Let's address each part of the problem, beginning with graphing the functions and then answering the questions.

1. Graphing the Functions

The given functions are of the form $y = 2^{x + c}$, where $c$ is a constant. We'll graph the following:

1. $y = 2^x$
2. $y = 2^{x - 1}$
3. $y = 2^{x + 2}$
4. $y = 2^{x - 3}$

Graph Interpretation:

• $y = 2^x$: This is the base function, an exponential curve passing through the point (0, 1).
• $y = 2^{x-1}$: This is the graph of $y = 2^x$ shifted to the right by 1 unit.
• $y = 2^{x+2}$: This graph shifts the base graph $y = 2^x$ to the left by 2 units.
• $y = 2^{x-3}$: This graph shifts the base graph $y = 2^x$ to the right by 3 units.

2. Answering the Questions

a) What effect does $c$ have on the position of the graph?

The constant $c$ horizontally shifts the graph of the base function $y = 2^x$:

• When $c$ is positive, the graph shifts to the left by $c$ units.
• When $c$ is negative, the graph shifts to the right by $|c|$ units.

b) What effect does $c$ have on the shape of the graph?

The value of $c$ does not change the shape of the graph. The shape remains the same as that of the base function $y = 2^x$. The graph remains an exponential curve, which grows rapidly as $x$ increases.

c) What is the horizontal asymptote of each graph?

For each of these graphs, the horizontal asymptote remains the same:

• Horizontal Asymptote: $y = 0$

Regardless of the value of $c$, as $x$ approaches negative infinity, $y$ approaches 0.

d) What transformation is used to graph $y = 2^{x-c}$ from $y = 2^x$?

The transformation used is a horizontal shift:

• If $c$ is positive, the graph shifts to the right by $c$ units.
• If $c$ is negative, the graph shifts to the left by $|c|$ units.

Questions for Further Exploration:

1. What happens to the graph if the base of the exponential function is changed from 2 to another positive number?
2. How does the graph of $y = 2^{-x}$ compare to $y = 2^x$?
3. How would a vertical shift be introduced into the graph of $y = 2^x$?
4. What is the effect of adding a coefficient to $2^x$ in the form $y = a \cdot 2^x$?
5. How does the rate of growth of $y = 2^x$ compare with $y = 3^x$?

Tip:

When analyzing the effect of different transformations on a graph, always consider how each parameter shifts or scales the function, whether horizontally, vertically, or in terms of stretching/compressing.