 
Welcome back to speller tutorial services today
everyone will work with quadratic equations
And we know from previous videos that a quadratic equation is nothing more than a polynomial of degree 2, so let's go ahead and begin
The height of an hour can be modeled by this function, and we see this function is a quadratic because it's highest exponent is 2
where the value of x is the arrows horizontal distance in feet so let's begin with Part A
We need to find a horizontal distance the arrow travels before it hits the ground before we start the problem
Let's take a little let's take a take a look at a quick little sketch. Let's say this here is our
XY coordinate, and we just want to
Just freehand how this arrow is traveling with on the travel and a position something like this
Which we know that is just a basic parabola
Alright when an arrow is shot. We have the x axis which is here and this x axis
What is going to be the horizontal distance?
That this particular arrow travel what they're asking in this question is how part of that our travel before it hits the ground
We know that when the arrow hits the ground its vertical height has to be 0
So in this example we're going to set f of X or the value of y to 0 so let's again
Want to begin with our?
with our a polynomial
I'm just doing a basic rewrite here
and
I'm setting that equal to 0 we know that f of X is
equal to Y and in this case that vertical distance or that vertical height we want to set equal to 0
Next for us to find the roots or the place of that
Or the places where this parabola touches the x axis in this case we have to use the quadratic equation
quadratic equation should be memorized
But and some in some testing areas or depending on what's going on or your teacher or instructor?
They may give you a a formula sheet for all
Right well the quadratic equation is negative B. Plus or minus the square root of B squared minus
4a see
where all of that is divided by the expression 2a in
This particular example we need to grab the coefficients for meet one of our terms the full vision for an x squared term
corresponds to a
And that's negative
5,000
the coefficient for the X term is 2 and
Our constant here is fine
What we need to do at this point is to?
Is to substitute those values into our quadratic equation?
Alright, so let's start with the value of B
Which we said was 2?
But the formula says I mean even if you need to have a negative in front
2 squared is 4 and then we're going to multiply 4 times
The negative 5 thousands times C. Which is 5?
Then all of that will be divided by 3 times negative
5 thousands and we
Continue to simplify, and we end up with negative 2 plus or minus 4 plus
110 when you multiply negative 4 times negative 5 thousands times 5 you end up with a positive
110 and then all that needs to be divided by
Our denominator once I simplify or resolve this denominator. I end up with a negative
100 all right and want to follow my calculator and continue to resolve this I want to end up with two solutions
One solution is negative 2 and 4,800
hundreds and the other solution is
402 and
4,800 since we're talking about the distance traveled on a real object we know that distance travel is not going to be negative so we
Could disregard that
so we come to the conclusion that when that arrow was shot it travels a horizontal distance or an X distance of
402 and
4800 feet
Now depending on the circumstance either you need to do a check or your teacher allows you to use calculated
Let's take a look at the calculator and see how we can do this using one of the Texas, Texas Instruments calculator all right
Let's begin
Okay, let's pull up the calculator and check this one
First let's go and go to window we know that we're gonna have an x value of
400 over expecting an x value of 4 to 2 feet and 1400 so let's set the maximum value. Let's say to
450
Not quite sure I'm not quite sure how high the arrow traveled at this moment, so let's set the maximum height
So let's set that to 300
All right now let's go back to
let's go to y equals and now let's enter in this particular this particular function here, so it's
It was negative
5 thousandths x squared
Plus 2x
Plus 5 so that's entered and now let's go ahead and graph this particular
As you suspected we see that it is graphing in the shape of a parabola again, which is a quadratic equation?
Why we want to do now second and then press the trace button so we can get to these calculation function for these calc function
Now a function that we want to get to is number two
Which would be zero we want to calculate the zero, which just means tell us where this particular function crosses the x axis
We're not interested in this in this
Crossing position here because that's where the arrow is it shot from we're interested in the x value over here
So let's find our left now when we see that the calculator is asking for that here
We want to be
To the left of this position where whether we're the parabola crosses the x axis
Let's get close to it because that gives me
That one out of the calculation of the calculator to happen at a faster rate
That's our left down, so let's go ahead and press ENTER now notice the calculator wants to write down
Let's move to the right of that if our little
Ball a little spider if that moves off the screen we know we're to the right well there. We are to the right
Let's press ENTER again. It wants us to enter a guess
We don't want to enter this will let the calculator just begin its calculations between these two values here
and
we see that the calculator comes up with a value of 402 and
48 hundredths just like we calculated by hand so you did a good job there. Let's scroll up and let's take a look at
Part B of this particular question
Right for Part B. We need to find the maximum height of the arrow
Well to find the maximum height we need to find this y-value to tell us what's going on up here first
Let's calculated by hand and then we'll go back and check our work using the calculator
So we begin
By knowing that X will be is is
found by the expression
negative B over 2a
where V and a are the same values that we used earlier for the quadratic equation and
we know that to find the value of y
y would be taking a function and
inputting that x value
In which we're using the expression negative B over 2a
Let's begin by doing our substitutions
We said earlier that the value of B. Would be 2 again. We use the negative 2
That's in our in our expression, and we're going to multiply that by 2 times negative
5 thousands
Once we do that we see that X comes out to a value of
200 and now we're going to take this value of 200 and put that back into our original function so
f of 200
Be equal to negative
5 thousands
times
200 squared plus 2 times
200 plus 5 and again all I did here was go back to this original equation that we had and
I input values of 200
For each place where I see X and once I resolve that that's gonna. Give me the height or the y-value for
The arrow and in this case that's well come out
205 feet
Now let's go to the calculator and let's verify
That our maximum
That the maximum height of this arrow is 205 feet. Let's begin by pressing 2nd and
Then callaghan and this time we want to go down so for we could be the maximum value or the maximum
Again the calculator is asking for the left bound
So let's move over far left
That far left is possible
But we want to move anywhere task what I would consider the vertex or the maximum value which I would
Has to be somewhere in this region, so I want to move it out of somewhere around here or inside the little
little pointer on the function somewhere over there a
Little bit more all right, that's good enough for the left bound
Press ENTER, and then let's move it. Let's move this little spider
to the right-hand side
That's what I would consider to be the maximum
And then let's go down here and press ENTER again
A little too aggressive there with that for that dinner with that right arrow now, it's again calculator once again. Let's press ENTER
and we
See that the Y value does come out to be 205 feet just like we calculated by him
but this here was a quick tutorial on calculating the distance something travels and
Also the height using the quadratic equation and other relationships that we know we're dealing with
Parabolas or quadratic equations, thank you again for watching this video. Please friend us on Facebook
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