Graphs of the form yy = 22xx+cc where c is from the set of real numbers
(5 marks)
Graph the following and consider the questions given.

• yy = 2xx
• yy = 2xx−1
• yy = 2xx+2
• yy = 2xx−3

a) What effect does c have on the position of the graph?
b) What effect does c have on the shape of the graph?
c) What is the horizontal asymptote of each graph?
d) What transformation is used to graph yy = 2xx−cc ffffffff yy = 2xx

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.

Explore graphs of exponential functions yy = 2xx+cc where c is a real number. Understand how different values of c affect the position and shape of the graph, and identify the horizontal asymptote for each function. Suitable for high school students studying exponential functions and transformations.

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