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

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

Step2: Find the vertex and axis of symmetry

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

Step3: Find the y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into \( h(x) \): \( h(0)=3\times(0)^{2}-12\times0 + 9=9 \). So the y - intercept is \( (0,9) \).

Step4: Find the 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 into \( h(x) \): \( h(4)=3\times(4)^{2}-12\times4 + 9=3\times16-48 + 9=48-48 + 9=9 \). So the point is \( (4,9) \).
Let's choose \( x = 1 \) (we already know it's an x - intercept, but we can also use it as a point). Or we can 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:

• Plot the vertex \( (2,-3) \), the axis of symmetry \( x = 2 \) (a vertical dashed line).
• Plot the y - intercept \( (0,9) \) and its mirror image across the axis of symmetry \( x = 2 \). The mirror image of \( (0,9) \) is \( (4,9) \) (since the distance from \( x = 0 \) to \( x = 2 \) is 2, so we move 2 units to the right of \( x = 2 \) to get \( x=4 \)).
• Plot the x - intercepts \( (1,0) \) and \( (3,0) \).
• Plot the other point \( (-1,24) \) and its mirror image \( (5,24) \) (distance from \( x=-1 \) to \( x = 2 \) is 3, so move 3 units to the right of \( x = 2 \) to get \( x = 5 \), and \( h(5)=3\times25-60 + 9=75-60 + 9=24 \)).
• Since \( a>0 \), draw a parabola opening upward passing through these points.

Answer:

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

To sketch the graph, plot these points and draw a parabola opening upward with the vertex at \( (2,-3) \), symmetric about \( x = 2 \).