Hello, Alley,
Please note that I assume that assume the data tabkle should read "Year/Number of people who moved to another state," and not "from" another state.
I suggest using DESMOS to plot the equation as well as the individual data points. Plot the equation y=0.03x2-0.63x+4.26, where x is the number of years after 1990. y is the millions of people that moved.
For example, x for 1995 would be 5 (1995 - 1990). You'll find a parabola, and the individual points from the table loosely (not exactly) fall on that line.
Using this graph, you will find a minimum at around x = 10.5, with a y value of 0.952. (0.952 million people moved 10.5 years after 1990, or 2000.5).
Plug 2000.5 into the original equation to find the millions of people that moved at that time:
A more precise method to find the minimum is to note that the slope of the parabola transitions from a negative value from x=0 to 10, to a positive value above x = 11. The point at which the parabola is a minimum is 0, as it goes from negative to positive slope. The first derivative of an equation provides the slope for any point x on the curve. So we can take the first derivative of the parabolic equation and set it equal to zero. The resulting value of x will be the years since 1990:
y=0.03x2-0.63x+4.26
y' = 0.06x - 0.63
We set value of y' (the slope) equal to zero, the minimum:
0 = 0.06x - 0.63
x = 10.5
This matches our visiual analysis of the graph, so we can confirm the value of 10.5 years after 1990 as the point that a minimum of people moved to another. Use x = 10.5 in the original equation to find the number of people who moved at that point in time.
I get y = 0.952 million people moved. To the nearest tenth would make it 1 million people.
Bob
