Question

write the equation of the parabola in standard form. solutions: {-5, 3} point: (2, 28)

Explanation:

Step1: Write factored - form of parabola

Since the solutions are \(x = - 5\) and \(x = 3\), the factored - form of the parabola is \(y=a(x + 5)(x - 3)\).

Step2: Find the value of \(a\)

Substitute the point \((2,28)\) into the equation \(y=a(x + 5)(x - 3)\). We get \(28=a(2 + 5)(2 - 3)\).
First, simplify the right - hand side: \((2 + 5)(2 - 3)=7\times(-1)=-7\).
So, \(28=-7a\). Solve for \(a\) by dividing both sides by \(-7\), \(a=-4\).

Step3: Expand the factored form to standard form

Substitute \(a = - 4\) into \(y=a(x + 5)(x - 3)\), we have \(y=-4(x + 5)(x - 3)\).
Expand \((x + 5)(x - 3)=x^{2}-3x + 5x-15=x^{2}+2x-15\).
Then \(y=-4(x^{2}+2x - 15)=-4x^{2}-8x + 60\).

Answer:

\(y=-4x^{2}-8x + 60\)