 + 3 = 4 \)[/tex].
3. Key Points and Behavior at Specific Values:
- When [tex]\( x = -10 \)[/tex], [tex]\( f(x) \approx 3.0001 \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 2^{-3} + 3 = \frac{1}{8} + 3 \approx 3.125 \)[/tex].
- When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 4 \)[/tex].
- When [tex]\( x = 10 \)[/tex], [tex]\( f(x) \approx 1283 \)[/tex].
4. General Shape of the Curve:
- The y-values start very close to 3 for very negative x-values. The function rises very slowly initially but starts increasing rapidly as x approaches larger values.
- As we go further to the right, the function grows rapidly without bound.
5. Characteristics of the Graph:
- The function has a horizontal asymptote [tex]\( y = 3 \)[/tex] as [tex]\( x \to -\infty \)[/tex].
- The curve sharply rises after moving past the point [tex]\( x = 3 \)[/tex].
Considering these points, we can summarize:
- The function is close to 3 for large negative values and rises steeply as x becomes positive, particularly past the point [tex]\( x = 3 \)[/tex].
- The function crosses the y-axis at around [tex]\( (0, 3.125) \)[/tex] and [tex]\( (3, 4) \)[/tex].
- The overall shape is an exponential curve that shifts upward by 3 units from the base exponential curve [tex]\( 2^{x-3} \)[/tex].
Given these characteristics of [tex]\( f(x) \)[/tex], the graph you are looking for will:
- Start close to the line [tex]\( y = 3 \)[/tex] on the left-hand side.
- Increase gradually at first.
- Rise steeply as [tex]\( x \to +\infty \)[/tex].
The graph that represents the function [tex]\( f(x) = 2^{x-3} + 3 \)[/tex] would show this exponential behavior with the described properties.
