Question

Write the equation of the parabola that passes through the points (
–
2,0), (1,
–
4), and (3,0).
Write your answer in the form y=a(x–p)(x–q), where a, p, and q are integers, decimals, or simplified fractions.

Answer 1

Answer:

$y=\dfrac{2}{3}(x+2)(x-3)$

Step-by-step explanation:

The factored form of a quadratic function is:

$y = a(x - p)(x - q)$

where a is the leading coefficient, and p and q are the zeros.

The zeros of a quadratic function are the values of x where the function equals zero, representing the points where the graph intersects the x-axis. In other words, they are the x-coordinates of the points where the y-coordinate is zero.

As the equation of the parabola passes through points (-2, 0) and (3, 0), then the zeros of the function are x = -2 and x = 3.

Substitute these values for p and q in the factored formula:

$y = a(x - (-2))(x - 3)\\\\\\y = a(x +2)(x - 3)$

To find the value of a, substitute the other point (1, -4) into the equation:

$-4 = a(1 +2)(1 - 3)\\\\\\-4 = a(3)(-2)\\\\\\-4=-6a\\\\\\a=\dfrac{-4}{-6}\\\\\\a=\dfrac{2}{3}$

Therefore, the equation of the parabola that passes through points (-2, 0), (1, -4) and (3, 0) in the form y = a(x - p)(x - q) is:

$\Large\boxed{\boxed{y=\dfrac{2}{3}(x+2)(x-3)}}$