To complete the table of values for the quadratic equation \( y = 3x^2 - 6x + 9 \), we first need to determine the vertex, which will give us the highest or lowest point of the parabola.
Step 1: Find the Vertex
The vertex of a parabola given by the equation in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
In this equation, \( a = 3 \) and \( b = -6 \).
\[ x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \]
Now, substitute \( x = 1 \) back into the equation to find \( y \):
\[ y = 3(1^2) - 6(1) + 9 \] \[ y = 3(1) - 6 + 9 \] \[ y = 3 - 6 + 9 = 6 \]
So, the vertex (the lowest point, since \( a > 0 \)) is at \( (1, 6) \).
Step 2: Create a Table of Values
Let's choose some values of \( x \) around the vertex to complete the table:
| \( x \) | \( y = 3x^2 - 6x + 9 \) | |---------|----------------------| | -1 | \( 3(-1)^2 - 6(-1) + 9 = 3 + 6 + 9 = 18 \) | | 0 | \( 3(0)^2 - 6(0) + 9 = 0 + 0 + 9 = 9 \) | | 1 | \( 6 \) (the vertex) | | 2 | \( 3(2^2) - 6(2) + 9 = 12 - 12 + 9 = 9 \) | | 3 | \( 3(3^2) - 6(3) + 9 = 27 - 18 + 9 = 18 \) |
So the completed table of values is:
| \( x \) | \( y \) | |---------|----------| | -1 | 18 | | 0 | 9 | | 1 | 6 | | 2 | 9 | | 3 | 18 |
Summary
- The vertex of the parabola \( y = 3x^2 - 6x + 9 \) is at the point \( (1, 6) \). This is the lowest point of the parabola.
- The parabola opens upwards (since \( a = 3 > 0 \)).
So, the low point is at \( (1, 6) \), and since it's an upward-opening parabola, there is no high point in the traditional sense.
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