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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) = x2+3x-7 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 x-intercepts, 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 x-intercepts and y- intercept.
Clearly label each .
Compute the vertex.

A quadratic function is defined by a second-degree polynomial in one variable of the form
f(x) = ax2+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 y-coordinate)

Where does this value occur?___ (the x-coordinate)

What is the domain of your function? (Remember that this is the set of x-values that may be
plugged into your equation.)____

What is the range of your function? (Remember that this is the set of y-values that may be
obtained from the x-values that are plugged into your equation.)___

******************************************************************************
Enter the information to let f(x) = -x2-5x-2.

Select Graph.

Show your algebraic work here to compute the x-intercepts, 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.

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?___

******************************************************************************
Let f(x) = 2x2-6x+10.

Select Graph.
Show work to determine the y-intercept and vertex. Show algebraic work to explain why there
are no x-intercepts. 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 x-intercept.

x-intercept: ( ___, 0)

x-intercept: (___ , 0)

Describe the real life meaning or insight to the problem that the x-intercepts 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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