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

Answers

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, andx_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).

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Step-by-step explanation:

y=3(x+4.5)(x+2.8) or  y=3x^2+21.9x+37.8

Step-by-step explanation:

we know that

The equation of a vertical parabola in factored form is equal to

y=a(x-x_1)(x-x_2)

where

a is a coefficient

x_1 and x_2 are the roots or x-intercepts of the quadratic equation

In this problem we have

x_1=-4.5\\x_2=-2.8

substitute

y=a(x+4.5)(x+2.8)

Find the value of a

we have the y-intercept (0,37.8)

substitute the value of x and the value of y of the y-intercept in the equation and solve for a

37.8=a(0+4.5)(0+2.8)

a=37.8/12.6\\a=3

so

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

Convert to expanded form

y=3(x^2+2.8x+4.5x+12.6)

y=3x^2+21.9x+37.8



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