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:

pythonCopy code
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.