Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
Quadratic and Square Root Functions
Activity:
Square Roots & Quadratics: What’s the Connection?
TEKS:
(2A.9) Quadratic and square root functions. The student formulates
equations and inequalities based on square root functions, uses a variety
of methods to solve them, and analyzes the solutions in terms of the
situation.
The student is expected to:
(G) connect inverses of square root functions with quadratic functions.
Overview:
In this activity, students will use their graphing calculators to investigate
the connections between square root functions and quadratic functions.
Students should be able to write the equations of functions by using
transformation of functions. Students should also be familiar with the
graph of quadratic and square root functions.
Materials:
Square Roots & Quadratics: What’s the Connection Handout -1 per
student
Transparencies of the activity
Graphing calculators
Map colors
Grouping:
2 or 3 students
Time:
1 class period
Lesson:
Procedures
1. Divide students into groups of 2 or 3
and distribute the activity worksheets,
calculators, and map colors.
Notes
Each student should receive his or her
own set of worksheets. Each student
needs to choose two different map color
pencils for the activity.
2. Have students complete the activity
worksheets in their groups. Allow 20 to
30 minutes for students to work in their
groups.
Circulate among the groups as they work
to ensure that students remain on task and
to answer questions as they arise.
You may need to go over how to find the
DrawInv function on their calculators and
give guidance on finding an appropriate
window to view both the function and its
inverse. Refer to the Calculator Notes for
Teachers pages at the end of this lesson if
you need help with the DrawInv function
on the calculator.
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Algebra II
Page 1
Mathematics TEKS Refinement 2006 – 9-12
Procedures
Tarleton State University
Notes
Detailed notes are written for examining
problem i.
You may want to have groups compare
and discuss the first graph and their
results to see that they are on the right
track, or you may just want to walk the
entire class through the analysis of the first
graph.
Guide students to make their sketches
with enough detail to be able to identify
ordered pairs especially the x- and yintercepts of both graphs. Many students
will try to copy the graph without marking
either axis’s units or labeling any ordered
pairs. They will need these details for their
work.
3. Have the groups share their results,
and hold a class discussion of the
observations that they have made
through this investigation.
Be sure to carefully guide the discussions,
especially the results from (4) and (5).
Discuss how to restrict the domain of a
quadratic function in order to have an
inverse that is a function.
This is a good opportunity to introduce the
term one-to-one function, and illustrate
how, without a restriction on the domain, a
quadratic function does not have an
inverse that is a function.
Emphasize that a square root function is
only the inverse of a quadratic function if
the domain is restricted appropriately (to
one side of the vertex); likewise, a
quadratic function is not the inverse of a
square root function unless the domain is
appropriately restricted.
At the end of the activity, you can assist
students in deducing a procedure on how
you can find the equation of an inverse of
a function (by switching the domain and
range values of a function).
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Algebra II
Page 2
Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
Homework:
Assign appropriate homework from the text, or provide some square
root and quadratic functions and ask students to find their inverses.
Extensions:
As an extension you can introduce the notation f −1 ( x ) .
You could also have them draw in the line y = x and explore the
relationship between it and the graphs of a function and its inverse.
Students could derive the geometric definition of inverse.
Resources:
This activity was adapted from some investigations in Discovering
Advanced Algebra, by Jerald Murdock, Ellen Kamischke, & Eric
Kamischke, published by Key Curriculum Press.
A good website for graph paper is
http://www.mathematicshelpcentral.com/graph_paper.htm. Teachers
may want to use a different template for the graph paper they use on
this activity, and this is a good place to find other options.
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Algebra II
Page 3
Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
Square Roots & Quadratics: What’s the Connection?
Several square root and quadratic functions are given below:
i.
f (x ) = (x + 6) + 2
ii.
f (x ) = 2 + x + 5
iii.
f ( x ) = −2 x
iv.
f ( x ) = .25 x 2
v.
f ( x ) = −6 + x − 2
vi.
f (x ) = (x − 2) − 5
2
2
1. For each of these functions, do the following:
a. Graph y1 = f(x) on your calculator; then use the DrawInv function
(under the DRAW menu) to draw the inverse of f on your calculator.
Sketch both of these graphs on the graph grids provided (see
following page) using one color to sketch the function and another
color for its inverse. Make sure you label at least three points on both
graphs.
b. Determine whether the inverse is or is not a function. (Remember that
a function passes the vertical line test.) Find the domain of the
inverse, and write the domain on the lines provided to the right of each
grid. Find an equation or equations for the inverse graphed and write
them on the lines to the right of the grid. Verify your response by
graphing the function in y1 and graphing your equation of the inverse
in y2 on your calculator.
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Algebra II
Page 4
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Algebra II
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Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
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Mathematics TEKS Refinement 2006 – 9-12
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Algebra II
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Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
v.
Mathematics TEKS Refinement 2006 – 9-12
Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
2.
Study your sketches and equations. What observations can you
make?
3.
Find the coordinates of the x-intercepts of each function above; then
find the y-intercepts of the inverse. What to you notice?
4.
Pair the functions given in (i) – (vi) then explain your rationale for
pairing them in this way.
5.
Are any of these functions inverses of each other? Justify your
answer.
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Algebra II
Page 7
Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
Teacher Solutions:
Square Roots & Quadratics: What’s the Connection?
Several square root and quadratic functions are given below:
i.
ii.
iii
f (x ) = (x + 6) + 2
2
f (x ) = 2 + x + 5
f ( x ) = −2 x
iv.
f ( x ) = .25 x 2
v.
f ( x ) = −6 + x − 2
f (x ) = (x − 2) − 5
2
vi.
1. Blue graphs are functions. Red graphs are inverses.
i.
ii.
Inverse is not a
function
Domain : x ≥ 2
Inverse
y = −6 ± x − 2
iii.
iv.
Inverse is a
function
Domain : x ≤ 0
Inverse :
f ( x ) = .25 x 2
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Inverse is not
a function
Domain : x ≥ 0
y = ±2 x
Algebra II
Page 8
Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
v.
vi.
Inverse is a
function
Domain : x ≥ −6
f ( x ) = ( x + 6) 2 + 2
b.
Determine whether the inverse is or is not a function. (Remember that a
function passes the vertical line test.) Find the domain of the inverse, and write
the domain on the lines provided on the right of each grid. Find an equation or
equations for the inverse graphed and write them on the lines to the right of the
grid. Verify your response by graphing the function in y1 and graphing your
equation of the inverse in y2 on your calculator.
When students write the equations for the inverse, if the inverse is a function
students can use function notation to write its equation. If the inverse is not a
function, then the equation needs to be written using y = .
2. Study your sketches and equations. What observations can you make?
Answers will vary.
• They may notice that a parabola’s inverse is not a function, unless you look at
only a portion of its original graph.
• The inverse of a square root function is always a function and is only one half of
a parabola.
• Students may also notice that the pattern in the coordinates of the inverse. Some
of its points are related to the original function. In the original function a point has
coordinates (x, y) and its inverse contains a point with coordinates (y, x)
3. Find the coordinates of the x-intercepts of each function above; then find the yintercepts of the inverse. What to you notice?
Answers will vary. Some possible answers.
• If the graph contains an x-intercept (a,0), then its inverse contains a yintercept (0,a).
• If the original function does not have an x-intercept, its inverse does not have
a y-intercept.
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Algebra II
Page 9
Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
4. Pair the functions given in (i) – (vi) then explain your rationale for pairing them in this
way.
Answers may vary.
i & v, ii & vi, and iii and iv.
5. Are any of these functions inverses of each other? Justify your answer.
Answers may vary.
If you look at the equations of the inverses for each square root function, they are
half of a parabola function. If you look at the equations of the inverses for the
parabolas, they are a pair of square root functions that each represents a different
half of the original parabola graph.
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Algebra II
Page 10
Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
Calculator Notes to Get Started ( Problem i)
Key strokes to draw the inverse. You have to input Y1 first.
The calculator just plots the points for the inverse of the function in Y1. It does not find
the equation of the inverse.
Students can input equations in Y2 that
they think is the inverse to see if it
matches the inverse drawn by the
calculator.
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Students can change the way the
calculator graphs their inverse function so
that can compare their equation’s graph
with the drawing of the inverse.
Algebra II
Page 11
Mathematics TEKS Refinement 2006 – 9-12
Tarleton State University
Students can see that this equation only matches half of the inverse. This inverse
cannot be described by just one function. You may want to ask students what to try to
find the equation for the bottom half of the function.
Ask students why they think this happens. (From the drawing of the inverse of the
original function students should have answered that it is not a function because it fails
the vertical line test. This is the reason why the inverse cannot be expressed by a
single function.)
The inverse can be expressed as y = ± x − 2 − 6 .
Before students graph the next function, students need to clear the drawing of the
inverse that is currently on the screen by doing the following
Quadratic and Square Root Functions
Square Roots & Quadratics: What’s the Connection?
Algebra II
Page 12