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Algebra Success
T873
LESSON 36: Graphing Quadratic Equations
[OBJECTIVE]
The student will find the equation for the axis of symmetry and the coordinates of
the vertex of a parabola. The student will find the x- and y-intercepts of a graph and
use them, along with the vertex, to graph a quadratic equation.
[MATERIALS]
Student pages S328–S340
Transparencies T886, T887, T888, T890, T892, T894, T896, T897, T899, T902
Paper for the foldable (1 sheet per student)
Scissors
Optional: graphing calculator
Wall-size four-quadrant grid
[ESSENTIAL QUESTIONS]
1. What is a quadratic equation?
2. How do we graph a quadratic equation?
[GROUPING]
Cooperative Pairs, Whole Group, Individual
[LEVELS
OF
TEACHER SUPPORT]
Modeling (M), Guided Practice (GP), Independent Practice (IP)
[MULTIPLE REPRESENTATIONS]
SOLVE, Table, Verbal Description, Graph, Algebraic Formula
[WARM-UP] (5 minutes – IP) S328 (Answers on T885.)
• Have students turn to S328 in their books to begin the Warm-Up. Students will
review factoring polynomials. Monitor students to see if any of them need help
durng the Warm-Up. Give students 3 minutes to complete the problems and then
spend 2 minutes reviewing the answers as a class. {Algebraic Formula, Verbal
Description}
[HOMEWORK]: (5 minutes)
Take time to go over the homework from the previous night.
[LESSON]: (48–55 minutes – M, GP, IP)
T874
Algebra Success
LESSON 36: Graphing Quadratic Equations
SOLVE Problem
(2 minutes – GP) T886, S329 (Answers on T887.)
Have students turn to S329 in their books, and place T886 on the overhead.
The first problem is a SOLVE problem. You are only going to complete the S step
with students at this point. Tell students that during the lesson they will learn
to graph a quadratic equation by finding the axis of symmetry and vertex. They
will use this knowledge to complete this SOLVE problem at the end of the lesson.
{SOLVE}
Graph Parabolas
(10 minutes – M, GP) T886–T888, S329–S331
(Answers on T889.)
Use the following activities to model for your students how to graph parabolas.
For each table on S329 (T886), work with students to find the corresponding yvalues for the given x-values, graph the points, and then connect the points to
form a parabola. {Table, Verbal Description, Graph, Algebraic Formula}
Algebra Success
T875
LESSON 36: Graphing Quadratic Equations
MODELING
Problem 1
Step 1: Direct students’ attention to the equation in Problem 1, y = x2 + 3x
x + 2,
and to the table of x-values. Have students plug in each x-value into
the equation and then use the order of operations to find and write the
corresponding y-value:
Find the value of y when x = -4.
• Substitute the value of -4 into
(-4)2 + 3(-4) + 2
• Simplify the exponent.
16 + 3(-4) + 2
• Multiply.
16 + -12 + 2
• Add.
6
the equation for each x.
So, y = 6 when x = -4.
Find the value of y when x = -3.
• Substitute the value of -3 into
(-3)2 + 3(-3) + 2
• Simplify the exponent.
9 + 3(-3) + 2
• Multiply.
9 + -9 + 2
• Add.
2
the equation for each x.
So, y = 2 when x = -3.
Continue this process to find the remaining y-values in the table.
Step 2: Graph the points on S330 (T887) with students and then connect the
points to form a parabola.
• Graph the point (-4, 6). Begin at the origin (0, 0). Move to the left
4 units and up 6 units. Make a point at (-4, 6) on the graph.
• Graph the point (-3, 2). Begin at the origin (0, 0). Move to the left
3 units and up 2 units. Make a point at (-3, 2) on the graph.
• Continue to graph the other points.
T876
Algebra Success
LESSON 36: Graphing Quadratic Equations
MODELING
Problem 2
Step 1: Direct students’ attention to the equation in Problem 2, y = -x2 – 2x
x + 3,
and to the table of x-values. Have students plug in each x-value into
the equation and then use the order of operations to find and write the
corresponding y-value:
Find the value of y when x = -4.
•
Substitute the value of -4 into
the equation for each x.
-(-4)2 – 2(-4) + 3
•
Simplify the exponent.
-16 – 2(-4) + 3
•
Multiply.
-16 + 8 + 3
• Add.
So, y = -5 when x = -4.
-5
Find the value of y when x = -3.
•
Substitute the value of -3 into
the equation for each x.
-(-3)2 – 2(-3) + 3
•
Simplify the exponent.
-9 – 2(-3) + 3
•
Multiply.
-9 + 6 + 3
• Add.
So, y = 0 when x = -3.
0
Continue this process to find the remaining y-values in the table.
Step 2: Graph the points on S331 (T888) with students and then connect the
points to form a parabola.
•
Graph the point (-4, -5). Begin at the origin (0, 0). Move to the left
4 units and down 5 units. Make a point at (-4, -5) on the graph.
•
Graph the point (-3, 0). Begin at the origin (0, 0). Move to the left
3 units. Make a point at (-3, 0) on the graph.
•
Continue to graph the other points.
Algebra Success
T877
LESSON 36: Graphing Quadratic Equations
Analyze Parabolas
(5 minutes – M, GP) T890, S332 (Answers on T891.)
Have students turn to S332 in their books, and place T890 on the overhead. Use
the following activity to analyze the equations that students graphed on S330
and S331. {Table, Verbal Description, Graph, Algebraic Formula}
MODELING
Maximums, Minimums, and Symmetry
Step 1: Complete the first two statements at the top of S332 (T890) with students.
Introduce students to the vocabulary words quadratic and parabola.
Add these words to your word wall.
Step 2: Use the rest of the page to analyze the two graphs on S330 and S331.
• Point to each coefficient of the equation y = x2 + 3x
x + 2 as you
identify a, b, and c. Remind students that if the variable does not
have a coefficient, or number in front of the variable, the coefficient
is 1.
• Have students look at the graph on S330 (T887) to determine
whether it opens upward or downward. (The graph opens upward.)
Have students circle their answers on S332.
• Explain to students that if a is positive, then there is a minimum
point. This point is called the vertex. Have students complete the
statements on S332 and label the vertex on S330.
• Point to each coefficient of the equation y = -x2 – 2x
x + 3 as you
identify a, b, and c. Remind students that if the variable does not
have a coefficient, or number in front of the variable, the coefficient
is 1.
• Have students look at the graph on S331 (T888) to determine whether
it opens upward or downward. (The graph opens downward.) Have
students circle their answers on S332.
• Explain to students that if a is negative, then there is a maximum
point. If a is positive, then there is a minimum point. This point is
called the vertex. Have students complete rest of the statements
on S332 and label the vertex on S331 and S330.
T878
Algebra Success
LESSON 36: Graphing Quadratic Equations
Step 3: Discuss the symmetry of parabolas with students. Have students draw
a line on each graph which represents the line of symmetry. Explain to
students that if they were to fold the parabola along a line down the
middle vertically so that each point on the two sides matches up, this line
would be the line of symmetry.
Analyze Parabolas
(5 minutes – M, GP) T892, S333 (Answers on T893.)
Have students turn to S333 in their books, and place T892 on the overhead. Use
the following activity to analyze the equations that students graphed on S330
and S331. {Table, Verbal Description, Graph, Algebraic Formula}
MODELING
Axis of Symmetry and Vertex
Step 1: Explain to students that they will now identify the equation of the line of
symmetry, or axis of symmetry, for each graph. Remind students that the
equations of vertical lines are always in the form of x = a number. Have
students use the coordinates of the points on each graph to identify the
equation for each axis of symmetry. Students can use their graphs on
S330 and S331 and fold them to see the lines of symmetry.
•
For the first graph, the axis of symmetry passes through the point
(-1.5, -0.25). The x-coordinate of this point is -1.5. So, the equation
for the axis of symmetry is x = -1.5.
•
For the second graph, the axis of symmetry passes through the
point (-1, 4). The x-coordinate of this point is -1. So, the equation
for the axis of symmetry is x = -1.
Step 2: Discuss the formula x = b , which is used to find the axis of symmetry
2a
for a parabola whose equation is written in the form ax2 + bx + c. Have
students use the formula to check the equation for the axis of symmetry
of each graph.
•
•
For the first equation, a = 1 and b = 3. Plug the values into the
formula:
x = b = 3 = 3 = -1.5. So the equation x = -1.5 is correct.
2a
2(1)
2a
2(-1)
2
For the second equation, a = -1 and b = -2. Plug the values into the
formula:
- x = b = ( 2) = 2 = -1. So the equation x = -1 is correct.
-2
Algebra Success
T879
LESSON 36: Graphing Quadratic Equations
Step 3: Discuss the vertex with your students. Have students plug in the value
of x from the axis of symmetry into the original equation to find the
y-coordinate for the vertex.
•
For the first graph, the axis of symmetry is x = -1.5. Plug in -1.5 for
x to find the y-coordinate of the vertex.
y
y
y
y
y
=
=
=
=
=
x2 + 3x
x+2
2
( 1.5) + 3(-1.5) + 2
2.25 + 3(-1.5) + 2
2.25 – 4.5 + 2
-0.25
Equation
Plug in x = -1.5.
Simplify exponent.
Multiply.
Subtract and add.
The vertex is at (-1.5, -0.25).
•
For the second graph, the axis of symmetry is x = -1. Plug in -1 for
x to find the y-coordinate of the vertex.
y
y
y
y
y
=
=
=
=
=
-x2 – 2x
x+3
-(-1)2 – 2(-1) + 3
-1 – 2(-1) + 3
-1 + 2 + 3
4
Equation
Plug in x = -1.
Simplify exponent.
Multiply.
Add.
The vertex is at (-1, 4).
Find Intercepts
(8 minutes – M, GP) T894, S334 (Answers on T895.)
Have students turn to S334 in their books, and place T894 on the overhead. Use
the following activity to model for students how to find the x- and y-intercepts of
parabolas. {Table, Verbal Description, Graph, Algebraic Formula}
T880
Algebra Success
LESSON 36: Graphing Quadratic Equations
MODELING
Factor
Explain to students that they will begin by factoring the trinomial which describes
the parabola, x2 + 3x
x + 2. Some students may want to factor using the box
method, some may want to factor using the grouping method, and some may
want to factor by creating two binomials.
--------------------------------------Box Method --------------------------------Step 1: Remind students that when a trinomial is written in the form ax2 + bx +
c, students can use two equations to help them create a fourth term:
______ • ______ = a • c
______+ ______ = b
Write these equations on T894 and have students do the same in their
books.
Step 2: Show students that the trinomial in Problem 1, x2 + 3x
x + 2, is in the form
2
ax + bx + c, and that a = 1, b = 3, and c = 2. Ask students to complete
the two equations, using the same two numbers.
1•2=a•c=1•2=2
1+2=b=3
Step 3: Explain to students that the numbers they use to complete the equations
tell them how to split the middle term. Since the numbers in this case are 1
and 2, students should split the middle term, 3x, into 1x
x + 2x, or x + 2x.
Step 4: Have students factor the polynomial x2 + x + 2x + 2 into two binomials
using the box method:
•
•
•
•
Find
Find
Find
Find
the
the
the
the
GCF
GCF
GCF
GCF
of
of
of
of
the
the
the
the
first row. The GCF of x2 and x is x.
second row. The GCF of +2x
x and +2 is 2.
2
first column. The GCF of x and +2x
x is x.
+
second column. The GCF of x and 2 is 1.
The answer is (x
x + 1)(x + 2).
Algebra Success
T881
LESSON 36: Graphing Quadratic Equations
MODELING
-------------------------------------- Grouping ----------------------------------Step 1: Remind students that when a trinomial is written in the form ax2 + bx +
c, students can use two equations to help them create a fourth term:
______ • ______ = a • c
______ + ______ = b
Step 2: Show students that the trinomial in Problem 1, x2 + 3x
x + 2, is in the form
ax2 + bx + c, and that a = 1, b = 3, and c = 2. Ask students to complete
the two equations, using the same two numbers.
1•2=a•c=1•2=2
1+2=b=3
Step 3: Remind students that the numbers they use to complete the equations tell
them how to split the middle term. Since the numbers in this case are 1
and 2, students should split the middle term, 3x, into 1x
x + 2x, or x + 2x.
x2 + 3x
x+2
x2 + x + 2x + 2
Given
Split the middle term
Step 4: Have students group the first two monomials and the last two monomials
with parentheses.
x2 + 3x
x+2
2
x + x + 2x + 2
(x2 + x) + (2x
x + 2)
Given
Split the middle term
Group the first two and last two terms
Step 5: Have students factor out the GCF of each binomial. The GCF of (x2 + x) is
x, and the GCF of (2x
x + 2) is 2.
x2 + 3x
x+2
x2 + x + 2x + 2
(x2 + x) + (2x
x + 2)
x(x
x + 1) + 2(x + 1)
Given
Split the middle term
Group the first two and last two terms
Factor out the GCF for each binomial
Step 6: Remind students that one of the binomials in the answer is the common
binomial (in this case, x + 1), and the other binomial in the answer is
made up of the terms left over (in this case, x + 2). So the final answer
is (x
x + 1)(x + 2), or (x + 2)(x + 1).
T882
Algebra Success
LESSON 36: Graphing Quadratic Equations
MODELING
---------------------------------- Two Binomials -------------------------------Have students look at the original equation x2 + 3x
x + 2 and create two binomials,
splitting the x2 between both binomials: (x
x + ?)(x + ?). Explain that the last two
numbers in the binomials must have a product of a • c and a sum of b. Have
students try different combinations of numbers to find the correct answer. Then
have students multiply the binomials using FOIL to check their answers.
Find the Intercepts
Step 1: After students factor the trinomial, have students set each binomial equal
to zero on S334 and solve each for x to find the x-intercepts of the
parabola described by the equation.
x+2=0
–2–2
x = -2
x+1=0
– 1 – 1 Subtract to isolate the variable.
x = -1
Step 2: Have students find the x-intercepts on the graph on S330. The parabola
crosses the x-axis at -2 and -1. Have students compare the x-intercepts
shown in the graph on S330 to the x-intercepts they found on S334 in
order to see that they are the same.
Step 3: Have students find the y-intercept of the graph on S330. The parabola
crosses the y-axis at 2. Have students compare the y-intercept to the
value of c in the equation in order to see that they are the same.
Repeat the steps above to factor the trinomial in Problem 2, -x2 – 2x
x + 3, on S334
and then find the x- and y-intercepts of the graph. Have students compare their
answers with the x- and y-intercepts shown in the graph on S331 in order to see
that they are the same.
Summary
(2 minutes – GP) T896, S335
Have students turn to S335 in their books, and place T896 on the overhead. Use
the page to summarize what students have learned about graphing parabolas.
{Algebraic Formula, Verbal Description}
Algebra Success
T883
LESSON 36: Graphing Quadratic Equations
Practice
(10 minutes – GP) T897, S336 (Answers on T898.)
Have students turn to S336 in their books, and place T897 on the overhead.
Use the page to practice graphing a parabola using what students learned in the
lesson. {Algebraic Formula, Verbal Description}
SOLVE Problem
(6 minutes – GP) T899, S337 (Answers on T900.)
Remind students that the SOLVE problem is the same one from the beginning of
the lesson. Complete the SOLVE problem with your students. {SOLVE, Algebraic
Formula}
Quadratics Foldable
(5 minutes – M, GP, IP)
Pass out one sheet of colored paper to each student. Use the following activity
to create the Quadratics Foldable with students. {Algebraic Formula, Verbal
Description}
MODELING
Quadratics Foldable
Step 1: Fold the piece of paper vertically, hotdog fold.
Step 2: Leave the paper folded and fold the piece of paper in half, hamburger
fold, and then fold the piece of paper in half again, hamburger fold.
Step 3: Open the paper up once until you have two rectangles. Use your finger to
find the corner of the paper which is folded on the top and side. Both of
the sides intersecting in the corner are folded.
Step 4: Cut a small strip ( 1 of an inch at most) off the top from the folded corner
16
to the middle crease.
Step 5: Open the sheet of paper up showing a long rectangle with four sections.
Find the crease in the middle of the sheet of paper. Hold the paper up,
slide your index fingers in the middle of the paper on the crease and pull
out creating four rectangular flaps.
Step 6: Fold the four flaps together forming a booklet. You should now have a
booklet with 8 pages.
Step 7: Label the outside of the foldable Quadratics, y = ax2 + bx + c. Use the
pages for the three examples. Create a transparency to model for students
what should be written on each page.
T884
Algebra Success
LESSON 36: Graphing Quadratic Equations
If time permits...
(8 minutes – IP) S338 (Answers on T901.)
Have students find all of the parts of the parabola and graph the parabola
on S338. Students may work in cooperative pairs or independently. Give your
students 6 minutes to complete the page. Use 2 minutes to review the answers.
{Algebraic Formula}
Instructions for finding the maximum or minimum value or an x-intercept for
a quadratic equation on the graphing calculator are included on S339 (T902).
Please use these if they are appropriate for your students period.
[CLOSURE]: (5 minutes)
• To wrap up the lesson, go back to the essential questions and discuss them with
students.
• What is a quadratic equation? (A quadratic equation is an equation with the
basic form y = ax2 + bx + c
c, whose graph is a parabola.)
• How do we graph a quadratic equation? (You
You can use the ax
axis of symmetry,
vertex, and x- and y-intercepts, or find points to graph a quadratic
equation.)
[HOMEWORK]: Assign S340 for homework. (Answers on T903.)
[QUIZ ANSWERS] T904–T905
1. C
2. C
3. C
4. A
5. C
6. B
7. A
8. D
9. B
10. D
The quiz can be used at any time as extra homework or to see how the students did
on understanding the graphs of quadratic equations.