Algebra Applications: Quadratic Functions

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Title: Algebra Applications: Quadratic Functions

Title: Introduction to Quadratics

Narrator: Quarterbacks have a intuitive grasp of

quadratic functions.

The path of a football is a parabola, which is the graph

of a quadratic function.

The quarterback knows that for short passes a straighter path

is needed. In mathematical terms a short parabola with a

vertex not to high above the horizontal axis; whereas for

longer passes a more arcing path is needed that means a

parabolas with a higher vertex, but you don't have to be a

quarterback to understand quadratic functions.

All quadratic functions are of the form ax^2+ bx+c.

The graph of the quadratic function has the familiar

parabolic arc but it can have different locations and

orientations depending on the values of a b and c.

in this program you will explore different applications

of quadratic functions.

In the first segment you will explore the parabolic paths

of fireworks.

Given the preferred locations of three fireworks explosions,

what are the equations that model the arcs of the

pyrotechnic paths.

In the next segment we become accident investigators

determining how fast a car was moving based on the skid marks

it left on the road.

A quadratic function is helpful in determining

this result.

In the last segment you will explore a quadratic model for

growth data as we look at the change in height as a child

goes from birth to being a toddler.

This program will cover the following concepts in

quadratic functions: the standard form of a quadratic

equation, the roots of a quadratic function, the vertex

of a parabola, solving a quadratic equation, quadratic

regression, graphs of parabolas.

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Narrator: fireworks displays are dramatic,

colorful, and mathematical.

Although you may not be able to see the path of the

fireworks in the night sky, the real pyrotechnics happens

at the algebraic level.

The path of fireworks is a parabola and a parabola is the

graph of a quadratic function.

A quadratic function in standard form is written as

y=ax^2+bx+c.

And the shape of the graph is a parabola.

a b and c represents different numbers and each has

a different effect on the graph of the function.

The constant a affects the width of a parabola and

its orientation.

The constant c is the y intercept where the parabola

intersects the y axis and the constant b affects the

horizontal location of the parabola.

So now we have a blueprint for creating the path

of fireworks.

So let's create a fireworks display.

Let's restrict our graphs off the parabolas to the

first quadrant.

In fact if we let c= 0 so that the y intercept is at the

origin then the graphs will be a model of fireworks launched

from the origin.

Graphs of this sort have this kind of equation y=ax^2+bx.

This equation is easily factored into vertex form.

The advantage of this form is that the roots of the factors

show where the zeros of the function are located.

This corresponds to where the fireworks are launched and

where they would land if they didn't explode.

But how can you control how high the fireworks go?

We know that fireworks launched from the origin will

land at x equals negative b over a.

Midway between these two points is the vertex or

highest point on the parabola.

The x coordinate of the vertex is therefore x equals negative

b over two a, and the maximum height of the fireworks is the

y coordinate at x equals negative b over two a.

We now have all the tools we need to plan our

fireworks display.

So suppose you want to launch three volleys of fireworks.

The first volley explodes at point twenty comma 75.

The second volley explodes at twenty comma one hundred.

And the third volley explodes at twenty comma one twenty five.

The three points have the same x coordinate and so they are

on the same vertical line.

What are the equations of the three parabolas that describe

the paths of the three volleys?

Lets use the TI Nspire to find out.

Turn on the TI Nspire, create a new document; you may need

to save a previous document.

Create a graph window.

Change the windows settings to show the first quadrant.

Press MENU and under view select WINDOW SETTINGS.

Change x min to zero, x max to fifty, y min to

negative ten, and y max to one hundred fifty.

Hide the entry line by pressing CONTROL and G,

add points to the first quadrant by pressing MENU, and under

POINTS AND LINES selecting POINT.

Move the pointer to the middle of the x axis, from there move

up to place three points equally spaced and

align vertically.

Press ENTER to create each point.

Add coordinate labels to each point.

Press MENU and under ACTIONS select COORDINATES

AND EQUATIONS.

Use the nav pad to move the pointer above each

of the points.

Press ENTER once to display the coordinates and press

ENTER again to place the label on the screen.

Repeat for the other two points.

You'll see that the coordinates don't match the

desired coordinates.

Hover over the x coordinate for one of the points and

Press ENTER twice to edit the coordinates.

Change the x coordinate to twenty.

Press ENTER.

Changing the coordinates changes the location

of the point.

Repeat this process with the y coordinate.

Repeat with these coordinates, twenty, one hundred and

twenty, one twenty five.

The coordinates are now the correct ones.

We now need to find the three quadratic functions.

We know the generic form of the equation is y equals ax

times the quantity x plus b over a where b over a is the

nonzero root of the function which happens to be at x

equals forty.

We substitute the xy coordinates with the three

points and solve three separate equations.

In each case solving for a, we get three solutions: a equals

negative three over sixteen, a equals negative one fourth,

and a equals negative five over sixteen.

Now input each function.

Press the TAB key to go to the entry line for f one,

input the first function negative three sixteenths x

times the quantity x minus forty.

Press the DOWN ARROW and input the second function, negative

one fourth x times the quantity x minus 40, press the

DOWN ARROW again and input the third function, negative

five sixteenths x times the quantity x minus forty.

Press ENTER to see the three parabolas.

Enjoy the fireworks.

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Narrator: Driving fast is exciting, but the faster you

drive the longer it takes for your car to come to a stop.

This pull of tires on the road lasts longer for a faster

moving car.

But why does it take longer for a faster car to come to

a stop?

Two words, kinetic energy.

The faster a car moves the more kinetic energy it has.

For a car of mass m and speed v, the kinetic energy or the

energy of motion is K E equals one half M V squared.

This is a quadratic equation.

In order for the car to stop, work needs to be done equal to

the amount of kinetic energy.

Work is force times distance, and force is mass

times acceleration.

So the work needed to stop a moving car is mass times

acceleration times the stopping distance.

So mathematically when a vehicle is brought to a

complete stop the following equation is generated, one

half m v squared equals m times a times d.

Notice how the value of the mass divides out.

Solve for d the distance traveled as the car comes

to a stop.

This is the stopping distance d equals v squared over two a,

but this equation assumes that you instantaneously step

on the brakes, but there is usually a split second between

the time you need to stop, and the time you step on

the brakes.

During that time the car is still moving at speed v.

Assume that it takes two thirds of a second to react.

Our stopping distance equation becomes d equals v squared

over two a plus two thirds v.

Since car speed is usually expressed in miles per hour,

we now need to adjust the equation so that the results

are in feet, not miles.

Multiply by thirty-six hundred over fifty-two eighty, which

converts from miles per hour to feet per second, simplify

to get the new equation.

The values of a vary from four to five.

If you're bringing a car to a stop on a dry road use the

higher value.

The stopping distance for a wet rainy road uses the

lower value.

You'll see from the graphs that because stopping distance

is based on the square of the, speed the faster you go, the

farther the car skids to a stop.

Accident investigators have to work backward; they know the

breaking distance by measuring the skid marks on the road.

By measuring the skid marks and using the simplified

formula d equals v squared over two a, they can calculate

the speed of the car at the time the brakes were applied.

Then taking these results and using the full formula, the

total stopping distance is found.

Alternatively, if an investigator knows the total

stopping distance they can find the speed of the car by

solving the quadratic equation or v.

Suppose that at an accident scene the total stopping

distance for a car is determined to be two hundred

and fifty feet.

Was the car going over the speed limit of sixty-five

miles an hour at the time of the accident?

Lets use the TI Nspire to solve.

Turn on the TI Nspire, create a new document; you may need

to save a previous document.

Create a graph window, input this quadratic function at the

f1 function entry line; y equals zero point zero six

eight two x squared, plus zero point four five four five x.

This is the equation of speed vs. stopping distance on a

dry road.

Press the DOWN ARROW and input this quadratic function at the

F two function entry line.

Y equals zero point zero eight five two x squared plus zero

point four five four five x, this is the equation of speed

vs. stopping distance on a wet road.

Press ENTER to graph both functions.

Change the windows settings.

Press MENU and under WINDOWS ZOOM select

WINDOWS SETTINGS, change the settings to these

values, x min equals negative ten x max equals one hundred,

y min equals negative ten, y max equals four hundred.

Now graph y equals two hundred and fifty, this is the

stopping distance.

Press the TAB key to access the function entry line.

Access the F three function entry line, input two

hundred and fifty and Press ENTER.

Add an intersection point where each parabola crosses

the horizontal line, Press MENU and under POINTS AND

LINES select INTERSECTION POINT.

Move the pointer over one of the parabolas and Press ENTER,

then move the pointer to the horizontal line and press

enter again.

You will see the coordinates of the intersection point.

Repeat with the other parabola and the same horizontal line.

Afterward, press ESCAPE and if necessary move the coordinate

labels to a clear part of the screen.

The X coordinate of each intersection point represents

the speed a car was going that had a stopping distance of two

hundred and fifty feet.

From the coordinates of the intersections we see that on

either a dry road, or a wet road, the speed of the car was

less than sixty five miles per hour.

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Narrator: From the time you were born to the time you were

a toddler, your body goes through dramatic change.

Your height and weight increase at such a pace that

the growth is non linear.

A graph of age in months vs. height in centimeters clearly

shows non linear growth.

The center for disease control has developed growth charts

that doctors use to monitor the growth and development of

newborns and toddlers.

Why are these charts important?

They help identify how a child's development

is progressing.

Since infants and toddlers vary in size and weight, a

particular child's measurements fall into

different categories known as percentiles.

For example the fiftieth percentile where most

infants and toddlers fall, has this graph for boys.

The fifth percentile has this graph.

The graphs have the same shape, but one is higher than

the other, the one for the fiftieth percentile.

In each case, the shape of the graph is similar and

non linear.

In each case, a quadratic function can be found that

matches the data.

Lets use the statistics capabilities of the TI Nspire

to find quadratic models for these data sets.

Turn on the TI Nspire, create a new document; you may need

to save a previous document.

Create a list and spreadsheet window; input the following

CDC data, the data consists of age and height values for

infants in the fifth percentile.

The age is measured in months and the height is measured in

centimeters, start with column a, use the up arrow to move

the cursor to the very top, input the column heading age,

then add the column heading fifth at the top of

the column.

Starting at cell a one input the column a data pressing

ENTER after each entry.

Then start at cell b one and input the b column data, pause

the video to input the data into each column.

Once you have input the data into columns a and b move to

column e, add the column heading age two, to the very

top of the column as you did with column a, then add the

column heading fiftieth to the very top of the next column,

copy the contents of column a to column e, since the age

data is identical for this data set, move the cursor to

cell a one, hold the SHIFT key and press the DOWN ARROW to

highlight multiple cells, highlight the entire set of

data, press CONTROL and c to copy the contents of these

cells, move the cursor to cell e one and press CONTROL and v,

Input the data from the second column of this table to column

f, the age data has already been pasted to column e, the

data in column f is the height for the fiftieth percentile,

pause the video to input the data.

Once you have input all the data move the cursor to the

very top of column a, press the UP ARROW one more time

to highlight the entire column, hold the SHIFT key and press

the RIGHT ARROW to highlight column b.

Perform a quadratic regression on the data in columns a and

b, Press MENU, and under statistics select stat

calculations, and under that select quadratic regression.

At the dialog box tab your way to the okay button and

Press ENTER.

You will see the results of the regression in column

c and d.

Next, move the cursor to the top of column e and highlight

the column.

Then highlight column f, perform a quadratic regression

of the data in columns e and f.

Now, let's look at the graphs of the data points and the

regression curves.

Create a graph window, press CONTROL and I to insert a new

window, select graphs, first graph the scatter plot data,

Press MENU, and under GRAPH TYPES select SCATTER PLOT.

For scatter plot one labeled s one, input the column headings

you used in columns a and b for x and y.

Input age, press the DOWN ARROW and input fifth.

Then press the DOWN ARROW to go to s two.

Input age two for X and fiftieth for Y, then press

ENTER, to see the data Press MENU, and under WINDOW/ZOOM

select ZOOM DATA.

You should now see the two data sets and the outlines of

two parabolas.

Now overlay the regression equations, Press MENU and

under GRAPH TYPE select FUNCTION.

As part of the quadratic regression, the function that

corresponds to the scatter plot s one is in the f one

field so use the up arrow ton identify f one, press ENTER.

You should now see the function graph overlapping the

data for s one, press the TAB key to bring back the function

entry line, use the DOWN ARROW to locate f two, which is

where the function corresponding to s two is

located, press ENTER.

You should now have overlapping graphs for

both data sets.

The advantage of the function graphs is that they provide

information for all intermediate age values, using

the trace feature shows a continues set of coordinates.

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