To find the high or low point of the parabola given by the equation \( y = 3x^2 - 6x + 9 \), we can first compute the vertex of the parabola. Since the coefficient of \( x^2 \) is positive (\( a = 3 \)), the parabola opens upwards, meaning it has a low point (minimum).
The x-coordinate of the vertex (the low point) is given by the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( b = -6 \) and \( a = 3 \):
\[
x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1
\]
Now we can find the corresponding y-coordinate by substituting \( x = 1 \) into the equation:
\[
y = 3(1)^2 - 6(1) + 9
\]
Calculating this:
\[
y = 3(1) - 6 + 9 = 3 - 6 + 9 = 6
\]
Therefore, the low point of the parabola is:
\[
(1, 6)
\]
Now we can complete the table for the specified values of \( x \):
-
For \( x = -3 \):
\[
y = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54
\]
-
For \( x = -2 \):
\[
y = 3(-2)^2 - 6(-2) + 9 = 3(4) + 12 + 9 = 12 + 12 + 9 = 33
\]
-
For \( x = -1 \):
\[
y = 3(-1)^2 - 6(-1) + 9 = 3(1) + 6 + 9 = 3 + 6 + 9 = 18
\]
-
For \( x = 0 \):
\[
y = 3(0)^2 - 6(0) + 9 = 0 + 0 + 9 = 9
\]
-
For \( x = 1 \):
\[
y = 6 \text{ (previous calculation)}
\]
-
For \( x = 2 \):
\[
y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9
\]
Now, we can fill in the completed table:
| x | y |
|----|-----|
| -3 | 54 |
| -2 | 33 |
| -1 | 18 |
| 0 | 9 |
| 1 | 6 |
| 2 | 9 |
The low point of the parabola is:
\[
\text{The low point of this parabola is } (1, 6).
\]
