write an equation for a parabola with x - intercepts (-1,0) and (4,0) which passes through the point (1…

Explanation:

Step1: Write the factored - form of the parabola

Since the x - intercepts are $x=-1$ and $x = 4$, the factored form of the parabola is $y=a(x + 1)(x - 4)$.

Step2: Substitute the given point into the equation

Substitute $x = 1$ and $y=-18$ into $y=a(x + 1)(x - 4)$. We get $-18=a(1 + 1)(1 - 4)$.

Step3: Solve for $a$

First, simplify the right - hand side: $(1 + 1)(1 - 4)=2\times(-3)=-6$. So the equation becomes $-18=a\times(-6)$. Divide both sides by $-6$ to find $a = 3$.

Step4: Write the final equation of the parabola

Substitute $a = 3$ back into the factored form $y=a(x + 1)(x - 4)$. The equation is $y = 3(x + 1)(x - 4)=3(x^{2}-3x - 4)=3x^{2}-9x - 12$.

Answer:

$y = 3x^{2}-9x - 12$