Question

1. write the equation of the parabola in vertex form: $y = a(x - h)^2 + k$.

vertex__________
equation__________
what are the coordinates of the roots?

Explanation:

Step1: Find the vertex

The vertex of a parabola is the highest or lowest point. From the graph, the vertex is at \((1, 1)\), so \(h = 1\) and \(k = 1\).

Step2: Determine the value of \(a\)

The parabola passes through the origin \((0, 0)\). Substitute \(x = 0\), \(y = 0\), \(h = 1\), and \(k = 1\) into the vertex form \(y = a(x - h)^2 + k\):
\(0 = a(0 - 1)^2 + 1\)
\(0 = a(1) + 1\)
\(a = -1\)

Step3: Write the vertex form equation

Substitute \(a = -1\), \(h = 1\), and \(k = 1\) into \(y = a(x - h)^2 + k\):
\(y = -1(x - 1)^2 + 1\) or \(y = -(x - 1)^2 + 1\)

Step4: Find the roots

The roots are the x - intercepts (where \(y = 0\)). From the graph, the parabola crosses the x - axis at \((0, 0)\) and \((2, 0)\). We can also solve \(0 = -(x - 1)^2 + 1\):
\((x - 1)^2 = 1\)
\(x - 1=\pm1\)
Case 1: \(x - 1 = 1\Rightarrow x = 2\)
Case 2: \(x - 1=-1\Rightarrow x = 0\)

Answer:

• Vertex: \((1, 1)\)
• Equation: \(y = -(x - 1)^2 + 1\)
• Coordinates of the roots: \((0, 0)\) and \((2, 0)\)