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

sketch the graph of the quadratic function $h(x) = 3x^2 - 12x + 9$ on the coordinate plane, ensuring to clearly label the vertex, axis of symmetry, x-intercepts, y-intercept, and at least two other distinct points that demonstrate the parabolas shape, and indicate whether it opens upward or downward.

sketch the graph of the quadratic function $h(x) = 3x^2 - 12x + 9$ on the coordinate plane, ensuring to clearly label the vertex, axis of symmetry, x-intercepts, y-intercept, and at least two other distinct points that demonstrate the parabolas shape, and indicate whether it opens upward or downward.

Answer

Explanation:

Step1: Determine the direction of opening

For a quadratic function (y = ax^{2}+bx + c), if (a>0), it opens upward. Here (a = 3>0), so the parabola opens upward.

Step2: Find the axis of symmetry

The formula for the axis of symmetry is (x=-\frac{b}{2a}). Given (a = 3) and (b=-12), then (x =-\frac{-12}{2\times3}=2).

Step3: Find the vertex

Substitute (x = 2) into (h(x)=3x^{2}-12x + 9). (h(2)=3\times(2)^{2}-12\times2 + 9=3\times4-24 + 9=12-24 + 9=-3). So the vertex is ((2,-3)).

Step4: Find the y - intercept

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

Step5: 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). Solve (x-1=0) or (x - 3=0). So (x=1) or (x = 3). The x - intercepts are ((1,0)) and ((3,0)).

Step6: Find two other points

Let (x=4), then (h(4)=3\times4^{2}-12\times4 + 9=3\times16-48 + 9=48-48+9 = 9). Let (x=-1), then (h(-1)=3\times(-1)^{2}-12\times(-1)+9=3 + 12+9=24). So two other points are ((4,9)) and ((-1,24)).

Answer:

  • The parabola opens upward.
  • Axis of symmetry: (x = 2).
  • Vertex: ((2,-3)).
  • y - intercept: ((0,9)).
  • x - intercepts: ((1,0)) and ((3,0)).
  • Two other points: ((4,9)) and ((-1,24)).