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

User Aepot
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1 Answer

5 votes

Answer:


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).

User Ravi Anand
by
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