To graph the parabola [tex]\( y = 3x^2 - 12x + 9 \)[/tex] and plot the five specific points, follow these steps:
1. Identify the Vertex:
The formula for the x-coordinate of the vertex in a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is given by:
[tex]\[
x_{\text{vertex}} = \frac{-b}{2a}
\][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 9 \)[/tex]. Plugging in these values:
[tex]\[
x_{\text{vertex}} = \frac{-(-12)}{2 \cdot 3} = \frac{12}{6} = 2
\][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 2 \)[/tex] back into the equation:
[tex]\[
y_{\text{vertex}} = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3
\][/tex]
Thus, the vertex is [tex]\( (2, -3) \)[/tex].
2. Calculate Points to the Left of the Vertex:
Choose [tex]\( x \)[/tex] values 1 and 0, which are to the left of [tex]\( x = 2 \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 3(1)^2 - 12(1) + 9 = 3 - 12 + 9 = 0
\][/tex]
So, the point is [tex]\( (1, 0) \)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 3(0)^2 - 12(0) + 9 = 9
\][/tex]
So, the point is [tex]\( (0, 9) \)[/tex].
3. Calculate Points to the Right of the Vertex:
Choose [tex]\( x \)[/tex] values 3 and 4, which are to the right of [tex]\( x = 2 \)[/tex].
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 3(3)^2 - 12(3) + 9 = 27 - 36 + 9 = 0
\][/tex]
So, the point is [tex]\( (3, 0) \)[/tex].
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = 3(4)^2 - 12(4) + 9 = 48 - 48 + 9 = 9
\][/tex]
So, the point is [tex]\( (4, 9) \)[/tex].
4. Plot the Points:
The five points to be plotted on the graph are:
- Vertex: [tex]\( (2, -3) \)[/tex]
- Points to the left of the vertex: [tex]\( (1, 0) \)[/tex], [tex]\( (0, 9) \)[/tex]
- Points to the right of the vertex: [tex]\( (3, 0) \)[/tex], [tex]\( (4, 9) \)[/tex]
These points can be plotted on a coordinate plane, and then draw a smooth curve through these points to form the graph of the parabola.
Remember, a parabola is symmetric about its vertex. Thus, points on both sides of the vertex should mirror each other. This symmetry is reflected in the calculated points here.
