Graph the equation $$ y=5 x^{2} $$

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UpStudy AI · Accepted Answer

Sure! Let's explore the graph of the quadratic equation $$ y = 5x^{2} $$.

### **1. Understanding the Equation**

The equation $$ y = 5x^{2} $$ represents a **parabola**. Here's a breakdown of its characteristics:

- **Standard Form:** $$ y = ax^{2} + bx + c $$
  
  In this case:
  - $$ a = 5 $$
  - $$ b = 0 $$
  - $$ c = 0 $$

- **Vertex:** The lowest or highest point of the parabola.
  
  For $$ y = 5x^{2} $$, the vertex is at **(0, 0)**.

- **Axis of Symmetry:** A vertical line that passes through the vertex.
  
  Here, it's the **y-axis** or $$ x = 0 $$.

- **Direction:** Since $$ a = 5 > 0 $$, the parabola **opens upwards**.

- **Width:** The coefficient $$ a = 5 $$ makes the parabola **narrower** compared to the standard parabola $$ y = x^{2} $$.

### **2. Plotting Key Points**

Let's calculate some points to help visualize the graph:

| $$ x $$ | $$ y = 5x^{2} $$ |
|---------|------------------|
| -3      | 45               |
| -2      | 20               |
| -1      | 5                |
| 0       | 0                |
| 1       | 5                |
| 2       | 20               |
| 3       | 45               |

### **3. Graph Description**

- **Shape:** A symmetric U-shaped curve (parabola) centered at the origin.
- **Narrowness:** The larger coefficient (5) makes it steeper compared to $$ y = x^{2} $$.
- **Points:** Passes through the origin and points (1,5), (-1,5), (2,20), (-2,20), etc.

### **4. Visual Representation**

While I can't display images directly, I can provide you with a simple ASCII representation:

```
       |
      45|          *       *
      40|                    
      35|                    
      30|                    
      25|                    
      20|      *             *
      15|
      10|
       5|    *         *
       0|----*----*----*----*----*
         -3   -2   -1    0    1    2    3
```

*Note: This is a simplistic representation. For a more accurate graph, consider using graphing software or plotting the points manually.*

### **5. Plotting with Python (Optional)**

If you're familiar with Python, you can use the `matplotlib` library to plot the graph:

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

# Define the function
def f(x):
    return 5 * x**2

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

# Create the plot
plt.figure(figsize=(8,6))
plt.plot(x, y, label='y = 5x²', color='blue')

# Adding key features
plt.scatter([0], [0], color='red')  # Vertex
plt.axhline(0, color='black', linewidth=0.5)  # X-axis
plt.axvline(0, color='black', linewidth=0.5)  # Y-axis

# Labels and title
plt.title('Graph of y = 5x²')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid(True)
plt.show()
```

**Steps to Plot:**

1. **Install Matplotlib:** If you don't have it installed, use `pip install matplotlib`.
2. **Run the Code:** This script will display the graph of $$ y = 5x^{2} $$ with key features highlighted.

### **6. Summary**

- **Equation:** $$ y = 5x^{2} $$
- **Type:** Parabola opening upwards
- **Vertex:** (0, 0)
- **Axis of Symmetry:** $$ x = 0 $$ (y-axis)
- **Width:** Narrower than $$ y = x^{2} $$ due to the coefficient 5
- **Key Points:** (±1, 5), (±2, 20), (±3, 45), etc.

Feel free to use graphing tools like [Desmos](https://www.desmos.com/calculator) or the Python script above to visualize the parabola more precisely!
The graph of $$ y = 5x^{2} $$ is a narrow, upward-opening parabola with its vertex at (0,0). It passes through points like (1,5), (-1,5), (2,20), and (-2,20).

Graph the equation \( y=5 x^{2} \)

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