graph the function state its domain and range y = -2^x + 3

To graph the function $ y = -2^x + 3 $, let's break down the process step by step and then state its domain and range.

### 1. **Understanding the Function:**
   - The function is an exponential function of the form $ y = a \cdot b^x + c $, where $ a = -1 $, $ b = 2 $, and $ c = 3 $.
   - The base of the exponent $ b = 2 $ is positive, and the coefficient $ a = -1 $ is negative, which means the graph will be a reflection of the standard exponential function $ 2^x $ across the x-axis, shifted upward by 3 units.

### 2. **Graphing the Function:**
   - **Asymptote:** The horizontal asymptote will be at $ y = 3 $ since the function is shifted up by 3 units.
   - **Behavior:** The graph will decrease rapidly as $ x $ increases and approach the horizontal asymptote $ y = 3 $ from below.
   - **Key Points:**
     - At $ x = 0 $: $ y = -2^0 + 3 = -1 + 3 = 2 $
     - As $ x $ increases positively: $ y $ decreases.
     - As $ x $ decreases negatively (for large negative values of $ x $): $ y $ increases towards 3.

### 3. **Domain and Range:**
   - **Domain:** The domain of the function is all real numbers since exponential functions are defined for all real $ x $. Thus, the domain is $ (-\infty, \infty) $.
   - **Range:** The range is determined by the horizontal asymptote and the direction in which the function moves. Since the graph approaches 3 but never reaches it, and since it decreases without bound as $ x $ increases, the range is $ (-\infty, 3) $.

### 4. **Plot the Graph:**
   - **x-values:** Choose a few key x-values (e.g., -2, -1, 0, 1, 2) and calculate the corresponding y-values to plot the curve.
   - **Graph Behavior:** The curve starts from below the asymptote at $ y = 3 $, passes through the point (0, 2), and continues decreasing as $ x $ increases.

Now, let's visualize the graph of this function:

I'll plot the graph for you:

```python
import numpy as np
import matplotlib.pyplot as plt

# Define the function y = -2^x + 3
def f(x):
    return -2**x + 3

# Create an array of x values
x = np.linspace(-3, 3, 400)
y = f(x)

# Plot the function
plt.figure(figsize=(8, 6))
plt.plot(x, y, label=r"$y = -2^x + 3$", color='blue')
plt.axhline(y=3, color='red', linestyle='--', label='Asymptote $y=3$')
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.title(r"Graph of $y = -2^x + 3$")
plt.xlabel("x")
plt.ylabel("y")
plt.ylim([-10, 5])
plt.xlim([-3, 3])
plt.legend()
plt.grid(True)
plt.show()
```

### Summary:
- **Domain:** $ (-\infty, \infty) $
- **Range:** $ (-\infty, 3) $

Would you like any more details or explanations?

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### Questions:
1. What is the effect of changing the base from 2 to 3 in the function $ y = -2^x + 3 $?
2. How does the graph of $ y = -2^x + 3 $ differ from $ y = 2^x + 3 $?
3. What happens if the function is $ y = -2^{-x} + 3 $?
4. How does the horizontal asymptote change if the constant term changes from 3 to 5?
5. What is the y-intercept of the function?
6. How would the graph change if the function were $ y = -2^x - 3 $?
7. How does the graph of $ y = -2^x + 3 $ relate to the graph of $ y = 2^x - 3 $?
8. What happens to the domain and range if the function is modified to $ y = -2^{(x-1)} + 3 $?

**Tip:** When analyzing exponential functions, always check for horizontal asymptotes, as they indicate the long-term behavior of the function.

Explore how to graph the exponential function y = -2^x + 3. Learn about its domain, range, and behavior through detailed step-by-step analysis. Suitable for Grades 9-12.

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