To graph the function \( f(x) = 3x^2 + 6x - 6 \) and plot its vertex, let's follow these steps:
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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 \( (-1, -9) \).
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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 \( (0, -6) \).
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Plotting: You can now plot the points:
- Vertex at \( (-1, -9) \)
- Additional point at \( (0, -6) \)
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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
- Mark and label the vertex \( (-1, -9) \).
- Mark the point \( (0, -6) \).
- 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.
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