Question

1. a parabola with x - intercepts at 0 and 1 and which passes through the point (2, - 2)

Explanation:

Step1: Write the factored - form of the parabola

Since the x - intercepts are at \(x = 0\) and \(x = 1\), the factored form of the parabola is \(y=a(x - 0)(x - 1)=ax(x - 1)\).

Step2: Substitute the given point into the equation

Substitute \(x = 2\) and \(y=-2\) into \(y = ax(x - 1)\). We get \(-2=a\times2\times(2 - 1)\).

Step3: Solve for \(a\)

Simplify the right - hand side of the equation: \(a\times2\times(2 - 1)=2a\). So, \(2a=-2\), and then \(a=-1\).

Step4: Write the equation of the parabola

Substitute \(a=-1\) back into the factored form \(y = ax(x - 1)\), we get \(y=-x(x - 1)=-x^{2}+x\).

Answer:

\(y=-x^{2}+x\)