MPM2D Lesson 20: Assessment of Learning Strategies

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning Lesson 20 Assessment of Learning Task 1: Knowledge (25 marks) 1. Complete the following table: (10 marks) Equation in vertex form 1 -1 3 0 -5 0 0 -3 4 Opens up or down? Up Down Up Compressed or stretched vertically? Neither Neither Stretched by a factor of 3 Vertex a minimum or maximum? Minimum Maximum Minimum Minimum or maximum value Equation of axis of symmetry Coordinates of vertex (0, 0) (-5, -3) (0, 4) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 1

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning 2. Expand and simplify the following product, using the distributive property: . (3 marks)  Expanding with proper signs  Collecting like term 3. Write the following equations in factored form. Remember to common factor first (if possible). a) (2 marks) b) (3 marks) Two numbers whose is sum -7 and whose product is -27 are -9 and 3. c) (4 marks) d) (3 marks) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 2

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning Task 2: Application (25 marks) 4. Given the following graph of a parabola, find its equation, in the form (7 marks) Actions Results  Vertex form. From graph:  Coordinates of the vertex (3, 1) and the given point (1, 0) are read directly from the given graph. Substitute into the equation:  Replace and in the equation by their values obtained from the graph to find . Simplifying : Therefore, it is  In brackets, Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 3

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TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning 5. A parabola is defined by the equation . a) Write the equation in factored form. (2 marks) The factored form of the equation is b) Identify the -intercepts of the parabola. (2 marks) The factors are and . Therefore, the -intercepts of the parabola are and . c) State the equation of the axis of symmetry. (1 mark) The axis of symmetry is located at the average of the zeros. Therefore, the equation of the axis of symmetry is . d) Determine the coordinates of the vertex. (2 marks) Since the vertex is located along the parabola at the axis of symmetry, the -value from the equation of the axis of symmetry can be substituted into the equation to determine the - coordinate of the vertex. Therefore, the coordinates of the vertex are (5, -9). e) Sketch the parabola. (2 marks) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 4

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning 6. Write the following expression in vertex form by completing the square. (4 marks) The vertex is located at . There is no vertical stretch or compression, and no reflection. 7. A golf ball is hit from the top of a tee. The quadratic equation describes its height, , in metres as time, , in seconds passes. Determine how long the ball is in the air. (5 marks) Hint: Use the quadratic formula. When the ball hits the ground, its height is 0. To determine how long the ballis in the air, use the quadratic formula to find the roots of the equation Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 5

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning and The first root, can be considered inadmissible because the sensor cannot detect the height of the ball is hit from the tree at 0 s and lands at approximately 4 s, it is in the air for approximately 4 s. Task 3: Communication (25 marks) 8. Explain in detail the role of , , and in the equation . You may use words, numerical examples, and sketches to support your explanation. (8 marks) = By changing the value of in the equation, , the parabola is either shifted up or down. The shape of the parabola stays the same, but the parabola is translated (shifted) vertically by the units. When is a positive number or value, the parabola shifts up. When is a negative number or value, the parabola shifts down. For example, in , the parabola shifts 7 units up, while, for example, in , the parabola shifts 8 units down. The -coordinate of the vertex matches the value of . = The effects of the value of in the equation are: 1. The shape of the transformed parabola is not congruent to the standard parabola. 2. Some of the parabolas open downward. (example: if the value of is negative, like -1, the parabola will open downwards) 3. Some of the parabolas are stretched, making them appear narrower, while others are flattened. (example: if the value of is greater than 1, like 3, the parabola`s shape will be stretched vertically. if the value of is in between than 0 and 1, like 0.5, the parabola`s shape will be compressed vertically and will look flattened.) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 6

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TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning = Changing the value of in the equation shifts the standard parabola to the left or to the right. The shape of the parabola still remains unchanged. The parabola is translated to the right when the value of is positive. For example, the value of is positive in . The parabola is shifted five units to the right. The parabola is shifted to the left when the value of is negative. For example, the value of is negative in . The parabola is shifted six units to the left. The value of represents the -coordinate of the vertex. The value of does not affect the -coordinate. 9. Think about the standard parabola defined by . How does the parabola defined by compare to the standard parabola? Describe all of the transformations. Then, draw a reasonable sketch of both parabolas. (6 marks) Compared to the standard parabola defined by the equation , is reflected across the x -axis and stretched vertically by a factor of 4. It also has a horizontal translation of 3 units to the left and a vertical translation of 7 units down. Its vertex is located at (-3, -7), and equation of the axis of symmetry is x = -3. (6 marks) 10. Factoring is an algebraic process, which is an essential one in the field of mathematics. Consider the following equation: This equation in factored form can be written as: Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 7

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning Explain using words and algebra, how the zeros, the axis of symmetry, and the vertex can be obtained. (6 marks) The zeros ( x- intercepts) can be determine from the factored form. The zeros of a function are located along the x- axis, where any point has a y- coordinate of 0. Set to find the corresponding x- coordinates. The expression on the right side of the equals sign is product of two factors: and . The product is equal to zero only when one of the factors is 0. This occurs when or Therefore, the x- intercepts are and The axis of symmetry is halfway between the x- intercepts. Therefore, its equation corresponds to the average of the zeros. The equation of the axis of symmetry is . To determine the coordinates of the vertex, substitute into the original equation. The coordinates of the vertex are (3, 7). 11. Consider the graph of : If the graph were extended much farther to the left from the interval shown, do you think the graph could ever touch or cross the -axis? Explain in detail. (3 marks) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 8

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning If the graph of is traced to the left, it will approach the -axis ever more closely , without either touching it or crossing it , because this type of graph always divides by 2, and dividing such positive quantities by 2 can never make a zero or negative result. can never be negative in the relation . 12. The graphs of and are shown here: Identify two major differences in behaviour between these graphs. (2 marks) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 9

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TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning Two major differences in behaviour between and are: i. has a line of symmetry, while does not. ii. passes through (0, 0), while passes through (0, 1). Task 4: Thinking (25 marks) 13. For the following graph of the parabola, find its equation in the form . Then, express the answer in standard or expanded form, . (9 marks) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 10

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning Actions Results  Vertex form. From graph:  Coordinates of the vertex (3, 1) and the given point (1, 0) are read directly from the given graph. Substitute into the equation:  Replace and in the equation by their values obtained from the graph to find . Simplifying : Therefore,  In brackets, Vertex form to standard form: Express the square as a product of the binomial with itself: Multiply each term of the first binomial with each term of the second binomial: Multiply with the trinomial: Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 11

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning So, the standard form of the equation is 14. Sketch the parabola described by the equation . (6 marks) Rewrite the equation in factored form: From the factored form, identifiy the zeros as and Find the average of the zeros to locate the axis of symmetry at Substitute into the original equation to determine the coordinates of the vertex: Therefore the coordinates of the vertex are (5, -8). 15. The amount of money that a local charity earns by selling T-shirts at a mall depends on the price of each T-shirt. The monthly profit, , in dollars is given by the quadratic equation where represents the price of each T-shirt. a) Determine the roots of the equation. (3 marks) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 12

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TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning The equation contains a common factor of Therefore, it can be written as: The roots of the equation can be determined by solving: Divide both sides by . Solve this equation using the quadratic formula: and The roots of the equation are and b) Determine the vertex of the parabola. (1 mark) Since the vertex is a point on the parabola at the axis of symmetry, its -coordinates can be determined by substituting into the original equation. The approximate location of the vertex is (17.86, 4660.71). c) Draw a sketch of the relationship. Label the axes and the vertex. (2 marks) Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 13

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning (17.86, 4660.71) d) Under what circumstances will the charity maximize its profit (make the most money)? (1 mark) The charity will make the most profit when each T-shirt is sold for around $17.86. The projected monthly profit under this condition is about $4660.71. e) Describe the significance of the roots in terms of the graph and in terms of the charity’s venture. (1 mark) The roots and can be seen on the graph as the -intercept of the parabola. The root shows that if each T-shirt is sold for $29.40, the charity will make a profit of $0. Similarly, if each T-shirt is sold at $6.32, the charity will also make no profit. f) Determine the range of prices that the charity could consider, in order to make a monthly profit of at least $3000. (2 marks) To determine the interval, or a range of prices, from the graph, I should draw a horizontal line representing a monthly profit of $3000. Then I should locate the points were this line intersects the parabola. The -values of these points will determine the range of prices. In other words, the charity can choose from prices between these two -values. Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 14

TVO ILC MPM2D Lesson 20 Lesson 20 Assessment of Learning So from this graph, it seems that the charity should consider prices from $11.00 to $24.75. Copyright © 2018 The Ontario Educational Communications Authority. All rights reserved. 15

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