# Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c

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2 Day Lesson Title Math Learning Goals Expectations 10, QR3.01, QR Solve Problems Involving Quadratic Relations (lessons not included) 12 Summative Assessment Solve problems involving quadratic relations by interpreting a given graph or a graph generated with technology from its equation. CGE 5g TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 2

4 7.1.1: Quadratic Equations (Teacher) Cut out each pair of quadratic equations and distribute to the students. Standard Form Factored Form y = x 2 + 3x + 2 y = (x + 1)(x + 2) y = x 2 + 2x 3 y = (x 1)(x + 3) y = x 2 1x 6 y = (x + 2)(x 3) y = x 2 + x 6 y = (x 2)(x + 3) y = x 2 + 5x + 4 y = (x + 1)(x + 4) y = x 2 + 2x 3 y = (x + 3)(x 1) y = x 2 x 6 y = (x 3)(x + 2) y = x 2 + 6x + 5 y = (x + 1)(x + 5) y = x 2 4x 5 y = (x + 1)(x 5) y = x 2 4x 12 y = (x + 2)(x 6) y = x 2 + 6x + 8 y = (x + 2)(x + 4) y = x 2 5x + 6 y = (x 3)(x 2) y = x 2 2x 3 y = (x 3)(x + 1) y = x 2 + 7x + 12 y = (x + 3)(x + 4) y = x 2 + 7x + 10 y = (x + 2)(x + 5) y = x 2 7x + 10 y = (x 2)(x 5) TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 4

5 7.1.2: Graphing Quadratic Equations Name: 1. Obtain a pair of equations from your teacher. 2. Press the Zoom button and press 6 (for ZStandard) to set the window to make the max and min on both axes go from 10 to Press the y = button and key in your two equations into Y 1 and Y To change the graph of Y 2 to animation : Move the cursor to the left of Y 2. Press Enter four times to toggle through different graph styles available. You should see 5. Press Graph. First the Y 1 quadratic will appear, then the Y 2 quadratic will appear and be traced by an open circle. 6. Complete the three columns of the table below. Our Two Equations What They Look Like What We Think It Means 7. Press 2 nd Graph so that you can look at the tables of values for the two curves. Discuss what you see and complete the table. What We Noticed About the Table of Values What We Think It Means TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 5

6 7.1.3: Algebra Tile Template 1. y = ( )( ) = 2. y = ( )( ) = 3. y = ( )( ) = 4. y = ( )( ) = 5. y = ( )( ) = 6. y = ( )( ) = TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 6

7 Unit 7: Day 2: Multiply a Binomial by a Binomial Using a Variety of Methods Math Learning Goals Expand and simplify second-degree polynomial expressions involving one variable that consist of the product of two binomials or the square of a binomial, using the chart method, and the distributive property. Grade 10 Applied Materials BLM 7.2.1, algebra tiles graphing calculators 75 min Minds On Whole Class Discussion Students expand (x + 1)(x + 1) with algebra tiles. Discuss the meaning of repeated multiplication, i.e., (x + 1)(x + 1) = (x + 1) 2. Individual Practice Students practise multiplication of a binomial with positives only, using BLM Part A and algebra tiles. Action! Consolidate Debrief Reflection Whole Class Guided Instruction Model the different combinations of multiplication, i.e., monomial monomial and monomial binomial with positive and negative terms, using the chart method. Connect the use of algebra tiles to the chart method. Model the chart method for multiplication of a binominal binomial. Students practise this method, BLM Part B. Use algebra tiles to show each of these and recall the distributive property. x(x + 3) 2(x + 3) Multiply (x+2)(x+3) using tiles, and show double distributive, (the algebraic model). Model the steps involved in the algebraic manipulation of multiplying a binomial by a binomial. Connect to distributive property by calling it double distributive. Using the Distributive Property Lead students to understand the connection between the chart method and the algebraic method of multiplying binomials. (x + 4)(x 3) x +4 = x (x 3) + 4(x 3) = x 2 x x 2 4x 3x + 4x 12 = x 2 + x x 12 Students practise this method, using BLM Part C. Learning Skills/Work Habits/Observation/Checklist: Assess how well students stay on task and complete assigned questions. Individual Journal In your journal, write a note to a friend who missed today s class. Summarize the three methods of multiplying binomials that you worked with. Use words, diagrams, and symbols in your explanation. Home Activity or Further Classroom Consolidation Solve the problems by multiplying the binomials. Assessment Opportunities Representation with algebra tiles is best for expressions with positive terms only. This representation can be used for binomials with positive and negative terms. x +3 x x 2 3x +2 2x 6 Use visual aid to show double distributive Provide chart template and algebra tile template 7.1.3, as needed. TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 7

8 7.2.1: Multiply a Binomial by a Binomial Name: Part A Use algebra tiles to multiply binomials and simplify the following: 1. y = (x + 1)(x + 3) = 2. y = (x + 2)(x + 3) = Part B Use the chart method to multiply and simplify the following: 1. y = (x + 1)(x + 3) = 2. y = (x + 2)(x +3) = 3. y = (x + 2)(x 1) = 4. y = (x 2)(x + 3) = 5. y = (x 1)(x 1) = 6. y = (x 1) (x 2) = TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 8

9 7.2.1: Multiply a Binomial by a Binomial (continued) Part C Multiply and simplify the two binomials, using the chart method and the distributive property. 1. (x + 4)(x 3) x +4 x 3 2. (x 3)(x 3) x 3 x 3 3. (x + 2) 2 x +2 x (x + 2)(x 1) x +2 x 1 5. (x 2) (x + 1) x 2 x (x 1) 2 x 1 x 1 7. (x 1)(x 2) x 1 x 2 8. (x 3)(x 4) x 3 x 4 TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 9

10 7.2.1: Multiply a Binomial by a Binomial (continued) Answers to Part B 1. y = x 2 + 4x + 3 x 1 x x 2 x 3 3x 3 2. y = x 2 + 5x+ 6 x 2 x x 2 2x 3 3x 6 3. y = x 2 + x 2 x 2 x x 2 2x 1 x 2 4. y = x 2 + x 6 x 2 x x 2 2x 3 3x 6 5. y = x 2 2x + 1 x 1 x x 2 x 1 x 1 6. y = x 2 3x +2 x 1 x x 2 x 2 2x 2 Answers to Part C 1. x 2 + x x 2 6x x 2 + 4x x 2 + x 2 5. x 2 x 2 6. x 2 2x x 2 3x x 2 7x + 12 TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 10

11 7.2.2: Chart Template for Distributive Property 1. y = ( )( ) = 2. y = ( )( ) = 3. y = ( )( ) = 4. y = ( )( ) = 5. y = ( )( ) = 6. y = ( )( ) = 7. y = ( )( ) = 8. y = ( )( ) = TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 11

13 7.3.1: Finding the y-intercept of a Quadratic Equation Name: 1. Use the graphing calculator to find the y-intercept for each of the equations: Note any patterns you see. Equation y-intercept 2 y = x x 2 2 y = x + 2x 8 2 y = x x + 6 ( 1)( 2) y = x x ( 4)( 3) y = x+ x + ( 3) y = x How can you determine the y-intercept by looking at a quadratic equation? 3. Which form of the quadratic equation is easiest to use to determine the y-intercept? Explain your choice. 4. Using your conclusion from question 2, state the y-intercept of each and check using a graphing calculator. Equation 2 y = x 2x 8 2 y = x x 6 2 y = x + 3x + 2 ( 4)( 1) y = x x ( 2)( 5) y = x x + y-intercept Does it check? Yes No 5. Explain the connection between the y-intercept and the value of y when x = 0. TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 13

14 7.3.2: Quadratic Equations Name: 1. Find the y-intercept for each of the following quadratic equations given in factored form. Write the equations in standard form. Show your work. a) = ( 5)( + 2) = = y x x standard form: y-intercept: b) = ( + 4)( 3) = y x x standard form: = y-intercept: c) = ( 4) = 2 y x standard form: = y-intercept: ) y = ( x+ ) d 5 = = 2 standard form: y-intercept: 2. Find the y-intercept for each of the following quadratic equations: a) y = ( x+ 4)( x + 2) b) y = ( x 6) 2 y-intercept: y-intercept: TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 14

16 7.4.1: Finding the x-intercepts of a Quadratic Equation Name: To find the x-intercepts: 1. Enter the equation in Y 1. y = x 2 x 6 2. Press ZOOM and 6 (Zstandard) to set the scale for your graph. The calculator will then show the parabola. 3. Press 2nd TRACE 1 to view the Calculate screen. 4. Select 2: ZERO. Your screen should be similar to the following screen. 5. You will be asked to enter a left bound. You can move the cursor to the left of one x-intercept (or just enter an x value that is to the left of the x-intercept). Press ENTER. 6. Repeat for the right bound, being sure that you are to the right of the same x-intercept. 7. The next screen will say guess. You can guess if you want but it is not necessary. Press ENTER. You will get one x-intercept. 8. Repeat steps 3 through 7 to get the other x-intercept. TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 16

17 7.4.1: Finding the x-intercepts of a Quadratic Equation (continued) 1. Use the graphing calculator to find the x-intercepts for each of the following: Equation 2 y = x 4x 12 2 y = x + 2x 8 2 y = x x + 6 ( 1)( 2) y = x x ( 4)( 3) y = x + x + ( 3)( 5) y = x x + First x-intercept Second x-intercept 2. Can you determine the x-intercepts by looking at a quadratic equation? Explain. 3. Which form of the quadratic equation did you find the easiest to use when determining the x-intercepts? Explain the connection between the factors and the x-intercepts. TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 17

18 7.4.2: Area with Algebra Tiles Name: Using algebra tiles create the rectangles for the following areas. Complete the following chart. Area of Rectangle Number of x 2 Tiles Number of x Tiles Number of Unit Tiles Sketch of Rectangle Length Width x 2 + 4x + 3 x 2 + 5x + 6 x 2 + 6x + 8 x 2 + 7x Find a relationship between the number of x tiles and the numbers in the expressions for the length and width. 2. Find a relationship between the number of unit tiles and the numbers in the expressions for the length and width. 3. If the area of a rectangle is given by x 2 + 8x + 15, what expression will represent the length and the width? TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 18

19 7.4.3: Factoring Using Algebra Tiles and Making Connections to the Graphic Name: Part A For each of the following, shade in the appropriate rectangular area. Then shade in the tiles that represent the length and width for each of those areas. Use the length and width to represent and state the factors. State the x-intercepts. Check using a graphing calculator. 1. y = x 2 + 3x + 2 = ( )( ) 2. y = x 2 + 5x + 4 = ( )( ) x-intercepts, Check with calculator. x-intercepts, Check with calculator. 3. y = x 2 + 6x + 5 = ( )( ) 4. y = x 2 + 4x + 4 = ( )( ) x-intercepts, Check with calculator. x-intercepts, Check with calculator. TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 19

20 7.4.3: Factoring Using Algebra Tiles and Making Connections to the Graphic (continued) Part B Using the diagrams in Part A, find the x- and y-intercepts for each quadratic relation. Use the information to make the sketch on the grid provided. 1. standard form: y = x 2 + 3x + 2 factored form: y-intercept: first x-intercept: second x-intercept: 2. standard form: y = x 2 + 5x + 4 factored form: y-intercept: first x-intercept: second x-intercept: TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 20

21 7.4.3: Factoring Using Algebra Tiles and Making Connections to the Graphic (continued) 3. standard form: y = x 2 + 6x + 5 factored form: y-intercept: first x-intercept: second x-intercept: 4. standard form: y = x 2 + 4x + 4 factored form: y-intercept: first x-intercept: second x-intercept: 5. In what way is the last example different from the others? TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 21

23 7.5.1: Four Corners y = x 2 + 4x + 3 y = (x + 1)(x + 3) y = x 2 + 3x + 2 TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 23

24 7.5.1: Four Corners (continued) y = (x + 2)(x + 1) y = x 2 + 5x + 6 y = (x + 3)(x + 2) TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 24

25 7.5.1: Four Corners (continued) y = x 2 + 6x +5 y = (x + 5)(x + 1) y = x 2 + 5x + 4 TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 25

26 7.5.1: Four Corners (continued) y = (x + 1)(x + 4) x +3 y = x 2 + 1x 6 x x 2 +3x 2 2x 6 y = (x + 3)(x 2) TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 26

27 7.5.1: Four Corners (continued) x 1 x x 2 1x y = (x 1)(x 3) 3 3x 3 y = x 2 4x + 3 x 3 y = (x 3)(x 4) x x 2 3x 4 4x +12 TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 27

28 7.5.1: Four Corners (continued) y = x 2 7x + 12 TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 28

29 7.5.2: Factored Form and x-intercepts Name: Use algebra tiles to find the length and width for each given area. Use the graphing calculator to find the x-intercepts of the corresponding quadratic relation. Graph both the area model and factored form of the quadratic relation to check that these are the same before finding the x-intercepts. Area Length Width Factored Form First x-intercept of corresponding relation Second x-intercept of corresponding relation x 2 + 4x + 3 x 2 + 5x + 6 x 2 + 6x + 8 x 2 + 7x What do you notice about the constant term in the length and width expressions and the coefficient of the x term in the area expressions? 2. What do you notice about the constant term in the length and width expressions and the constant term in the area expressions? 3. If an area is expressed as x x + 21, what must be true of the constant terms in the length and width expressions? 4. If the standard form of a quadratic relation is y = x 2 + bx + c, and it has x-intercepts of r and s, then the same relation would then be y = (x r)(x s). How would you find the value of r and s? TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 29

30 BLM 7.5.3: Match It! Name: Match each pair of numbers on the left with the correct product and sum on the right. Pair of Numbers 1. r = 1, s = 5 a. Product and Sum r s = 5 r + s = 6 2. r = 1, s = 6 b. r s = 5 r + s = 4 3. r = 2, s = 3 c. r s = 5 r + s = 4 4. r = 1, s = 5 d. r s = 5 r + s = 6 5. r = 3, s = 4 e. r s = 6 r + s = 7 6. r = 1, s = 5 f. r s = 6 r + s = 7 7. r = 6, s = 2 g. r s = 6 r + s = 1 8. r = 1, s = 5 h. r s = 12 r + s = r = 1, s = 6 i. r s = 12 r + s = r = 1, s = 12 j. r s = 12 r + s = 1 TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 30

32 7.6.1: Use Intercepts to Graph It! Given the standard form of the quadratic relation, identify the value of the sum and product needed to factor. Express the relation in factored form, identify the x-intercepts and y-intercept, and use these results to make a sketch of each parabola. Standard Form A y = x 2 + 6x B y = x 4x 5 C y = x 2 + 4x 5 D y = x 2 6x+ 5 E y = x 2 + 7x+ 6 F y = x 2 6x+ 9 G y = x 2 x 6 H y = x x+ 12 I y = x 2 4x 12 J y = x 2 + x 12 Product and Sum r s = 5 r + s = 6 r s = 5 r + s = 4 r s = 5 r + s = 4 r s = 5 r + s = 6 r s = 6 r + s = 7 r s = r + s = r s = r + s = r s = r + s = r s = r + s = r s = r + s = Pair of Numbers r = 1, s = 5 Factored Form x-intercepts y-intercept ( 1)( 5) y = x+ x + 1 and 5 5 Sketch of the relation TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 32

33 7.6.1: Use Intercepts to Graph It! (Answers) Standard Form Product and Sum Pair of Numbers Factored Form x-intercepts y-intercept A y = x 2 + 6x+ 5 r s = 5 r + s = 6 r = 1 s = 5 ( 1)( 5) y = x+ x + 1 and 5 5 B y = x 2 4x 5 r s = 5 r + s = 4 r = 1 s = 5 ( 1)( 5) y = x+ x 1 and 5 5 C y = x 2 + 4x 5 r s = 5 r + s = 4 r = 1 s = 5 ( 1)( 5) y = x x+ 1 and D y = x 6x+ 5 r s = 5 r + s = 6 r = 1 s = 5 ( 1)( 5) y = x x 1 and 5 5 E y = x 2 + 7x+ 6 r s = 6 r + s = 7 r = 1 s = 6 ( 1)( 6) y = x+ x + 1 and 6 6 F y = x 2 6x+ 9 r s = 9 r + s = 6 r = 3 s = 3 ( 3)( 3) y = x x 3 9 G y = x 2 x 6 r s = 6 r + s = 1 r = 2 s = 3 ( 2)( 3) y = x+ x 2 and 3 6 H y = x x+ 12 r s = 12 r + s = 13 r = 1 s = 12 ( 1)( 12) y = x+ x+ 1 and I y = x 2 4x 12 r s = 12 r + s = 4 r = 6 s = 2 ( 6)( 2) y = x x+ 6 and 2 12 J y = x 2 + x 12 r s = 12 r + s = 1 r = 3 s = 4 ( 3)( 4) y = x x+ 3 and 4 12 TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 33

35 7.7.1: Investigate Relations of the Form y = ax 2 + b Name: 1. Obtain a graphing calculator and equation from your teacher. 2. Type in the equation (using the Y = button on your calculator). Key in zoom 6 to get the max and min from 10 to 10 on your window. 3. Fill in the table. Your equation A sketch of your quadratic equation (including the x- and y-intercepts) y = Coordinates of x-intercepts (, ) and (, ) and the coordinate of the y-intercept (, ) 4. In your group, sketch all four graphs. 5. Identify what is the same and what is different in these four graphs. TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 35

36 7.7.1: Investigate Relations of the Form y = ax 2 + b (continued) 6. Fill in the following table. Standard form y = x 2 + 4x y = x 2 + 6x y = x 2 2x y = x 2 5x From the graph, identify the x-intercepts r and s r = s = r = s = r = s = r = s = Write each equation in factored form y = (x r)(x s) 7. Clear the y = screen on one of the four calculators, enter the factored form of the four equations, and graph. Are these graphs the same as the ones in your sketch? If yes, continue to question 8. If no, revise and check. Ask your teacher for assistance, if needed. 8. Can the equations in the third column of your table be simplified? Explain. 9. Record the simplified versions of your relation in factored form. Standard Form Factored Form y = x 2 + 4x y = x 2 + 6x y = x 2 2x y = x 2 5x TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 36

37 7.7.2: Factoring x 2 + bx Consider the outer portion of the algebra tile representation as the length and width of a room. The rectangle is the carpet. Colour in as many cells as required for each example to form a rectangle. To factor, form a rectangle using the tiles, then determine the length and width of the room. Example 1: x 2 + 2x Example 2: x 2 + 3x factored form ( ) ( ) the coordinates of the x-intercepts of y = x 2 + 2x (, ) and (, ) the coordinate of the y-intercept (, ) factored form ( ) ( ) the coordinates of the x-intercepts of y = x 2 + 3x (, ) and (, ) the coordinate of the y-intercept (, ) Use your algebra tiles to factor the following: 1. x 2 + 4x 2. x 2 + 1x factored form ( ) ( ) the coordinates of the x-intercepts of y = x 2 + 4x (, ) and (, ) the coordinate of the y-intercept (, ) 3. x 2 + 5x 4. x 2 + x factored form ( ) ( ) the coordinates of the x-intercepts of y = x 2 + 1x (, ) and (, ) the coordinate of the y-intercept (, ) factored form ( ) ( ) the coordinates of the x-intercepts of y = x 2 + 5y (, ) and (, ) the coordinate of the y-intercept (, ) factored form ( ) ( ) the coordinates of the x-intercepts of y = x 2 + x (, ) and (, ) the coordinate of the y-intercept (, ) TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 37

38 7.7.3: Two Methods for Factoring x 2 + bx (Teacher) 1. Algebra Tile Model for Common Factoring Use overhead tiles to demonstrate the concept of common factoring for the example x 2 + 6x. Explanation This is the only rectangle that can be formed with algebra tiles. Factoring requires identifying the length and the width that results in this rectangle s area. In this case the dimensions are (x) and (x + 6). Alternative Explanation Separate the x 2 tile from the x tiles to show the length and width of each part of the expression. x 6 x x In the square, the area is given by x times x. In the rectangle, the area is given by x times 6. The total area is given by (x)(x) + (x)(6), we say that x is a common factor of each term. Explain x multiplies x, and x multiplies 6, thus x multiplies x and 6; that is, x multiplies (x + 6), or x (x + 6) Repeat the process with x 2 + 3x. 2. Algebraic Model for Common Factoring Use the principle of what is common to each term to factor the right-hand side of each equation. Lead the class through algebraic thinking in factoring the following examples: x 2 1x x 2 + 5x x 2 2x = x (x 1) = x (x + 5) = x (x 2) TIPS4RM: Grade 10 Applied: Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c 38

40 7.8.1: Graphing Relationships of the Form y = x 2 a 2 Name: 1. Working with a partner and one graphing calculator, set your Window: Xmin = 10; Xmax = 10; Xscl = 1; Ymin = 36; Ymax = 10; Yscl = 1; Xres = 1. Complete the following table. Relation in standard form Sketch each graph. Label as A, B, C, D, or use different colours. y- intercept x- intercept (r) x- intercept (s) Relation in factored form y = (x r) (x s) A y = x 2 4 B y = x 2 9 C y = x 2 16 D y = x 2 25 Consider your results from question 1 and answer the following questions. 2. What is the same about the relations? 3. What is the same about the graphs? 4. What is the same about the vertex of each graph? 5. What do you notice about the r and s values of each relation? 6. Solve this puzzle. How can you find the y-intercept and the x-intercepts of the graph of a quadratic relation of the form y = x 2 a 2? TIPS4RM: Grade 10 Applied: Unit 7 Quadric Relations 40

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