Question

1. given the graph below, identify the properties in the chart below. 5

\

$$\begin{tabular}{|l|l|} \\hline direction of opening & \\\\ \\hline vertex & \\\\ \\hline number of x-intercepts & \\\\ \\hline x-intercepts & \\\\ \\hline equation of axis of symmetry & \\\\ \\hline \\end{tabular}$$

1. rewrite the following quadratic function \\(y = 3x^2 - 6x + 1\\) in the vertex form using the method of completing square. 3

Explanation:

Analyze the graph properties

We observe the parabola's features:

• It opens upward.
• The lowest point (vertex) is at \((3, -2)\).
• It crosses the x-axis at two points: \((2, 0)\) and \((4, 0)\).
• The vertical line of symmetry passes through the vertex: \(x = 3\).

Complete the square for the function

We rewrite \(y = 3x^2 - 6x + 1\):

• Factor out the leading coefficient \(3\) from the variable terms:

$$y = 3(x^2 - 2x) + 1$$

• Complete the square inside the parentheses by adding and subtracting \(( -2/2 )^2 = 1\):

$$y = 3(x^2 - 2x + 1 - 1) + 1$$

• Expand and simplify:

$$y = 3(x - 1)^2 - 3 + 1$$

$$y = 3(x - 1)^2 - 2$$

Answer:

Question 5

Property	Value
Vertex	\((3, -2)\)
Number of x-intercepts	2
X-Intercepts	\((2, 0)\) and \((4, 0)\)
Equation of axis of symmetry	\(x = 3\)

Question 6

$$y = 3(x - 1)^2 - 2$$