algebra success t873 - ntn math homework and lesson notes for website… · algebra success t873...

Algebra Success T873

LESSON 36: Graphing Quadratic Equations

[OBJECTIVE]

The student will fi nd the equation for the axis of symmetry and the coordinates of the vertex of a parabola. The student will fi nd the x- and y-intercepts of a graph and use them, along with the vertex, to graph a quadratic equation.

[MATERIALS]

Student pages S328–S340Transparencies T886, T887, T888, T890, T892, T894, T896, T897, T899, T902Paper for the foldable (1 sheet per student)ScissorsOptional: graphing calculatorWall-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)

Algebra SuccessT874

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 fi rst 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 fi nding 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 fi nd the corresponding y-values for the given x-values, graph the points, and then connect the points to form a parabola. {Table, Verbal Description, Graph, Algebraic Formula}



MODELING

Problem 1

Step 1: Direct students’ attention to the equation in Problem 1, y = y = y x2x2x + 3x + 2x + 2x , 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 fi nd and write the corresponding y-value:

Find the value of y when y when y x = x = x -4.

• Substitute the value of -4 into (-4)2 + 3(-4) + 2the equation for each x.

• Simplify the exponent. 16 + 3(-4) + 2

• Multiply. 16 + -12 + 2

• Add. 6

So, y = 6 when y = 6 when y x = x = x -4.


• Substitute the value of -3 into (-3)2 + 3(-3) + 2the equation for each x.

• Simplify the exponent. 9 + 3(-3) + 2

• Multiply. 9 + -9 + 2

• Add. 2

So, y = 2 when y = 2 when y x = x = x -3.

Continue this process to fi nd 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.


Algebra SuccessT876

MODELING

Problem 2

Step 1: Direct students’ attention to the equation in Problem 2, y = y = y -x2x2x – 2x + 3x + 3x , 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 fi nd and write the corresponding y-value:


• Substitute the value of -4 into -(-4)2 – 2(-4) + 3the equation for each x.

• Simplify the exponent. -16 – 2(-4) + 3

• Multiply. -16 + 8 + 3

• Add. -5So, y = y = y -5 when x = x = x -4.


• Substitute the value of -3 into -(-3)2 – 2(-3) + 3the equation for each x.

• Simplify the exponent. -9 – 2(-3) + 3

• Multiply. -9 + 6 + 3

• Add. 0So, y = 0 when y = 0 when y x = x = x -3.

Continue this process to fi nd 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.



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 fi rst 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 coeffi cient of the equation y = y = y x2 + 3x + 2 as you x + 2 as you xidentify a, b, and c. Remind students that if the variable does not have a coeffi cient, or number in front of the variable, the coeffi cient 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 coeffi cient of the equation y = y = y -x2 – 2x + 3 as you x + 3 as you xidentify a, b, and c. Remind students that if the variable does not have a coeffi cient, or number in front of the variable, the coeffi cient 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.


Algebra SuccessT878

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 x = a number. Have xstudents 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 fi rst 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 = x = 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 = x = x -1.

Step 2: Discuss the formula x = x = x-b2a

, which is used to fi nd the axis of symmetry

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 fi rst equation, a = 1 and b = 3. Plug the values into the formula:

x = x = x-b2a

= -3

2(1) =

-32

= -1.5. So the equation x = x = x -1.5 is correct.

• For the second equation, a = -1 and b = -2. Plug the values into the formula:

x = x = x-b2a

= -(-2)2(-1)

= 2-2

= -1. So the equation x = x = x -1 is correct.



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 fi nd the x from the axis of symmetry into the original equation to fi nd the xy-coordinate for the vertex.

• For the fi rst graph, the axis of symmetry is x = x = x -1.5. Plug in -1.5 for x to fi nd the x to fi nd the x y-coordinate of the vertex.

y = y = y x2 + 3x + 2 Equation x + 2 Equation xy = (y = (y -1.5)2 + 3(-1.5) + 2 Plug in x = x = x -1.5.y = 2.25 + 3(y = 2.25 + 3(y -1.5) + 2 Simplify exponent.y = 2.25 – 4.5 + 2 Multiply.y = 2.25 – 4.5 + 2 Multiply.yy = y = y -0.25 Subtract and add.

The vertex is at (-1.5, -0.25).

• For the second graph, the axis of symmetry is x = x = x -1. Plug in -1 for x to fi nd the x to fi nd the x y-coordinate of the vertex.

y = y = y -x2 – 2x + 3 Equation x + 3 Equation xy = y = y -(-1)2 – 2(-1) + 3 Plug in x = x = x -1.y = y = y -1 – 2(-1) + 3 Simplify exponent.y = y = y -1 + 2 + 3 Multiply.y = y = y 4 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 fi nd the x- and y-intercepts of parabolas. {Table, Verbal Description, Graph, Algebraic Formula}


Algebra SuccessT880

MODELING

Factor

Explain to students that they will begin by factoring the trinomial which describes the parabola, x2 + 3x + 2. Some students may want to factor using the box x + 2. Some students may want to factor using the box xmethod, 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 + bx + 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 + 2, is in the form x + 2, is in the form xax2 + bx + c, and that a = 1, b = 3, and c = 2. Ask students to complete c = 2. Ask students to complete cthe two equations, using the same two numbers.

1 • 2 = a • c = 1 • 2 = 2a • c = 1 • 2 = 2a • c

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 + 2x + 2x x, or x + 2x + 2x x.

Step 4: Have students factor the polynomial x2 + x + 2x + 2x x + 2 into two binomials x + 2 into two binomials xusing the box method:

• Find the GCF of the fi rst row. The GCF of x2 and x is x is x x.• Find the GCF of the second row. The GCF of +2x and x and x +2 is 2.• Find the GCF of the fi rst column. The GCF of x2 and +2x is x is x x.• Find the GCF of the second column. The GCF of x and x and x +2 is 1.

The answer is (x + 1)(x + 1)(x x + 2).x + 2).x



MODELING

-------------------------------------- Grouping -----------------------------------

Step 1: Remind students that when a trinomial is written in the form ax2 + bx + bx + bxc, 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 + 2, is in the form x + 2, is in the form xax2 + bx + c, and that a = 1, b = 3, and c = 2. Ask students to complete c = 2. Ask students to complete cthe two equations, using the same two numbers.

1 • 2 = a • c = 1 • 2 = 2a • c = 1 • 2 = 2a • c

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 + 2x + 2x x, or x + 2x + 2x x.

x2 + 3x + 2 Givenx + 2 Givenx x2 + x + 2x + 2x x + 2 Split the middle termx + 2 Split the middle termx

Step 4: Have students group the fi rst two monomials and the last two monomials with parentheses.

x2 + 3x + 2 Given x + 2 Given x x2 + x + 2x + 2x x + 2 Split the middle termx + 2 Split the middle termx (x2 + x) + (2x + 2) Group the fi rst two and last two termsx + 2) Group the fi rst two and last two termsx

Step 5: Have students factor out the GCF of each binomial. The GCF of (x2 + x) is x, and the GCF of (2x + 2) is 2.x + 2) is 2.x

x2 + 3x + 2 Givenx + 2 Givenx x2 + x + 2x + 2x x + 2 Split the middle termx + 2 Split the middle termx (x2 + x) + (2x + 2) Group the fi rst two and last two termsx + 2) Group the fi rst two and last two termsx x(x + 1) + 2(x + 1) + 2(x x + 1) Factor out the GCF for each binomialx + 1) Factor out the GCF for each binomialx

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 x + 1), and the other binomial in the answer is xmade up of the terms left over (in this case, x + 2). x + 2). x So the fi nal answeris (x + 1)(x + 1)(x x + 2), or (x + 2), or (x x + 2)(x + 2)(x x + 1).x + 1).x


Algebra SuccessT882

MODELING

---------------------------------- Two Binomials --------------------------------

Have students look at the original equation x2 + 3x + 2 and create two binomials, x + 2 and create two binomials, xsplitting the x2 between both binomials: (x + ?)(x + ?)(x x + ?). Explain that the last two x + ?). Explain that the last two xnumbers in the binomials must have a product of a • c and a sum of c and a sum of c b. Have students try different combinations of numbers to fi nd 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 fi nd the x to fi nd the x x-intercepts of the parabola described by the equation.

x + 2 = 0 x + 2 = 0 x x + 1 = 0x + 1 = 0x – 2 – 2 – 1 – 1 Subtract to isolate the variable. x = x = x -2 x = x = x -1

Step 2: Have students fi nd 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 fi nd 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.c in the equation in order to see that they are the same.c

Repeat the steps above to factor the trinomial in Problem 2, -x2 – 2x + 3, on S334 x + 3, on S334 xand then fi nd 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}



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 fi nger to fi nd 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 ( 116

of an inch at most) off the top from the folded corner

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 fi ngers in the middle of the paper on the crease and pull out creating four rectangular fl aps.

Step 6: Fold the four fl aps together forming a booklet. You should now have a booklet with 8 pages.

Step 7: Label the outside of the foldable Quadratics, y = y = 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.


Algebra SuccessT884

If time permits... (8 minutes – IP) S338 (Answers on T901.)

Have students fi nd 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 fi nding 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 = y = y ax2 + bx + bx + bx c, whose graph is a parabola.)c, whose graph is a parabola.)c

• How do we graph a quadratic equation? (You can use the axis of symmetry, You can use the axis of symmetry, You can use the axvertex, and x- and y-intercepts, or fi nd 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.


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