Question

write the equation of the parabola in factored form. (y = a(x - p)(x - q)) points: ((3,0),(-1,9)), and ((-10,0))

Explanation:

Step1: Identify the x - intercepts

The x - intercepts of the parabola are the values of x for which y = 0. Given the points (3,0) and (-10,0), we have p = 3 and q=-10 in the factored - form equation \(y = a(x - p)(x - q)\). So the equation becomes \(y=a(x - 3)(x + 10)\).

Step2: Substitute the non - x - intercept point

We substitute the point (-1,9) into the equation \(y=a(x - 3)(x + 10)\). When \(x=-1\) and \(y = 9\), we get \(9=a(-1 - 3)(-1+10)\).

Step3: Simplify the right - hand side

First, simplify \((-1 - 3)(-1 + 10)\): \((-1 - 3)=-4\) and \((-1 + 10)=9\). So the equation is \(9=a(-4)\times9\), which simplifies to \(9=-36a\).

Step4: Solve for a

Divide both sides of the equation \(9=-36a\) by - 36. We get \(a=-\frac{9}{36}=-\frac{1}{4}\).

Step5: Write the final equation

Substitute \(a =-\frac{1}{4}\) into \(y=a(x - 3)(x + 10)\). The equation of the parabola in factored form is \(y=-\frac{1}{4}(x - 3)(x + 10)\).

Answer:

\(y =-\frac{1}{4}(x - 3)(x + 10)\)