Question

write an equation for each parabola. see example 6 37. a parabola with x - intercepts at (-1, 0) and (3, 0) which passes through the point (1, -8)

Explanation:

Step1: Write the factored - form of the parabola

Since the x - intercepts are at $x=-1$ and $x = 3$, the factored - form of the parabola is $y=a(x + 1)(x - 3)$.

Step2: Substitute the given point into the equation

Substitute the point $(1,-8)$ into $y=a(x + 1)(x - 3)$. We get $-8=a(1 + 1)(1 - 3)$.

Step3: Solve for $a$

First, simplify the right - hand side of the equation: $a(1 + 1)(1 - 3)=a(2)(-2)=-4a$. So, $-8=-4a$. Divide both sides by $-4$ to find $a = 2$.

Step4: Write the equation of the parabola

Substitute $a = 2$ back into the factored - form $y=a(x + 1)(x - 3)$. We get $y=2(x + 1)(x - 3)$. Expand it: $y=2(x^{2}-3x+x - 3)=2(x^{2}-2x - 3)=2x^{2}-4x - 6$.

Answer:

$y = 2x^{2}-4x - 6$