https://file.aiquickdraw.com/mathbot/file/173030778493866hbk5y5.jpg

To graph the given parabola \( y = 3x^2 + 6x - 2 \), let's start by identifying key characteristics, including the vertex and additional points.

### Step 1: Rewrite in Vertex Form
Rewrite the equation \( y = 3x^2 + 6x - 2 \) to identify the vertex. This can be done by completing the square:

1. Factor out 3 from the \( x \)-terms:
   \[
   y = 3(x^2 + 2x) - 2
   \]
2. Complete the square inside the parentheses:
   - Half of 2 is 1, and \( 1^2 = 1 \).
   - Add and subtract 1 inside the parentheses:
     \[
     y = 3(x^2 + 2x + 1 - 1) - 2
     \]
   - Simplify:
     \[
     y = 3((x + 1)^2 - 1) - 2
     \]
     \[
     y = 3(x + 1)^2 - 3 - 2
     \]
     \[
     y = 3(x + 1)^2 - 5
     \]

Now, the equation is in vertex form \( y = 3(x + 1)^2 - 5 \), where the vertex is at \( (-1, -5) \).

### Step 2: Calculate Additional Points
We’ll find two points to the left and two points to the right of the vertex by substituting values of \( x \) around \( x = -1 \):

1. **At \( x = -3 \):**
   \[
   y = 3(-3)^2 + 6(-3) - 2 = 3(9) - 18 - 2 = 27 - 18 - 2 = 7
   \]
   So, the point is \( (-3, 7) \).

2. **At \( x = -2 \):**
   \[
   y = 3(-2)^2 + 6(-2) - 2 = 3(4) - 12 - 2 = 12 - 12 - 2 = -2
   \]
   So, the point is \( (-2, -2) \).

3. **At \( x = 0 \):**
   \[
   y = 3(0)^2 + 6(0) - 2 = -2
   \]
   So, the point is \( (0, -2) \).

4. **At \( x = 1 \):**
   \[
   y = 3(1)^2 + 6(1) - 2 = 3 + 6 - 2 = 7
   \]
   So, the point is \( (1, 7) \).

### Step 3: Plot the Points
You should plot these five points on the graph:

- Vertex: \( (-1, -5) \)
- Points to the left: \( (-3, 7) \) and \( (-2, -2) \)
- Points to the right: \( (0, -2) \) and \( (1, 7) \)

After plotting, connect these points to create the parabola shape.

Would you like further assistance with any part of this process?

---

### Related Questions

1. How do we find the axis of symmetry for a parabola?
2. What does completing the square achieve in terms of graphing parabolas?
3. Why is the vertex of a parabola important in graphing it?
4. What is the impact of the coefficient of \( x^2 \) on the parabola's shape?
5. How do we determine if a parabola opens upwards or downwards?

### Tip
For parabolas in the form \( y = ax^2 + bx + c \), you can use the formula \( x = -\frac{b}{2a} \) to find the \( x \)-coordinate of the vertex quickly.

Graph the parabola y = 3x^2 + 6x - 2 by plotting five points on the graph: the vertex, two points to the left of the vertex, and two points to the right of the vertex.

Learn how to graph the parabola y = 3x^2 + 6x - 2 by plotting the vertex and additional points. This problem includes concepts such as completing the square and vertex form of a quadratic equation, suitable for Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation