Pre-statistics. Using the quadratic
formula to solve quadratic equations
part five" lesson objective. Example 12 -
the number of children with autism who
are enrolled in special education
classes has increased greatly in recent
years, see Table seven.
Possibly due to redefine diagnosis and
public awareness, so we have data from
the years 1995 to 2003 and from this
data we see that the number in the
thousands is increasing each year. Let n
be the number in thousands of special
education students with autism at t
years since 1995. A linear model of the
situation is n= L (t) = 12t+ 14.
A quadratic model of the
situation is n equals Q(t) =
t-square + 4t + 22: a use a graphing
calculator to draw the graphs of both
models and in the same viewing window a
scatter plot of the data which model
describes the situation better? To put
the data in the calculator, I go to STAT
and number one for edit or hit enter I
have the number of years since 1995 in
list 1 and then let's do I have the
dependent variable the number of
students in special education with
autism. To see the scatterplot, I go
second y= which is stat plot,
currently the stat plots turned off. I
hit enter,
enter again to turn it on and we can see
the scatter plot is on, list 1 is our X
list list 2 is our Y list and then I
hit zoom and number 9 for Zoomstat. Now
this could be modeled by a linear or
quadratic we're gonna try both and see
which one according to the scatter plot
is doing a better job so I'll go to the
y equals I first put the linear function
in L(t)= 12t plus 14
a graph
and we can see that the linear model is
doing a pretty good job of modeling that
data let's look at the quadratic. So we
go back to the y=  . . . x squared
+ 4t+ 22
graphed
it appears according to the scatterplot
that the quadratic is doing a better job
of modeling this data.  Part B determine
which model predicts the larger number
of special education students with
autism for the years 2000 to 2016. tTo
answer this problem, we're going to use
our table command, second and graph, we want
the years for 2000 to 2016,
so 2000 would be t would be equal to 5, Y1is the linear model it's estimating
74,000 and Y2 is the quadratic it's
estimating 67,000, so for 2000 and also
for 2001 when t is 6 the linear model is
larger, after 2006 Y2 is larger than Y1.
 
So, to answer the question, we would say
for the year 2000 and 2001, the linear
model is estimating a larger number,
after that the quadratic up to 2016, is
estimating the larger number.
Part C, predict the number of special
education students with autism in 2012.
2012,  t is 17, 17 years beyond 1995, so to
do this algebraically, we start with our
quadratic model, and everywhere we see a
t we substitute 17, and we get 289 plus
68 plus 22 which equals 379. To interpret
this number we would say according to
the quadratic model in 2012 there were
three hundred and seventy nine thousand
special education students with autism.
Part D, estimate when there were forty
three thousand special education
students with autism. In this case we are
given Q(t) equals forty-three we
substitute that into our quadratic model
and the first step to solve this
algebraically, we would subtract 43 to both sides. To put this in
standard form, so now we have 0= 
t squared plus forty minus twenty one.
Using the quadratic formula we have a = 1, b=4, and c = -21
Substituting the
values of a, b, and c, in the quadratic
formula, we now have -4 plus or
minus the square root of b square, or 4
squared, which is 16 plus 84, and that
would be 4 times 1 times a negative 21,
and if I subtract a negative number
that becomes plus, divided by, 2 times 1, which is 2.  16
+ 84 is 100, now we take the square root
of 100, and we get 10, so we have negative
4 plus or minus 10 divided by 2. So
splitting up the plus and the minus, we
have negative 4 plus 10 divided by 2,
which becomes 6 divided by 2, which is 3.
And negative 4 minus 10 divided by 2
which becomes negative 14 divided by 2
which is negative 7. So we would
interpret this according to the
quadratic model, there were 43,000
special ed students with autism in both
1988 is when t is - 7 and in 1998
that's when tT was equal to 3. Verifying
those two parts using our graphing
calculator, we see that when t or in this
case x, was 17, we had 379 for Y2 and if
we go
back, when t was was 3 we have 43 for
our Y2, and if we go back to
-7,
and again Y2 is = 43 so we have
verified both parts of the question.
Thanks for watching.
