The function h of t equals
negative 0.23t squared plus 1.5t plus 19
where h of t is the height
in feet models the height
of an angry bird shot into the sky
as a function of time t in seconds.
We want to use this function
to answer the following questions.
Let's first graph this quadratic function
on the TI84 graphing calculator.
So for the first type we
want to enter the function,
though we'll use x instead of t.
So we'll press y equals.
To save some time I've already
entered the function y one,
let's go ahead and check it.
Negative 0.23x squared plus 1.5x plus 19.
Next we want to adjust the window
so we get a nice view of the graph.
Again, I've already
done this to save time.
If we press WINDOW, I've
changed the horizontal axis
to go from negative five to 20
and the vertical axis to go
from negative five to 25.
So if we press GRAPH, we have
a nice view of our function.
The first question is how high
above the ground was the bird
when it was launched?
When the bird is launched,
time t equals zero,
so we want to find the
function value h of zero.
On the calculator, we can do
this by pressing 2ND, TRACE
for the calculation menu,
select option one for value
and your next value of zero
which is really t equals
zero and press Enter.
Notice how this shows us
that the vertical intercept
is the point zero 19,
and therefore the initial
height was 19 feet.
Also, whenever we have
a quadratic function
in general formats we have here,
the vertical intercept is
always the point zero comma c,
where the quadratic
function is in the form
ax squared plus bx plus c.
And that should make sense,
because notice how h of zero
would just be zero plus zero plus 19,
giving us our initial height of 19 feet.
Next we're asked, after how many seconds
does the bird reach its highest point?
The next question is how high
is the angry bird at its highest point?
So we could answer both of these questions
by determining the vertex
of our quadratic function.
So going back to the
graph, the vertex should be
the coordinates of this high point.
So to find this point on the calculator,
we press 2ND, TRACE,
option four for maximum,
since the vertex is a high point.
So we press four.
Now it's going to ask us for
left bound and right bound.
We want to move the cursor
to the left of the vertex,
maybe somewhere in here, press Enter,
then for the right bound,
we move to the right of the vertex.
So we move to the right side of the vertex
by pressing the right arrow.
Maybe somewhere here,
Enter, and then Enter again.
And notice how the coordinates
of the vertex show below,
to two decimal places, the
vertex is 3.26 comma 21.45.
Remember the first coordinate or the input
would be the time it takes
to reach the highest point
and the output of the function value here
would be the actual
height at the high point.
So the bird reaches its highest
point after 3.26 seconds
and the bird is approximately 21.45 feet
above the ground at its highest point.
So 3.26 seconds
and a height of 21.45 feet.
If we wanted to find these values by hand,
and of course, we could use
our formula given here below
where time t would be equal to
negative b divided by two a,
and the height, or the
function output would be equal
to h of negative b divided by two a.
And if we did this, it
would look like this.
And notice how we get the
same result for the vertex.
Next we're asked, after how many seconds
does the angry bird hit the ground?
Well, we need to recognize here
that the bird hits the ground when h of t,
the height, is equal to zero.
To do this by hand we'd have
to use the quadratic formula.
We're going to do this graphically.
So we'll go back to our graph.
There's a couple ways of doing this.
We're going to use the
intersection method,
so we're going to press y
equals, press Enter at Y one,
and enter y two equals zero
which represents h of t equals zero.
And now when we graph this,
we're not going to see
the second function,
because it's actually right
along the horizontal axis.
But now because we have
the horizontal line,
h of t equals zero graphed,
we can find the coordinates
of this point here
where the angry bird hits the ground
using the intersection
feature of the calculator.
So we can press 2ND, TRACE,
option five for intersection,
and then simply press Enter three times,
Enter, Enter, Enter, and
notice how this tells us
that at time, approximately 12.92 seconds,
the height would be zero.
So, the bird hits the
ground after 12.92 seconds.
And the next question says
if the bird is traveling
at 18 feet per second, how
far does the angry bird travel
before it hits the ground?
Well, this is an
application of the formula
distance equals rate times time.
We know the bird is in
the air for 12.92 seconds,
so if the bird is traveling
at 18 feet per second,
the distance traveled,
d, is equal to the rate
which is 18 feet per second
times the time in seconds
which is 12.92.
So going back to the calculator,
let's go back to the home
screen by pressing 2ND, MODE,
and we have 18 times 12.92.
So the bird traveled
approximately 232.56 feet.
Now we still have two more questions.
We're asked to determine
the practical domain
and practical range of our function.
We want to express these
using interval notation.
So looking at the graph
of our quadratic function,
we found the vertical
intercept, we found the vertex,
and we found the horizontal intercept.
The practical domain is going to be
the set of all possible inputs,
which, in this case, would be
all possible values of time.
And the practical range is going to be
the set of all possible outputs,
in this case, all possible heights.
Of course, this is in feet.
And time is in seconds.
So looking at the graph, notice
how all the possible inputs
would be from zero all the way
to the horizontal intercept of 12.92.
So the practical domain
is the closed interval
from zero to 12.92.
Remember the square brackets indicate
that the endpoints are included.
And the practical range, or
the set of all possible outputs
would be from zero all the
way up to the highest point
of the vertex which is 21.45.
So the practical range
would be the closed interval
from zero to 21.45.
Okay, I hope you found this helpful.
