MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

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1 FOM 11 T0 INTRODUCING PARABOLAS 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) QUADRATIC FUNCTION = a statement containing different kinds of variables where one variable lacks a visible exponent and the largest exponent on the other variable is a. e.g. y = x + x +1 ; Μ = p p +1 ; z = 3b 8b. ) STANDARD FORM = the standard form of a quadratic function is written in this pattern: y = ax +bx +c or f ( x) = ax +bx +c where a, b and c are real numbers and a 0. 3) VERTEX FORM = the vertex form of a quadratic function is written in this pattern: y = a ( x p) +q or f ( x) = a ( x p ) +q where a, p and q are real numbers and a 0. 4) PARABOLA = the name of the graph of a quadratic function. INTRODUCING QUADRATIC FUNCTIONS I) A QUADRATIC FUNCTION IS A STATEMENT CONTAINING DIFFERENT KINDS OF VARIABLES WHERE ONE VARIABLE LACKS A VISIBLE EXPONENT AND THE LARGEST EXPONENT ON THE OTHER VARIABLE IS A. e.g. y = x + x +1 ; Μ = p p +1 ; z = 3b 8b. A) Quadratic functions can also be written using function notation. The function notation symbol (i.e. f ( x)) simply replaces the variable that lacks a visible exponent. e.g. f ( x) = x + x +1 ; Μ( p) = p p +1 ; z( b) = 3b 8b. B) Quadratic functions can also be written in very different forms: STANDARD FORM and VERTEX FORM. 1) The standard form of a quadratic function is written in this pattern: y = ax +bx +c or f ( x) = ax +bx +c where a, b and c are real numbers and a 0. e.g. y = x + x +1 ; Μ( p) = p +1 ; z( b) = 3b 8b. ) The vertex form of a quadratic function is written in this pattern: y = a x p where a, p and q are real numbers and a 0. e.g. y = ( x 3) +1 ; Μ( p) = ( p + ) 8 ; z( b) = 3 ( b 8 ). ( ) +q or f x C) The graphs of quadratic functions are called PARABOLAS. Parabolas are or shaped graphs. ( ) +q ( ) = a x p MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) LEADING COEFFICIENT = the coefficient of the term having the largest exponent. Given the standard form of a quadratic function y = ax +bx +c, a = leading coefficient. ) VERTEX = the highest point of a parabola that opens down, or the lowest point of a parabola that opens up. Its coordinates give the horizontal location and the value of the parabola's maximum or minimum. 3) MAXIMUM = the highest point of a parabola. The value of the maximum is the y-coordinate of the vertex. 4) MINIMUM = the lowest point of a parabola. The value of the minimum is the y-coordinate of the vertex. 5) AXIS OF SYMMETRY = a vertical line passing through the vertex that divides the parabola into two halves, each half being a mirror reflection of the other. THE AXIS OF SYMMETRY IS ALWAYS WRITTEN AS AN EQUATION: x = the x-coordinate of the vertex. 6) Y-INTERCEPT = the point where the graph touches or crosses the y-axis. Parabola s have exactly one y- intercept. ALL y-intercepts HAVE AN x-coordinate OF x = 0.

2 FOM 11 T0 INTRODUCING PARABOLAS 7) ZERO = the point where the graph touches or crosses the x-axis. Parabola's can have 0, 1 or zeros. Zeros are also known as x-intercepts. CHARACTERISTICS OF PARABOLAS I) GRAPHS OF QUADRATIC FUNCTIONS ARE CALLED PARABOLAS. All parabolas have the seven characteristics discussed below. Study them carefully. Be sure you understand and memorize them. A) DIRECTION OF OPENING: The DIRECTION OF OPENING describes a parabola's up ( ) or down ( ) orientation. 1) INVESTIGATION 1: Determining the direction of opening of a parabola. INSTRUCTIONS: Remove page 10 of these notes and use the graphs numbered 1through 4 to answer these questions. {Answers are on page 6 of these notes.} 1) State the sign of the leading coefficient of the quadratic functions in graphs 1 &? ) What is the direction of opening of the parabolas? 3) State the sign of the leading coefficient of the quadratic functions in graphs 3 & 4? 4) What is the direction of opening of the parabolas? 5) Describe the relationship between the sign of a quadratic function s leading coefficient and the direction of opening of its parabola. ) DIRECTION OF OPENING: The DIRECTION OF OPENING describes a parabola's up ( ) or down ( ) orientation. The leading coefficient (a) determines the parabola's direction of opening. If a is positive, the PARABOLAS opens up ( ). If a is negative, the PARABOLAS opens down ( ). 3) REQUIRED PRACTICE 1: Identify the DIRECTION OF OPENING of the parabolas for these quadratic functions. {Answers are on page 6 of these notes.} 1) y = x + 4x 1 ) f ( x) = x + 3x + 5 3) y = 8 31x 4x 4) H ( x) = 5 11x + x 5) y = 6 +11x 3x 6) y = 6( x 6) 1 7) y = x 8 8) y = 0.5( x + 3) 9) y = 7 5x B) VERTEX: All parabolas have a VERTEX, which is the highest point, the maximum, or the lowest point, the minimum, of the parabola. The coordinates of the vertex are the x and y-coordinates of the parabola s MAXIMUM or MINIMUM. 1) REQUIRED PRACTICE : Answer these questions. {Answers are on page 6 of these notes.} a) State the coordinates of the vertex of the graphs numbered 1 through 4 on page 10 of these notes. b) Page 393: State the coordinates of the vertex of the parabolas found in question 11(a, c & d). c) Page 370: State the coordinates of the vertex of the parabolas found in question 6.

3 FOM 11 T0 INTRODUCING PARABOLAS 3 C) MAXIMUM or MINIMUM: All parabolas have either a MAXIMUM, the highest point of a parabola, or a MINIMUM, the lowest point of a parabola. 1) INVESTIGATION : Determining a parabola s maximum or minimum value. INSTRUCTIONS: Use the graphs numbered 1 through 4 on page 10 of these notes to answer these questions. {Answers are on page 6 of these notes.} 1) What is the sign of the quadratic function s leading coefficient of graphs 1 and? ) What is the direction of opening of the parabolas of graphs 1 and? 3) Do the parabolas in graphs 1 and have a maximum or minimum value? 4) What is the sign of the quadratic function s leading coefficient of graphs 3 and 4? 5) What is the direction of opening of the parabolas of graphs 3 and 4? 6) Do the parabolas in graphs 3 and 4 have a maximum or minimum value? 7) Describe the relationship between the sign of a quadratic function s leading coefficient and whether its parabola has a maximum or minimum. 8) State the value of the maximum or minimum of each parabola: i.e. minimum of y = 4 1) ) 3) 4) ) MAXIMUM or MINIMUM: All parabolas have a MAXIMUM, the highest point of a parabola, or a MINIMUM, the lowest point of a parabola. The leading coefficient determines a parabola's direction of opening, which determines whether the parabola has a maximum or a minimum. This means a quadratic function s leading coefficient determines whether the parabola has a maximum or minimum. If a is positive, the parabola opens up ( ) and it has a MINIMUM. If a is negative, the parabola opens down ( ) and it has a MAXIMUM. 3) Maximum and minimum refer to up and down which relates to the y-axis of a grid. This means the value of the maximum or minimum is the y-coordinate of the parabola's vertex. 4) REQUIRED PRACTICE 3: Answer these questions. {Answers are on page 6 of these notes.} a) State whether the parabolas for these quadratic functions have a maximum or minimum value. Justify your answer. 1) y = x + 4x 1 ) f ( x) = x + 3x + 5 3) y = 8 31x 4x 4) H ( x) = 5 11x + x 5) y = 6 +11x 3x 6) y = 6( x 6) 1 7) y = x 8 8) y = 0.5( x + 3) 9) y = 7 5x b) Page 370: Question 6. {Answers are on page 565 of the text.} D) AXIS OF SYMMETRY: All parabolas have an axis of symmetry, which is a vertical line passing through its vertex. The axis of symmetry divides the parabola into two identical halves, each a symmetrical mirror reflection of the other. Since the axis of symmetry is a vertical line passing through the parabola's vertex, it is described with a vertical constant equation x = x-coordinate of the vertex.

4 FOM 11 T0 INTRODUCING PARABOLAS 4 1) REQUIRED PRACTICE 4: Determining the equation for a parabola s axis of symmetry. {Answers are on page 6 of these notes.} a) INSTRUCTIONS: Draw a dashed line representing the axis of symmetry on the graphs numbered 1 through 4 on page 10 of these notes then state the value of the equation of the axis of symmetry of each parabola: i.e. Axis of symmetry: x =7. 1) ) 3) 4) b) Page 369: State the equation of the axis of symmetry of the parabolas found in questions 3 & 4. E) y-intercept: The y-intercept is the point on the y-axis where the parabola touches or crosses. Every parabola has exactly one y-intercept. EVERY y-intercept HAS AN x-coordinate = 0. 1) SAMPLE PROBLEM 5: Study the parabola found on page 386 of your text. Its y-intercept is y = 1. ) INVESTIGATION 3: Determining the relationship between a parabola s y-intercept and the constant term of its quadratic function. INSTRUCTIONS: Use the graphs numbered 1 through 4 on page 10 of these notes to answer these questions. {Answers are on page 6 of these notes.} 1) Complete this table. Quadratic function s Parabola s Graph Constant term y-intercept ) What form (pattern) are the quadratic functions written in? 3) Describe the relationship between the constant term of a quadratic function written in standard form and its y-intercept. 3) REQUIRED PRACTICE 5: State the y-intercept of the parabola s defined by these quadratic functions. {Answers are on page 6 of these notes.} 1) y = x + 4x 1 ) f ( x) = x + 3x + 5 3) y = 8 31x 4x 4) H ( x) = 5 11x + x 5) y = 6 +11x 3x 6) y = 6x 6x 1 7) y = x 8 8) y = 0.5x + x 9) y = 7x 5x 10) Page 369: State the y-intercept of the parabolas defined by the functions in question. F) ZERO: A ZERO is the point where the parabola touches or crosses the x-axis. A parabola will have either 0, or 1 or zeros. Study the parabolas found at the top of page 359 of your text. The left most parabola has two zeros, the middle parabola has one zero, while the right most parabola does not have any zeros. 1) SAMPLE PROBLEMS 6: Study the parabola on page 386 of your text, it has two zeros which are recorded as zeros: x = 1, 4 or x-intercepts = ( 1, 0), ( 4, 0). Study the parabola on page 411 of your text, it has does not have zeros because it does not touch or cross the x-axis. ) REQUIRED PRACTICE 6: Answer these questions. {Answers are on page 6 of these notes.} a) State the zeros of each parabola in the graphs numbered 1 through 4 on page 10 of these notes. b) Page 369: State the zeros of the parabolas found in questions 3 & 4.

5 FOM 11 T0 INTRODUCING PARABOLAS 5 G) DOMAIN and RANGE: Every parabola has a DOMAIN which consists of the values of the x-axis that are used to create it, and a RANGE which consists of the values of the y-axis that are used to create it. 1) REQUIRED PRACTICE 7: Use the graphs numbered 1 through 4 on page 10 of these notes to answer these questions. {Answers are on page 7 of these notes.} 1) Complete this table. Graph Vertex Domain Range ) Page 369: State the range of the parabolas found in questions 3 & 4 3) What is the relationship between the vertex of a parabola and its range? H) REQUIRED PRACTICE 8: For these graphs, state the information indicated. {Answers are on page 7 of these notes.} 1) Direction of opening: Vertex: Equation of Axis of sym: x = Maximum or Minimum: Value of Max or Min: y= Zero(s): x = y-intercept: y-int = Domain: Range: ) Direction of opening: Vertex: Equation of Axis of sym: x = Maximum or Minimum: Value of Max or Min: y= Zero(s): x = y-intercept: y-int = Domain: Range:

6 FOM 11 T0 INTRODUCING PARABOLAS 6 ANSWERS TO THE REQUIRED PRACTICE INVESTIGATION 1 from page 1) positive ) up 3) negative 4) down 5) When the function s leading coefficient is positive the parabola opens up, when the function s leading coefficient is negative the parabola opens down. Required Practice 1 from page 1) up ) down 3) down 4) up 5) down 6) up 7) up 8) down 9) down Required Practice from page Graph1) ( 4, 9) Graph) 1, 11d) ( 3, 9) 6a) (, 4) 6b) 0, 3 ( ) Graph3) ( 3, 4) Graph4) ( 1, 3) 11a) (, 8) 11c) ( ) 6c) ( 5, 10) (, 4) INVESTIGATION from page 3 1) positive ) up 3) minimum 4) negative 5) down 6) maximum 7) When the quadratic function s leading coefficient is positive the parabola opens up and it has a minimum, when the quadratic function s leading coefficient is positive the parabola opens down and it has a maximum. 8) 1) Minimum of y = 9 ) Minimum of y = 1) Maximum of y = 4 ) Maximum of y = 3 Required Practice 3 from page 3 a1) minimum because a is positive causing the parabola to open up a) maximum because a is negative causing the parabola to open down a3) maximum because a is negative causing the parabola to open down a4) minimum because a is positive causing the parabola to open up a) maximum because a is negative causing the parabola to open down a6) minimum because a is positive causing the parabola to open up a7) minimum because a is positive causing the parabola to open up a8) maximum because a is negative causing the parabola to open down a9) maximum because a is negative causing the parabola to open down Required Practice 4 from page 4 Graph1) Axis of symmetry: x = 4 Graph) Axis of symmetry: x = 1 3a) Axis of symmetry: x = 1 3b) Axis of symmetry: x =.5 4a) Axis of symmetry: x = 4b) Axis of symmetry: x = 3 4c) Axis of symmetry: x = 4 4d) Axis of symmetry: x =.5 INVESTIGATION 3 from page 4 1) Complete this table? Quadratic function s Parabola s Graph Constant term y-intercept 1 7 y -int =7 0 y -int =0 3 5 y-int = y -int =0 ) Standard form 3) When a quadratic function is written in standard form its y-intercept is equal to its constant term. Required Practice 5 from page 4 1) y - int = 1 ) y - int = 5 3) y - int = 8 4) y - int = 5 5) y - int = 6 6) y - int = 1 7) y - int = 8 8) y - int = 0 9) y - int = 0 Text question a) y - int = 8 b) y - int = 0 Required Practice 6 from page5 Graph1) zeros: x = 7, 1 Graph) zeros: x =, 0 Graph3) zeros: x = 1, 5 Graph4) zeros: x = 0, 3a) zeros: x = 0, 3b) zeros: x! 1, 6 4a) zeros: x = 1, 3 4b) zeros: x! 1.4, 9. 4c) zeros: none 4d) zeros: x! 1, 6

7 FOM 11 T0 INTRODUCING PARABOLAS 7 Required Practice 7 from page 5 1) Complete this table. Graph Vertex Domain Range 1 ( 4, 9) x R y 9 ( 1, ) x R y 3 3, 4 4 1, 3 ( ) x R y 4 ( ) x R y 3 ) Page 369: Q 3 & 4. Question Vertex Domain Range ( ) x R y (.5, 9.) x R y 9. 3a 1, 3b 4a ( 3, 4) x R y 4 4b ( 4, 8) x R y 8 4c ( 3, 1) x R y 1 ( ) x R y 1.5 4d.5, 1.5 3) The starting value of the range is the same number as the y-coordinate of the vertex. Required Practice 8 from page 5 1) up V 3, 4 ( ) A of sym: x = 3 Min of y = 4 zeros: x = 5, 1 y-int = 5 domain: x R range: y 4 ) down V( 1.5,9.5) A of sym: x = 1.5 Max of y = 9.5 zeros: x! 1.7, 4.6 y-int! 7. domain: x R range: y 9.5

8 FOM 11 T0 INTRODUCING PARABOLAS

9 FOM 11 T0 INTRODUCING PARABOLAS 9 ASSIGNMENT: PRINT THIS INFORMATION ON YOUR OWN GRID PAPER LAST then FIRST Name T0 INTRODUCING PARABOLAS Block: Show the process required to complete each problem to avoid receiving a zero grade. Neatness Counts!! (Marks indicated in italicized brackets.) REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! Copy the sentences numbered 1-6 then match it with the correct term listed below. (3) Vertex Constant term Leading Coefficient Domain Axis of symmetry Coefficient Zero y-intercept Range Parabola 1) Name of the graph of a quadratic function. ) Point of the graph where the graph touches or crosses the y-axis. 3) The values of the x-axis that are used to make a graph. 4) Another name for the x-intercept of a quadratic function. 5) The number in front of the variable having the largest exponent. 6) Vertical line passing through the vertex. State a - g for questions 7 & 8. a) Coordinates of the vertex: i.e. V 5, 8 ( ) (1 each) b) The value of the maximum or minimum: i.e. The minimum is y = 8 (1 each) c) Equation of axis of symmetry: i.e. Axis of Sym: x = 5 (1 each) d) Zeros: i.e. Zeros: x = 1, 5 (1 each) e) y-intercept: i.e. y-int =5 (1 each) f) Domain: i.e. Domain: x R (1 each) g) Range: i.e. Range: y 4 (1 each) 7) 8) /17