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Quadratic Functions
About this Laboratory
A quadratic function in the variable x is a polynomial where the highest power
of x is 2. We will
explore the domains, ranges, and graphs of quadratic functions. After completing
this laboratory,
you should be able to graphically approximate and algebraically determine such
features as the
intercepts, vertex, and shape of a quadratic function.
******************************************************************************
Enter the numerical coefficients of f(x) = x^{2}+3x7 into the boxes for
f(x) = .
Select Graph. You will see the graph of the function. If possible, you
will see the function in
factored form and the coordinates of the vertex. The factored form will allow
you to easily
determine the intercepts if they exist.
Show your algebraic work here to compute the xintercepts, y intercept and
vertex. If
necessary, refer to your notes or book and clearly label each result.
Algebraically determine the
intercepts (as decimals to hundredths ) and the vertex ( coordinates as improper
fractions ) of your
function.
Sketch your function here:
Compute the xintercepts and y intercept. Clearly label each . 

Compute the vertex. 
A quadratic function is defined by a seconddegree
polynomial in one variable of the form
f(x) = ax^{2}+bx+c. The parabola will open up when there is a minimum
functional value and it
will open down when there is a maximum functional value. If the leading
coefficient is positive ,
there will be a minimum functional value. If the leading coefficient is
negative, there will be a
maximum functional value.
Which do you have a maximum or minimum value associated with the function?___
What is this value? ___(the ycoordinate)
Where does this value occur?___ (the xcoordinate)
What is the domain of your function? (Remember that this is the set of xvalues
that may be
plugged into your equation.)____
What is the range of your function? (Remember that this is the set of yvalues
that may be
obtained from the xvalues that are plugged into your equation.)___
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Enter the information to let f(x) = x^{2}5x2.
Select Graph.
Show your algebraic work here to compute the xintercepts, yintercept and
vertex. If necessary,
refer to your notes or book and clearly label each result. Algebraically
determine the intercepts
(as decimals to hundredths) and the vertex (coordinates as improper fractions)
of your function.
Compute the intercepts and vertex. Clearly lab el each . 
Sketch your function here: 
Which do you have a maximum or minimum value associated with the function?___ What is this value?___ Where does this value occur?___ What is the domain of your function?___ What is the range of your function?___ 
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Let f(x) = 2x^{2}6x+10.
Select Graph.
Show work to determine the yintercept and vertex. Show algebraic work to
explain why there
are no xintercepts. Give the domain, range, and axis of symmetry. If necessary,
refer to your
notes or book and clearly label each result. Algebraically determine the
intercepts (as decimals
to hundredths) and the vertex (coordinates as improper fractions) of your
function.
Clearly label each . 
CP2**************************************************************************
Use what you have learned about quadratic equations to solve the following
problem:
A rain gutter is to be made of a metal sheet that is 10 inches wide by turning
up the edges 90°.
What depth will provide the maximum cross sectional area to allow the most water
flow? What
is the maximum cross sectional area?
Let x be the depth of the gutter and sketch a picture of the gutter with the
length and depth
labeled in terms of x.
Develop a formula for cross sectional area by letting x
define the unknown quantity of depth that
is to be the independent value.
f(x) =___
The order of the following steps is important:
Select Draw Gutter in the applet. Fill in the rectangle to set the sheet
width in the applet as 10
and press Enter on the keyboard, then choose inches as the unit being
used. Select Refresh.
Point the arrow to the model and drag until you have depths that are
approximately equal to the
values listed in the table below. Select Capture Data Point to choose and
then record the area
values by filling in the table below.
depth (in.)  Area (in.^{2}) 
0  
1  
2  
3  
4  
5 
What happens to the cross sectional area as the depth size increases; give a complete answer.
Select Plot Data Points and then enter the area expression from the previous page into the equation box. Select Plot Function. Sketch a “complete graph” showing such things as intercepts and turning points. (Label the axes with appropriate words such as area, volume, ...) 
What does the x variable represent in this
example?___
What does the f(x), value represent in this example?___
What is the domain of the function?___
(Without respect to the problem.)
What is the range of the function?___
(Without respect to the problem.)
What x values make sense for the problem we are solving?___
These values make up the restricted domain. (Hint: Some values for x and y do
not make sense
for the box problem. Make a guess and come back to check this answer later.)
What y values make sense for the problem we are solving?___
These values make up the restricted range. (Hint: Some values for x and y do not
make sense
for the box problem. Make a guess and come back to check this answer later.)
Solve to find the exact value of each xintercept.
xintercept: ( ___, 0)
xintercept: (___ , 0)
Describe the real life meaning or insight to the problem that the xintercepts
provide; that is, how
do the coordinates relate to the numbers found in your gutter model and the
restricted domain?
Solve to find the exact coordinates of the vertex. (
___,___ ).
Describe the real life meaning of the vertex coordinates.
What depth will provide the maximum cross sectional area
to allow the most water flow?
(Include dimensional units such as inches, centimeters, square inches, ...)
___
What cross sectional dimensions will provide the maximum area to allow the most
water flow?
(Include dimensional units such as inches, centimeters, square inches, ...)
___
What is the maximum cross sectional area? (Include dimensional units such as
inches,
centimeters, square inches, ...)
___
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