Question

1. given that the general equation of a quadratic in factored form is ( f(x) = a(x - p)(x - q) ), give the equation of the graph below. assume ( a = \frac{1}{2} ) or ( -\frac{1}{2} ):

Explanation:

Step1: Identify x-intercepts

The graph crosses the x-axis at $x=2$ and $x=2$ (a repeated root at the vertex), so $p=2$, $q=2$.

Step2: Determine sign of $a$

The parabola opens upwards, so $a$ is positive. Given $a=\frac{1}{2}$ or $-\frac{1}{2}$, we choose $a=\frac{1}{2}$.

Step3: Substitute into factored form

Substitute $a=\frac{1}{2}$, $p=2$, $q=2$ into $f(x)=a(x-p)(x-q)$.
<Expression>
$f(x)=\frac{1}{2}(x-2)(x-2)$
</Expression>

Answer:

$f(x)=\frac{1}{2}(x-2)^2$