To graph the cubic function $y = x^3 - 2x^2 + x - 3$, we will follow these steps:

1. Identify key features:

• Intercepts:

  • Y-intercept: Set $x = 0$ to find $y$. $y = (0)^3 - 2(0)^2 + (0) - 3 = -3.$ So the y-intercept is at $(0, -3)$.

  • X-intercepts (Roots of the equation): Set $y = 0$, and solve $x^3 - 2x^2 + x - 3 = 0$.

2. Derivative for slope and critical points:

• To find where the function increases or decreases, we take the derivative $f'(x) = 3x^2 - 4x + 1$. Set $f'(x) = 0$ to find critical points.

3. Behavior at infinity:

• As $x \to \infty$, $y \to \infty$ (since the leading term is $x^3$).
• As $x \to -\infty$, $y \to -\infty$.

4. Plotting values:

• Pick a few values of $x$ (such as $x = -2, -1, 1, 2$) and calculate the corresponding $y$-values to sketch the curve.

I'll generate the graph for you now. numpy as np import matplotlib.pyplot as plt

Define the function y = x^3 - 2x^2 + x - 3

def f(x): return x3 - 2*x2 + x - 3

Generate x values from -3 to 3

x = np.linspace(-3, 3, 400) y = f(x)

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r'$y = x^3 - 2x^2 + x - 3$', color='blue')

Highlight the y-intercept (0, -3)

plt.scatter(0, -3, color='red', zorder=5) plt.text(0, -3, ' (0, -3)', verticalalignment='bottom', horizontalalignment='right')

Add labels and title

plt.axhline(0, color='black',linewidth=0.7) plt.axvline(0, color='black',linewidth=0.7) plt.title('Graph of y = x^3 - 2x^2 + x - 3') plt.xlabel('x') plt.ylabel('y') plt.grid(True) plt.legend()

Show the plot

plt.show()Here is the graph of $y = x^3 - 2x^2 + x - 3$. The key features include:

• The y-intercept at $(0, -3)$.
• The curve has a general cubic shape, with increasing values for large positive $x$ and decreasing values for large negative $x$.

Would you like further details or an analysis of the graph, such as finding the critical points or checking the roots?

Related Questions:

1. How can we find the roots of the cubic equation $x^3 - 2x^2 + x - 3 = 0$?
2. What is the meaning of critical points in the context of graphing functions?
3. How do the first and second derivatives help in determining the nature of the graph?
4. What is the end behavior of cubic functions?
5. How do you factor cubic polynomials?

Tip:

When graphing cubic functions, always check for local minima and maxima by finding critical points (where the derivative is zero) to better understand the turning points of the curve.