Answer:
see the attachments for the graph(s)
- y = -1/6(x -3)^2 +6
- y = -1/6(x +3)(x -9)
- y = -1/6x^2 +x +9/2
Step-by-step explanation:
1) The point at (3, 6) is on the vertical line that is halfway between the zeros at x=-3 and x=9, so it represents the vertex of the function. That knowledge, with any of the other points, lets you write the vertex form of the equation.
y = a(x -3)^2 +6
Using the point (0, 4.5), we can find the value of 'a':
4.5 = a(0 -3)^2 +6
-1.5 = 9a
-1.5/9 = a = -1/6
So, the vertex form of the equation is ...
y = -1/6(x -3)^2 +6
A graph of this is shown in the attachment.
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2) Now that we know the leading coefficient is -1/6, we can write the equation in "intercept form" (factored form) as ...
y = -1/6(x +3)(x -9)
In this form, each zero (p) gives rise to a factor (x-p).
The second attachment shows the graph of this.
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3) We can also write the equation in standard form, by expanding the one in (2) above:
y = -1/6(x^2 -6x -27)
y = -1/6x^2 +x +9/2
The third attachment shows the graph of this.
