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Answer: See the image below for the filled out table.
The other root is x = -2
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Explanation:
The turning point is at (1, -45) which is the vertex. This is where the graph goes downhill, and then turns around to go uphill, or vice versa. Depending on the direction, the vertex is the lowest point or the highest point on the parabola.
We have (h,k) = (1,-45) as the vertex, so h = 1 and k = -45
y = a(x-h)^2 + k
y = a(x-1)^2 + (-45)
y = a(x-1)^2 - 45
Now plug in any other point from the table. You cannot pick (1,-45) or else you won't be able to solve for the variable 'a'. Let's go for (0,-40)
We'll plug x = 0 and y = -40 into the equation above to solve for 'a'
y = a(x-1)^2 - 45
-40 = a(0-1)^2 - 45
-40 = a(-1)^2 - 45
-40 = a - 45
a-45 = -40
a = -40+45
a = 5
Therefore, the equation for this parabola is
y = 5(x-1)^2 - 45
As a way to check, we can plug in something like x = -3 to find that...
y = 5(x-1)^2 - 45
y = 5(-3-1)^2 - 45
y = 5(-4)^2 - 45
y = 5(16) - 45
y = 80 - 45
y = 35
Which matches what the table shows in the first column. I'll let you verify the other columns. As you can probably guess at this point, we'll plug in the x values to get the corresponding y values.
So for x = -2, we get...
y = 5(x-1)^2 - 45
y = 5(-2-1)^2 - 45
y = 5(-3)^2 - 45
y = 5(9) - 45
y = 45 - 45
y = 0
The result of 0 here indicates we have a root at x = -2. This is the other x intercept. The x intercept already given to us was x = 4.
The rest of the table is filled out using the same idea. You should get what you see below.
