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

Answer:

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

Step-by-step 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 $a$:

$\displaystyle a = \frac{37.8}{4.5\times 2.8} = 3$.

Hence the equation for this parabola:

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