Question

Write the equation for a parabola that has x-intercepts (−4.5, 0) and (−2.8, 0) and y-intercept (0, 37.8).

Answer 1

`y = 3(x + 4.5)(x + 2.8)`.

Explanation:

Start with the two
`x`-intercepts. The two zeros of the quadratic equation for this parabola are:

• 
  `x_1 = -4.5`, and
• 
  `x_2 = 2.8`.

(These are the
`x`-coordinates of the two
`x`-intercepts.)

By the factor theorem,
`x = k` (where
`k` is a real number) is a zero of a polynomial if and only if
`(x - k)` is a factor of that polynomial.

A quadratic equation is also a polynomial. In this case, the two zeros would correspond to the two factors

• 
  `(x - (-4.5)) = (x + 4.5)`.
• 
  `(x - (-2.8)) = (x + 2.8)`

A parabola could only have up to two factors. As a result, the power of these two factor should both be one. Hence, the equation for the parabola would be in the form

`y = a \, (x + 4.5)(x + 2.8)`,

where
`a` is the leading coefficient that still needs to be found. Calculate the value of
`a` using the
`y`-intercept of this parabola. (Any other point on this parabola that is not one of the two
`x`-intercepts would work.)

Since the coordinates of the
`y`-intercept are
`(0,\, 37.8)`,
`x = 0` and
`y = 37.8`. The equation
`y = a \, (x + 4.5)(x + 2.8)` becomes:

`37.8 = a \, (0 + 4.5)(0 + 2.8)`.

Solve for
`a`:

`\displaystyle a = (37.8)/(4.5* 2.8) = 3`.

Hence the equation for this parabola:

`y = 3(x + 4.5)(x + 2.8)`.