sketch the graph of the quadratic function ( h(x) = 3x^{2}-12x + 9 ) on the coordinate plane, ensuring to…

Explanation:

Step1: Find the vertex

For a quadratic function (y = ax^{2}+bx + c), the (x) - coordinate of the vertex is (x=-\frac{b}{2a}). Here (a = 3), (b=-12), so (x =-\frac{-12}{2\times3}=2). Substitute (x = 2) into (h(x)): (h(2)=3\times2^{2}-12\times2 + 9=3\times4-24 + 9=12-24 + 9=-3). The vertex is ((2,-3)).

Step2: Find the axis of symmetry

The axis of symmetry is (x =-\frac{b}{2a}). Since (a = 3), (b=-12), the axis of symmetry is (x = 2).

Step3: Find the (x) - intercepts

Set (h(x)=0), so (3x^{2}-12x + 9=0). Divide through by (3): (x^{2}-4x + 3=0). Factor: ((x - 1)(x - 3)=0). Solutions are (x=1) and (x = 3). The (x) - intercepts are ((1,0)) and ((3,0)).

Step4: Find the (y) - intercept

Set (x = 0), then (h(0)=3\times0^{2}-12\times0+9=9). The (y) - intercept is ((0,9)).

Step5: Find two other points

Let (x=4), then (h(4)=3\times4^{2}-12\times4 + 9=3\times16-48 + 9=48-48+9 = 9). So the point is ((4,9)). Let (x=-1), then (h(-1)=3\times(-1)^{2}-12\times(-1)+9=3 + 12+9=24). So the point is ((-1,24)).

Step6: Determine the direction of opening

Since (a = 3>0), the parabola opens upward.

Answer:

• Vertex: ((2,-3))
• Axis of symmetry: (x = 2)
• (x) - intercepts: ((1,0)) and ((3,0))
• (y) - intercept: ((0,9))
• Other points: ((4,9)) and ((-1,24))
• Direction of opening: Upward.

You can then plot these points on the coordinate plane and draw a smooth parabola passing through them.