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 coefficient of \(x^2\) is \(3\), which is positive. So the parabola opens upward.

Step2: Find vertex and axis of symmetry

For a quadratic function \(ax^2 + bx + c\), the x - coordinate of the vertex (and axis of symmetry) is \(x=-\frac{b}{2a}\). Here, \(a = 3\), \(b=- 12\). So \(x =-\frac{-12}{2\times3}=\frac{12}{6} = 2\).
To find the y - coordinate of the vertex, substitute \(x = 2\) into \(h(x)\): \(h(2)=3\times(2)^2-12\times2 + 9=3\times4-24 + 9=12-24 + 9=-3\). So the vertex is \((2,-3)\) and the axis of symmetry is \(x = 2\).

Step3: Find x - intercepts

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

Step4: Find 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 two other points

Let's choose \(x=4\): \(h(4)=3\times(4)^2-12\times4 + 9=3\times16-48 + 9=48-48 + 9=9\). So the point is \((4,9)\).
Let's choose \(x=-1\): \(h(-1)=3\times(-1)^2-12\times(-1)+9=3 + 12+9=24\). So the point is \((-1,24)\).

To sketch the graph:

• Draw the axis of symmetry \(x = 2\) (a vertical line).
• Plot the vertex \((2,-3)\), x - intercepts \((1,0)\), \((3,0)\), y - intercept \((0,9)\), and the other points \((4,9)\), \((-1,24)\).
• Since the coefficient of \(x^2\) is positive, draw a parabola opening upward passing through these points.

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. When sketching, plot the points and draw the curve connecting them symmetric about \(x = 2\).