Question

Graph the function f(x)=3x^2+6x–6.
Plot the vertex. Then plot another point on the parabola. If you make a mistake, you can erase your parabola by selecting the second point and placing it on top of the first.

graph gose 0-10 on x and y axis

Answer 1

To graph the function \( f(x) = 3x^2 + 6x - 6 \) and plot its vertex, let's follow these steps:

1. **Find the Vertex:**
The vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( a = 3 \) and \( b = 6 \).

\[
x = -\frac{6}{2 \times 3} = -\frac{6}{6} = -1
\]

Now we can find the \( y \)-coordinate of the vertex by substituting \( x = -1 \) into the original function:
\[
f(-1) = 3(-1)^2 + 6(-1) - 6 = 3(1) - 6 - 6 = 3 - 6 - 6 = -9
\]

So, the vertex is at \( (-1, -9) \).

2. **Choose Another Point:**
Let's calculate \( f(0) \) to find another point on the parabola:
\[
f(0) = 3(0)^2 + 6(0) - 6 = -6
\]
Therefore, another point on the parabola is \( (0, -6) \).

3. **Plotting:**
You can now plot the points:
- Vertex at \( (-1, -9) \)
- Additional point at \( (0, -6) \)

4. **Sketch the Parabola:**
Since the function opens upwards (as \( a = 3 > 0 \)), the graph will be a U-shaped curve. Using the vertex and the additional point, you can sketch the parabola.

Here’s a summary of the important points:
- Vertex: \( (-1, -9) \)
- Another point on the graph: \( (0, -6) \)

### Graph Visualization
1. Mark and label the vertex \( (-1, -9) \).
2. Mark the point \( (0, -6) \).
3. Draw the parabola opening upwards, passing through these points.

### Note:
Unfortunately, I can't draw the graph physically, but I recommend using graphing software or graph paper to visualize it based on the points provided. You can also use online graphing tools or graphing calculators to assist with this process.