This worksheet asks for graphing polynomial functions. It includes three specific functions:

1. $f(x) = x^3 + x^2 - 3$
2. $f(x) = 4 - x^3$
3. $f(x) = x^3 - x^2 + x - 1$

Let me provide step-by-step instructions and the graphs for each function:

1. $f(x) = x^3 + x^2 - 3$

• Identify key points:
  • For $x = 0$: $f(0) = -3$.
  • For $x = 1$: $f(1) = 1^3 + 1^2 - 3 = -1$.
  • For $x = -1$: $f(-1) = (-1)^3 + (-1)^2 - 3 = -3$.
  • For $x = 2$: $f(2) = 2^3 + 2^2 - 3 = 13$.
• Shape: Since the leading term is $x^3$, it has a cubic shape that falls to the left and rises to the right.

2. $f(x) = 4 - x^3$

• Identify key points:
  • For $x = 0$: $f(0) = 4$.
  • For $x = 1$: $f(1) = 4 - 1^3 = 3$.
  • For $x = -1$: $f(-1) = 4 - (-1)^3 = 5$.
  • For $x = 2$: $f(2) = 4 - 2^3 = -4$.
• Shape: Since the leading term is $-x^3$, it flips the typical cubic function, rising to the left and falling to the right.

3. $f(x) = x^3 - x^2 + x - 1$

• Identify key points:
  • For $x = 0$: $f(0) = -1$.
  • For $x = 1$: $f(1) = 1^3 - 1^2 + 1 - 1 = 0$.
  • For $x = -1$: $f(-1) = (-1)^3 - (-1)^2 + (-1) - 1 = -4$.
  • For $x = 2$: $f(2) = 2^3 - 2^2 + 2 - 1 = 5$.
• Shape: Similar cubic shape.

I can plot these functions for you. Let me generate graphs.import numpy as np import matplotlib.pyplot as plt

Define the functions

x = np.linspace(-3, 3, 500) # Values for x

f1 = x3 + x2 - 3 f2 = 4 - x3 f3 = x3 - x**2 + x - 1

Plot each function

plt.figure(figsize=(10, 8))

Plot f(x) = x^3 + x^2 - 3

plt.subplot(3, 1, 1) plt.plot(x, f1, label=r"$f(x) = x^3 + x^2 - 3$") plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.title("Graph of $f(x) = x^3 + x^2 - 3$") plt.legend() plt.grid()

Plot f(x) = 4 - x^3

plt.subplot(3, 1, 2) plt.plot(x, f2, label=r"$f(x) = 4 - x^3$", color='orange') plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.title("Graph of $f(x) = 4 - x^3$") plt.legend() plt.grid()

Plot f(x) = x^3 - x^2 + x - 1

plt.subplot(3, 1, 3) plt.plot(x, f3, label=r"$f(x) = x^3 - x^2 + x - 1$", color='green') plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.title("Graph of $f(x) = x^3 - x^2 + x - 1$") plt.legend() plt.grid()

Adjust layout and show

plt.tight_layout() plt.show() <<ImageDisplayed>>successHere are the graphs for the three polynomial functions:

1. $f(x) = x^3 + x^2 - 3$: This cubic function rises to the right and falls to the left with a distinct turning point.
2. $f(x) = 4 - x^3$: This inverted cubic function rises to the left and falls to the right.
3. $f(x) = x^3 - x^2 + x - 1$: A cubic function with multiple turning points, rising to the right and falling to the left.

Do you need further details on any specific graph or function?

Questions to Explore:

1. How can we find the exact turning points of these cubic functions?
2. What is the domain and range of each polynomial function?
3. How does the leading term of the polynomial influence the shape of the graph?
4. Can you verify the zeros of each polynomial graphically and algebraically?
5. What are the intervals of increase and decrease for each function?

Tip:

To analyze polynomial graphs effectively, always start by examining the degree and leading coefficient for general shape and behavior.