How to Determine if Quadratic Equations are Correct

When working with quadratic equations, especially in vertex form, it’s essential to understand how transformations affect their graphs. The vertex form of a quadratic equation is given by:

**y = a(x - h)² + k**

In this formula, **(h, k)** represents the vertex of the parabola, while **a** determines the direction and shape of the graph. Let's break down the transformations that affect the graph:

1. **Horizontal Shifts:**
- The term **(x - h)** shifts the graph to the right by **h** units. Conversely, **(x + h)** moves it to the left by **h** units.

2. **Vertical Shifts:**
- The **+k** in the equation raises the graph by **k** units, while **-k** lowers it by **k** units.

3. **Vertical Stretch/Compression & Reflection:**
- The coefficient **a** affects the graph’s steepness. If **|a| > 1**, the graph stretches vertically, making it narrower. If **0 < |a| < 1**, it compresses, resulting in a wider graph.
- If **a < 0**, the graph reflects over the x-axis, flipping it upside down.

Let’s examine three provided equations to assess their correctness based on these principles.

### Equation A: y = (x + 1)² + 5
This equation indicates a shift to the left by 1 unit (due to **x + 1**) and a vertical shift up by 5 units. The analysis confirms that it is indeed correct.

### Equation B: y = -3x²
Starting with the basic quadratic function **y = x²**, we reflect it over the x-axis (which gives **y = -x²**) and then stretch it vertically by a factor of 3 (resulting in **y = -3x²**). This transformation is accurate, confirming the correctness of this equation as well.

### Equation C: y = (0.5x - 3)²
To analyze this equation, we note that the coefficient 0.5 suggests a vertical compression (since **0 < 0.5 < 1**), and the term **(x - 3)** indicates a shift to the right by 3 units. However, the expression needs further clarity in terms of its intended transformation description.

In general, understanding these transformations allows students to verify and create quadratic equations accurately. Quadratics are prevalent not only in algebra but also in various real-world applications, such as physics for projectile motion and economics for profit maximization. Mastering these concepts will empower students in their mathematical journey, enhancing their problem-solving skills and confidence in math-related subjects.