iv. What are the practical domain and range of the function as shown
on the graph?
v. What kind of function rule would you expect to provide a good
algebraic model of the relationship shown in the graph?
vi. What would be the theoretical domain and range of such
a function?
2
I nvesti
nvestigg ation
Designing Parabolas
Unit 5
In the Quadratic Functions unit of Course 1, you explored quadratic patterns
of change and developed skill in reasoning with the various representations
of those patterns—tables, graphs, and symbolic rules. In this investigation,
you will extend your understanding and skill in use of quadratic functions.
As an example, consider the type of problem often faced by architects.
Developers of a new Magic Moments restaurant were intrigued by the
design of a restaurant at the Los Angeles International Airport. In a meeting
with their architect, they showed her a picture of the airport restaurant and
asked if she could design something similar for them.
Like the airport structure, the Magic Moments restaurant was to be
suspended above the ground by two giant parabolic arches—each 120 feet
high and meeting the ground at points 200 feet apart. To prepare plans for
the restaurant building, the designers had to develop and use functions
whose graphs would match the planned arches.
One way to tackle this design problem is to imagine a parabola drawn on
a coordinate grid as shown below. Any parabola can be described as the
graph of some quadratic function.
160 y
(x, 120)
120
80
40
0
x
0
40
80
120
160
200
As you work on the problems of this investigation and the next, look for
answers to this question:
What strategies can be used to find functions that
model specific parabolic shapes?
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1
Using ideas from your earlier study of quadratic functions and their
graphs, write the rule for a function with parabolic graph that
contains points (0, 0), (200, 0), and a maximum point whose
y-coordinate is 120. Use the hints in Parts a–d as needed.
a. The graph of the desired function has x-intercepts (0, 0) and
(200, 0). How do you know that the graph of the function
f(x) = x(x - 200) has those same x-intercepts?
b. What is the x-coordinate of the maximum point on this graph?
c. Suppose that g(x) has a rule in the form g(x) = k[x(x - 200)], for
Unit 5
some particular value of k. What value of k will guarantee that
g(x) = 120 at the maximum point of the graph?
d. Write the rule for g(x) in equivalent expanded form using the
k value you found in Part c.
2
The logo chosen for Magic Moments continued the parabola theme
with a large letter M drawn using two intersecting parabolas. The idea
is shown in the next graph.
10 y
8
6
4
2
0
x
0
2
4
6
8
10
a. Modify the strategy outlined in Problem 1 to find a quadratic
function that will produce the leftmost parabola in the M.
• Start with a function f(x) that has x-intercepts at (0, 0) and (6, 0).
• Find coordinates of the maximum point on the graph.
• Find the rule for a related function g(x) that has the same
x-intercepts as f(x) but passes through the desired maximum point.
• Write the function rule for f(x) using an equivalent expanded
form of the quadratic expression involved.
b. Use a similar strategy to find a function g(x) that will produce the
rightmost parabola.
• Start with a function that has a graph with x-intercepts (4, 0)
and (10, 0).
• Then adjust the rule so that the function graph also passes
through the required maximum point.
• Finally, write the function rule using an equivalent expanded
form of the quadratic expression involved.
LESSON 1 • Quadratic Functions, Expressions, and Equations
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Your solutions to Problems 1 and 2 illustrate important and useful
connections between rules and graphs for quadratic functions. Work on
Problems 3, 4, and 5 will develop your understanding and skill in using
those ideas.
3
Explain why the next diagram does or does not show the graph of
f(x) = (x - 3)(x + 1).
Unit 5
y
(3, 0) x
(-1, 0)
(0, -3)
(1, -4)
4
Use reasoning alone to sketch graphs of the following functions. Label
these key points with their coordinates on the graphs:
• x-intercept(s),
• y-intercept, and
• maximum or minimum point.
Then check results of your reasoning using a graphing tool.
a.
b.
c.
d.
e.
5
f(x) = (x + 3)(x - 1)
f(x) = (x - 1)(x - 3)
f(x) = (x + 1)(x + 5)
f(x) = -2(x + 2)(x - 3)
f(x) = 0.5(x - 6)2
Write rules for quadratic functions whose graphs have the following
properties. If possible, write more than one function rule that meets
the given conditions.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
x-intercepts at (4, 0) and (-1, 0)
x-intercepts at (7, 0) and (1, 0) and graph opening upward
x-intercepts at (7, 0) and (1, 0) and minimum point at (4, -10)
x-intercepts at (-5, 0) and (0, 0) and graph opening downward
x-intercepts at (3, 0) and (-5, 0) and maximum point at (-1, 8)
x-intercepts at (3.5, 0) and (0, 0) and graph opening upward
x-intercepts at (4.5, 0) and (1, 0) and y-intercept at (0, 9)
x-intercepts at (m, 0) and (n, 0)
only one x-intercept at (0, 0)
only one x-intercept at (2, 0) and y-intercept at (0, 6)
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Summarize
the Mathematics
In this investigation, you explored the ways in which factored forms, graphs, and
intercepts are related for quadratic functions.
a How can the factored expression of a quadratic function be used to locate the
x-intercept(s) and y-intercept of its graph?
Unit 5
b How can the factored expression of a quadratic function be used to locate the maximum
or minimum point of its graph?
c
How can the x-intercept(s), y-intercept, and maximum or minimum point of a parabola be
used to write a rule for the corresponding quadratic function?
Be prepared to share your ideas, strategies, and reasoning with the class.
Check Your Understanding
Use your understanding of connections between rules and graphs for
quadratic functions to complete the following tasks.
a. Use reasoning alone to sketch graphs of these functions. Label
x-intercepts, y-intercepts, and maximum or minimum points with their
coordinates. Then check your sketches with a graphing tool.
i. f(x) = x(x + 6)
ii. f(x) = -(x + 3)(x - 5)
iii. f(x) = 3(x - 2)(x - 6)
b. Find rules for quadratic functions with graphs meeting these conditions.
In any case where it is possible, write more than one rule that meets the
given conditions.
i. x-intercepts at (4, 0) and (-1, 0) and opening upward
ii. x-intercepts at (2, 0) and (6, 0) and maximum point at (4, 12)
iii. only one x-intercept at (-3, 0) and y-intercept at (0, 18)
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