03.04 lc)
write the equation of the graph shown below in factored form.
a graph that starts at the top left and continues down through the x axis at negative four to a minimum around y equals negative two point eight and goes up to touch the x axis at negative two and then goes back down to a minimum around y equals negative zero point four and then goes back up to cross the x axis at negative one.

  y = (x +4)(x +2)²(x +1)

Step-by-step explanation:

The function has a generally U shape, so is of even degree. The zero where the graph touches, but does not cross, the x-axis is a zero of even multiplicity. The minimum degree of a function that matches this description is 4. It will have zeros at x=-4, -2 (multiplicity 2), and -1. The constants in the corresponding binomial factors will be the opposite of these values, so the factored form is ...

  y = (x +4)(x +2)²(x +1)

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The multiplicity of a zero is the power of the corresponding factor.

if the eqn of the parabola is f(x) = (x-4)(x+2), we need to mult. this out to find the coordinates of the vertex.

then f(x) = x^2 + 2x - 4x - 8, and f(x) = x^2 - 2x - 8.

one way in which you can find the vertex is to calculate x = -b / (2a).   here, that comes to

        )

x = = 1.   f(1) = 1^2 - 2(1) - 8 = - 9.   vertex is at (1, -9).

        2(1) 

find another point on the curve:   let x=0, for which y=-8.   then (0, -8) is on the curve.   plot the vertex and this point and draw a smooth parabola thru both.

hope this