Question

1. consider the parabola ( y=(x - 2)(x - 6) ).

1a sketch the graph of the parabola.

Explanation:

Step1: Find the roots

Set \(y = 0\), then \((x - 2)(x - 6)=0\).
By the zero - product property \(x-2 = 0\) or \(x - 6=0\), so \(x=2\) and \(x = 6\). The roots (x - intercepts) are \((2,0)\) and \((6,0)\).

Step2: Find the axis of symmetry

The formula for the axis of symmetry of a parabola in factored form \(y=a(x - r_1)(x - r_2)\) is \(x=\frac{r_1 + r_2}{2}\). Here \(r_1 = 2\) and \(r_2=6\), so \(x=\frac{2 + 6}{2}=4\).

Step3: Find the vertex

Substitute \(x = 4\) into \(y=(x - 2)(x - 6)\). Then \(y=(4 - 2)(4 - 6)=2\times(-2)=-4\). The vertex is \((4,-4)\).

Step4: Find the y - intercept

Set \(x = 0\), then \(y=(0 - 2)(0 - 6)=12\). The y - intercept is \((0,12)\).

Answer:

Plot the points \((2,0)\), \((6,0)\), \((4,-4)\) and \((0,12)\) and draw a smooth parabola passing through these points. The parabola opens upwards (since the coefficient of \(x^{2}\) when expanded \(y=x^{2}-8x + 12\) is \(a = 1>0\)).