Please help me with this question, and please show your thinking

  y = (1/10)(x +4)(x +2)(x -3)²

Step-by-step explanation:

We observe that the function intercepts the x-axis at -4, -2, and +3. It crosses the x-axis at the first two of these intercepts, and touches the x-axis at x=3. x-axis intercepts correspond to factors (x -p), where "p" is the x-value of the intercept point.

A factor with odd degree will cause the function to change its sign as the x-value crosses the "p" value. If the degree is even, the sign of the function on either side of "p" is the same.

This means the factors (x+4) and (x+2) will have odd degree, and the factor (x-3) will have even degree. Since there are no "flat" spots* at the intercept points, we take the degrees to be the minimum possible values, 1, 1, and 2.

For some multiplier "a", the function can be written ...

  y = a(x +4)(x +2)(x -3)²

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Then for x=0, the value of the function is ...

  y = a(4)(2)(-3)² = 72a

We note that the graph appears to have a y-intercept of slightly more than 7, so an appropriate value of "a" is 1/10. That gives a y-intercept of 7.2, a good match to the graph.

Then the factored function is ...

  y = (1/10)(x +4)(x +2)(x -3)²

* When the degree of a factor is more than 1, the derivative of the function at the x-intercept will be zero. This means the graph will have a horizontal tangent ("flat spot") at the x-intercept. The higher the degree of the factor, the "flatter" the graph will be at the x-intercept.

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

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step-by-step explanation: