Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
1
TASK 3.5.1: CURVES AHEAD — TEACHER VERSION
Solutions
1. Given a function y = x 2 ! 4x ! 12 . Graph this function rule and identify its zeros.
zeros: x = -2, 6
2.
a. Design a story problem involving the area of a rectangle that would
depend upon this function and its zeros.
Answers will vary. An example solution might be similar to the following: Sadie is
designing a rectangular garden that has an area of 12 square feet. One side is 4 feet
shorter than the length of the other side. Find the dimensions of the garden.
b. Design a story problem involving the area of the triangles that would
depend upon this function and its zeros.
Answers will vary. An example solution might be
similar to the following: Toby is designing a patio.
He wants to include a bowtie accent piece in the
center of the patio (see picture). If the bowtie is
composed of two equilateral triangles whose height is
four feet less than the length of the base and the area
of the bowtie is 12 square feet, find the dimensions of
the equilateral triangles.
3. The function that you were given in (1) was copied incorrectly. It was supposed
to be y = x 2 ! 4x ! 21 . Adjust your story problems accordingly. How do your
original story problems have to change to match the new rule?
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
2
Story 1: Sadie is designing a rectangular garden that has an area of 21 square feet. One
side is 4 feet shorter than the length of the other side. Find the dimensions of the garden.
Story 2: Toby is designing a patio. He wants to include a bowtie accent piece in the
center of the patio. If the bowtie is composed of two equilateral triangles whose height is
four feet less than the length of the base and the area of the bowtie is 21 square feet, find
the dimensions of the equilateral triangles.
In both stories, only the area had to be changed.
4. The function that you were given in (3) was also copied incorrectly. It was
supposed to be y = x 2 ! 4x ! 5 . Adjust your story problems accordingly. How
do your original story problems have to change to match the new rule?
Story 1: Sadie is designing a rectangular garden that has an area of 5 square feet. One
side is 4 feet shorter than the length of the other side. Find the dimensions of the garden.
Story 2: Toby is designing a patio. He wants to include a bowtie accent piece in the
center of the patio. If the bowtie is composed of two equilateral triangles whose height is
four feet less than the length of the base and the area of the bowtie is 5 square feet, find
the dimensions of the equilateral triangles.
As in (3) only the areas had to be altered.
5. Graph the three functions on the same
axis. What do you notice about the
graphs? Include the vertices and yintercepts and relate them to the function
rules.
Parabola (2) is parabola (1) shifted down 9
units. Parabola (3) is parabola (1) shifted up 7
units. This can be seen by simply noting that
the axes of symmetry of each of the parabolas is
the line y=2. The y-intercept is clearly revealed
by the graph. The C is the y-intercept of each
respective rule. The vertices are not clearly
revealed by the functions. We would need to
write the function rules in the vertex form
y = a(x ! h)2 + k before the vertices can be
easily seen. Or, if you have the line of symmetry, you know the x-coordinate of the
vertices will be 2, so solve the rule for y using 2 for x.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
3
6. In general, what effect does changing the C in the standard form of the quadratic
equation y = Ax 2 + Bx + C have on the graph?
The C determines the vertical placement of the parabola. Changing C will shift the
graph upward or downward.
7. Suppose the original function y = x 2 ! 4x ! 12 is changed again, this time to
y = 2x 2 ! 4x ! 12 . Modify your story problems to match it and discuss what
changed.
Story 1: Sadie is designing a rectangular garden that has an area of 12 square feet. One
side is 4 feet shorter than twice the length of the other side. Find the dimensions of the
garden.
Story 2: Toby is designing a patio. He wants to include a bowtie accent piece in the
center of the patio. If the bowtie is composed of two equilateral triangles whose height is
four feet less than twice the length of the base and the area of the bowtie is 12 square
feet, find the dimensions of the equilateral triangles.
This time the areas do not change but the length of one of the sides of the rectangle in
story 1 and the length of the height of the triangle in story 2 do.
8. Again the function rule changes to y = 3x 2 ! 4x ! 12 . Modify your story
problems and discuss what changed.
Story 1: Sadie is designing a rectangular garden that has an area of 12 square feet. One
side is 4 feet shorter than three times the length of the other side. Find the dimensions of
the garden.
Story 2: Toby is designing a patio. He wants to include a bowtie accent piece in the
center of the patio. If the bowtie is composed of two equilateral triangles whose height is
four feet less than three times the length of the base and the area of the bowtie is 12
square feet, find the dimensions of the equilateral triangles.
Again, the areas do not change but the length of one of the sides of the rectangle in story
1 and the length of the height of the triangle in story 2 do.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
4
9. Graph the last two function rules on the same coordinate plane with the original.
What do you notice about the graphs? Include the vertices and y-intercepts and
relate them to the function rule.
The parabolas are becoming “skinnier”
with each change in the function rule, i.e.
for each x-value, the corresponding yvalue of graph (8) is greater than the
corresponding y-value of graph (7) which
is greater than the corresponding y-value
of our original graph. . The y-intercept is
clearly revealed by the graph. The C is the
y-intercept of each respective rule. The yintercept of each of the graphs remains (0,
12) for all three parabolas. (Why?) The
vertices are not clearly revealed by the
function rules. We would need to write
the function rules in the vertex form
y = a(x ! h)2 + k before the vertices can
be easily seen.
10. In general, what effect does
changing the A in the standard form of the quadratic equation
y = Ax 2 + Bx + C have on the graph?
For A>0, as A increases, the resulting parabolas become skinnier meaning that for each
x-value, the corresponding y-values will be greater than for the original A.
11. Suppose the original function y = x 2 ! 4x ! 12 is changed once again, this time to
y = x 2 ! 5x ! 12 . Modify your story problems accordingly. How do your
original story problems have to change to match the new function?
Story 1: Sadie is designing a rectangular garden that has an area of 12 square feet. One
side is 5 feet shorter than the length of the other side. Find the dimensions of the garden.
Story 2: Toby is designing a patio. He wants to include a bowtie accent piece in the
center of the patio. If the bowtie is composed of two equilateral triangles whose height is
five feet less than the length of the base and the area of the bowtie is 12 square feet, find
the dimensions of the equilateral triangles.
Again, the length of one of the sides of the rectangle in story 1 and the length of the
height of the triangle in story 2 do.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
5
12. For a final time, the function changes and is now y = x 2 ! 11x ! 12 . Modify your
story problems and discuss what changed.
Story 1: Sadie is designing a rectangular garden that has an area of 12 square feet. One
side is 11 feet shorter than the length of the other side. Find the dimensions of the
garden.
Story 2: Toby is designing a patio. He wants to include a bowtie accent piece in the
center of the patio. If the bowtie is composed of two equilateral triangles whose height is
eleven feet less than the length of the base and the area of the bowtie is 12 square feet,
find the dimensions of the equilateral triangles.
Again, the areas do not change but the length of one of the sides of the rectangle in story
1 and the length of the height of the triangle in story 2 do.
13. Graph the last two functions on the same coordinate planewith the original. What
do you notice about the graphs?
Include the vertices and yintercepts and relate them to the
function.
This time the parabolas are not changing
size, i.e. the parabolas are not becoming
skinnier or fatter. They are however
moving to the right and in a downward
direction. The y-intercept is clearly
revealed by the graph. The C is the yintercept of each respective rule. The yintercept of each of the graphs remains (0,
12) for all three parabolas. (Why?). The
vertices are not clearly revealed by the
functions. We would need to write the
function rules in the vertex form
y = a(x ! h)2 + k before the vertices can
be easily seen.
14. In general, what effect does changing the B in the standard form of the quadratic
function y = Ax 2 + Bx + C have on the graph?
For B<0, as B decreases, the resulting parabolas will stay the same size but will move in
a southeastward direction. If B>0, as B increases, the resulting parabolas will move in a
southwestward direction. Examine some graphs for proof of this.
Teacher notes
This activity is not appropriate for Algebra 1 students. Algebra 1 students only examine
the changes in a and c in the function rule y = ax2 + c.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
TASK 3.5.1: CURVES AHEAD — TEACHER VERSION
1. Given a function y = x 2 ! 4x ! 12 . Graph this function and identify its zeros.
2.
a. Design a story problem involving the area of a rectangle that would
depend upon this function and its zeros.
b. Design a story problem involving the area of triangle that would depend
upon this function and its zeros.
3. The function that you were given in (1) was copied incorrectly. It was supposed
to be y = x 2 ! 4x ! 21 . Adjust your story problems accordingly. How do your
original story problems have to change to match the new function?
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
7
4. The function rule that you were given in (3) was also copied incorrectly. It was
supposed to be y = x 2 ! 4x ! 5 . Adjust your story problems accordingly. How
do your original story problems have to change to match the new function?
5. Graph the three functions on the same axis. What do you notice about the graphs?
Include the vertices and y-intercepts and relate them to the function.
6. In general, what effect does changing the C in the standard form of the quadratic
equation y = Ax 2 + Bx + C have on the graph?
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
8
7. Suppose the original function y = x 2 ! 4x ! 12 is changed again, this time to
y = 2x 2 ! 4x ! 12 . Modify your story problems to match it and discuss what
changed.
8. Again the function changes to y = 3x 2 ! 4x ! 12 . Modify your story problems
and discuss what changed.
9. Graph the last two functions on the same coordinate planewith the original. What
do you notice about the graphs? Include the vertices and y-intercepts and relate
them to the function.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
9
10. In general, what effect does changing the A in the standard form of the quadratic
equation y = Ax 2 + Bx + C have on the graph?
11. Suppose the original function y = x 2 ! 4x ! 12 is changed once again, this time to
y = x 2 ! 5x ! 12 . Modify your story problems accordingly. How do your
original story problems have to change to match the new function?
12. For a final time, the function changes and is now y = x 2 ! 11x ! 12 . Modify your
story problems and discuss what changed.
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 5. Capstone Problem —
Curves Ahead; Task 3.5.1
10
13. Graph the last two functions on the same coordinate plane with the original.
What do you notice about the graphs? Include the vertices and y-intercepts and
relate them to the function.
14. In general, what effect does changing the B in the standard form of the quadratic
equation y = Ax 2 + Bx + C have on the graph?
November 24, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.