Different between quadratic and polynomial? What is rule and standard to be a quadratic. Draw graph.

A quadratic equation is a specific type of polynomial equation. Here's a detailed comparison:

### Quadratic Equation:
- **Form**: A quadratic equation is a polynomial of degree 2. It has the form $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are constants, and $ a \neq 0 $.
- **Degree**: The highest power of the variable (usually $ x $) is 2.
- **Graph**: The graph of a quadratic equation is a parabola. It opens upwards if $ a > 0 $ and downwards if $ a < 0 $.

### Polynomial Equation:
- **Form**: A polynomial equation can have various forms depending on its degree. A general polynomial equation of degree $ n $ is $ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0 $, where $ a_n, a_{n-1}, \ldots, a_1, a_0 $ are constants, and $ a_n \neq 0 $.
- **Degree**: The degree of the polynomial is the highest power of the variable in the equation.
- **Graph**: The graph of a polynomial equation varies greatly depending on its degree and the coefficients. For example, cubic equations (degree 3) have an "S" shaped curve, quartic equations (degree 4) can have up to three turning points, and so on.

### Rules and Standards to be a Quadratic:
1. **Degree**: The equation must be of degree 2.
2. **Coefficient**: The coefficient of $ x^2 $ (i.e., $ a $) must be non-zero.
3. **Form**: It should fit the form $ ax^2 + bx + c = 0 $.

### Graph of a Quadratic Equation:
Let's consider the quadratic equation $ y = x^2 - 4x + 3 $.

I will now draw the graph of this quadratic equation.

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

# Coefficients of the quadratic equation y = ax^2 + bx + c
a = 1
b = -4
c = 3

# Generate x values
x = np.linspace(-1, 5, 400)
# Calculate y values
y = a * x**2 + b * x + c

# Plot the graph
plt.figure(figsize=(8, 6))
plt.plot(x, y, label='$y = x^2 - 4x + 3$')
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5)
plt.title('Graph of Quadratic Equation $y = x^2 - 4x + 3$')
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.legend()
plt.show()
```

This code will produce a graph of the quadratic equation $ y = x^2 - 4x + 3 $, showing the parabolic shape characteristic of quadratic functions.

Would you like more details or have any questions? Here are some questions you might find interesting:

1. How do you solve a quadratic equation using the quadratic formula?
2. What are the differences between a cubic and a quartic polynomial?
3. How do polynomial roots affect the graph of the polynomial function?
4. What are the methods to factorize a quadratic equation?
5. How can you determine the vertex of a parabola from its quadratic equation?

**Tip:** When analyzing a quadratic equation, always check the discriminant $ b^2 - 2ac $. It helps to determine the nature of the roots of the equation (real and distinct, real and equal, or complex).

Understand the difference between quadratic and polynomial equations with detailed explanations and graphical representations. Learn about the rules and standards to identify a quadratic equation, including the forms and graphs of these equations. Suitable for students in Grades 9-12.

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