Question

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.

Explanation:

Step1: Determine Direction of Opening

The quadratic function is in the form \( h(x) = ax^2 + bx + c \), where \( a = 3 \), \( b = -12 \), \( c = 9 \). Since \( a = 3>0 \), the parabola opens upward.

Step2: Find Vertex and Axis of Symmetry

The x - coordinate of the vertex (and the equation of the axis of symmetry) is given by \( x=-\frac{b}{2a} \).
Substitute \( a = 3 \) and \( b=- 12 \) into the formula: \( x =-\frac{-12}{2\times3}=\frac{12}{6} = 2 \).
To find the y - coordinate of the vertex, substitute \( x = 2 \) into \( h(x) \):
\( h(2)=3\times(2)^2-12\times(2)+9=3\times4 - 24 + 9=12-24 + 9=-3 \).
So the vertex is \( (2,-3) \) and the axis of symmetry is \( x = 2 \).

Step3: Find y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into \( h(x) \):
\( h(0)=3\times(0)^2-12\times(0)+9 = 9 \). So the y - intercept is \( (0,9) \).

Step4: Find x - intercepts

Set \( h(x)=0 \), so \( 3x^2-12x + 9 = 0 \). Divide the entire equation by 3: \( x^2-4x + 3=0 \).
Factor the quadratic: \( (x - 1)(x - 3)=0 \).
Set each factor equal to zero: \( x - 1=0\) or \( x - 3=0 \). So \( x = 1 \) or \( x = 3 \). The x - intercepts are \( (1,0) \) and \( (3,0) \).

Step5: Find Two Other Points

Let's choose \( x = 4 \). Substitute \( x = 4 \) into \( h(x) \):
\( h(4)=3\times(4)^2-12\times(4)+9=3\times16-48 + 9=48-48 + 9 = 9 \). So the point is \( (4,9) \).
Let's choose \( x=-1 \). Substitute \( x=-1 \) into \( h(x) \):
\( h(-1)=3\times(-1)^2-12\times(-1)+9=3 + 12+9 = 24 \). So the point is \( (-1,24) \).

To sketch the graph:

1. Draw the coordinate plane.
2. Plot the vertex \( (2,-3) \), the axis of symmetry \( x = 2 \) (a vertical dashed line through \( x = 2 \)).
3. Plot the y - intercept \( (0,9) \), x - intercepts \( (1,0) \) and \( (3,0) \), and the other points \( (4,9) \) and \( (-1,24) \).
4. Since the parabola opens upward, draw a U - shaped curve passing through these points, symmetric about \( x = 2 \).

Answer:

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

(The graph is a parabola opening upward with the above - mentioned features plotted and connected.)