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QUADTRATIC RELATIONS

QUADTRATIC RELATIONS. QUADTRATIC RELATIONS. A relation which must contain a term with x2 It may or may not have a term with x and a constant term (a term without x) It can be written in 3 forms: Standard form Vertex form Zeros form. 1) Standard form. y= ax 2 + bx + c

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QUADTRATIC RELATIONS

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  1. QUADTRATIC RELATIONS

  2. QUADTRATIC RELATIONS • A relation which must contain a term with x2 • It may or may not have a term with x and a constant term (a term without x) • It can be written in 3 forms: • Standard form • Vertex form • Zeros form

  3. 1) Standard form y= ax2 + bx + c Where a, b, and c are any number

  4. 2) VERTEX form y= a (x-h)2+ k Where a, h, and k are any number

  5. 3) Zeros form y= a(x-b)(x-c) Where a, b, and c are any number

  6. A quadratic relation always has the shape of a parabolawhich can open up or down(see next slide)

  7. The line which seems to cut in half is called the line of symmetry

  8. The Vertex is the point which is the highest or lowest point on the graph

  9. Opens up- When A > 0Opens DOWN- When A < 0

  10. Characteristics of a quadratic relation written in vertex form • y= a (x-h)2 + k • LINE OF SYMMETRY • To find the line of symmetry, the expression in the brackets equals 0. Then solve for x. If there are no brackets, the line of symmetry is x=0. Note the value of h is always the opposite value (negative of) the number inside the bracket. (the sign of h is always the opposite of the operation in the brackets). • If the x intercept is not 0, it will always be the same number in the bracket but have the OPPOSITE sign.

  11. Examples: • A) y= -2 (x+3)2 – 4 • LINE OF SYMMETRY IS x+3=0 or x=-3 • B) y= 5(x-1)2 + 2 • LINE OF SYMMETRY IS x-1=0 or x=1 • C) y= 0.5x2+ 5 • LINE OF SYMMETRY IS x=0 • D) y= 2(x-7)2 • LINE OF SYMMETRY IS x-7=0 or x=7

  12. Last number (h, k) VERTEX Line of symmetry • To find the vertex, the x coordinate must be the same as the line of symmetry, and the y coordinate must be the last number (k) A) y=-2(x+3)2 -4 LINE OF SYMMETRY IS x=-3, Vertex is (-3, -4) B) y=5(x-1)2 +2 LINE OF SYMMETRY IS x=1, Vertex is (1, 2) C) y=0.5x2 +5 LINE OF SYMMETRY IS x=0, Vertex is (0, 5) D) y=-2(x-7)2 LINE OF SYMMETRY IS x=7, Vertex is (7, 0)

  13. GrAPHING QUADRATICS IN VERTEX FORM • Step 1: determine the line of symmetry • Step 2: Calculate the y coordinates of the vertex • Step 3: draw x, y grid, label the x and y axis, and then plot the vertex • Step 4: Now use a table of values to determine points near the vertex (2 numbers before and 2 numbers after the line of symmetry) • Step 5: Plot the points from your table of values and join them with a curved line. Add arrows to each end and label the graph with the equation.

  14. Last class

  15. GrAPHING QUADRATICS IN Standard form y= ax2 + bx + c • Step 1: determine the line of symmetry and the vertex • This is a little more difficult since you now have the term bx. • A. To find the line of symmetry, you use the formula: B. To find the y value of the vertex, substitute this into the equation. y= ax2 + bx + c

  16. GrAPHING QUADRATICS IN Standard form y= ax2 + bx + c B. To find the y value of the vertex, substitute this into the equation. Example: Determine the line of symmetry and the coordinates of the vertex. y= 4x2 – 2x + 5 Line of symmetry: a=4, b=-2, c=5

  17. GrAPHING QUADRATICS IN Standard form y= 4x2 – 2x + 5 B. To find the y value of the vertex, substitute this into the equation. At x=0.25, y= 4(0.25)2– 2(0.25) + 5 = 0.5 - 0.5 + 5 =5 ∴, the coordinates of the vertex are (0.25, 5)

  18. GrAPHING QUADRATICS IN Standard form COMPLETE QUESTION 4! Determine the line of symmetry and the coordinates of the vertex for a - f

  19. Steps for graphing quadratic equations in standard form: • Step 1: determine the line of symmetry • Step 2: Calculate the y coordinates of the vertex • Step 3: draw x, y grid, label the x and y axis, and then plot the vertex • Step 4: Now use a table of values to determine points near the vertex (2 numbers before and 2 numbers after the line of symmetry) • Step 5: Plot the points from your table of values and join them with a curved line. Add arrows to each end and label the graph with the equation.

  20. GrAPHING QUADRATICS IN Standard form y= ax2 + bx + c 1) Find the Line of Symmetry y= 3x2 - 6x + 5 Identify the variables a=3, b=-6, c=5 Substitute into the formula:

  21. GrAPHING QUADRATICS IN Standard form y= 3x2- 6x + 5 2) Find the Coordinates of the Vertex At x=1, y= 3(1)2 - 6(1) + 5 =3 – 6 +5 =2 ∴, the coordinates of the vertex are (1, 2)

  22. 3) Graph the vertex y= 3x2- 6x + 5 (1, 2)

  23. 4. Use a table of values to determine points near vertex Use a table of values with x=1,2,3 and x=0,-1 to find more points close to the vertex. y= 3x2 - 6x + 5 =3(-1)2 - 6(-1) + 5 =13 y= 3x2 - 6x + 5 =3(0)2 - 6(0) + 5 =4

  24. Steps for graphing quadratic equations in standard form: • Step 1: determine the line of symmetry • Step 2: Calculate the y coordinates of the vertex • Step 3: draw x, y grid, label the x and y axis, and then plot the vertex • Step 4: Now use a table of values to determine points near the vertex (2 numbers before and 2 numbers after the line of symmetry) • Step 5: Plot the points from your table of values and join them with a curved line. Add arrows to each end and label the graph with the equation.

  25. GrAPHING QUADRATICS IN Standard form y= ax2 + bx + c 1) Find the Line of Symmetry y= -2x2 + 8x - 3 Identify the variables a=-2, b=8, c=-3 Substitute into the formula:

  26. GrAPHING QUADRATICS IN Standard form y=-2x2+ 8x - 3 2) Find the Coordinates of the Vertex At x=1, -2(2)2+ 8(2) - 3 =-8+16-3 =5 ∴, the coordinates of the vertex are (2, 5)

  27. 3) Graph the vertex y=-2x2+ 8x - 3 (2, 5)

  28. 4. Use a table of values to determine points near vertex Use a table of values with x=2,3,4 and x=1,0 to find more points close to the vertex. y=-2(2)2+ 8(2) - 3 =-8+16-3 =5

  29. (0, -3) y=-2x2+ 8x - 3 (1, 3) (2, 5) (3, 3) (4, -5) 5) Plot & connect the points, arrows and label

  30. GrAPHING QUADRATICS IN Standard form COMPLETE QUESTION 5! Graph a-h quadratic relations using: 1) Find the line of symmetry 2) Calculate the y coordinates of the vertex 3) Graph the vertex 4) Use a table of values to determine points near the vertex.

  31. Opens up- When A > 0Opens DOWN- When A < 0

  32. Optimal Value • The height of the highest or lowest point • Always the last number • That is the maximum value if the graph opens down • That is the minimum value if the graph opens up. • The OPTIMAL VALUE always corresponds to the y coordinate of the vertex. To find the value of the optimal value: • A) Find the line of symmetry • B) find the vertex, by substitution (This is the optimal value)

  33. y a > 0 x a < 0 OPTIMAL VALUE The standard form of a quadratic function is: y = ax2 + bx + c The parabola will open up when the a value is positive. OPENS UP-When A > 0 If the parabola opens up, the lowest point is called the vertex (minimum). The parabola will open down when the a value is negative. Opens DOWN- When A < 0 If the parabola opens down, the vertex is the highest point (maximum).

  34. GrAPHING QUADRATICS IN Standard form COMPLETE QUESTION 6 and 7! Find the maximum and minimum values

  35. QUESTION 9 MAXIMUM MINIMUM (X-INTERCEPTS)

  36. QUESTION 10

  37. POSSIBLE QUIZ • SEE FILE!!!! INTRODUCING PARABOLAS

  38. STEP PATTERNS http://www.youtube.com/watch?v=4gMaw64RDlchttp://www.youtube.com/watch?v=JPorKyVh58Q • The first differences show us the step pattern of the parabola. (I.e. in the case of y = x2 it would have a 1,3,5 step pattern) • More importantly, all parabolas with ‘a’ values of 1 or (-1) will have 1,3,5 step patterns • It also tells us the direction of opening • If the second differences are (+) the parabola opens up • If the second differences are (-) the parabola opens down OVER 1 (right or left) from the vertex point, UP 1² = 1 from the vertex point OVER 2 (right or left) from the vertex point, UP 2² = 4 from the vertex point OVER 3 (right or left) from the vertex point, UP 3² = 9 from the vertex point OVER 4 (right or left) from the vertex point, UP 4² = 16 from the vertex point

  39. Question 11(A) • A ball is thrown into the air. The height of the ball after x seconds in the air is given by the quadratic equation h= -5x2 + 30x + 3, where h is the height in metres. Find the maximum height of the ball.

  40. Question 11(B) • Alvin shoots a rocket into the air. The height of the rocket is h=-5x2+200x, where h is the height in metres. Find the maximum height of the rocket.

  41. Question 11(D) • The cost, C, in dollars, to hire workers to build a new playground at a park can be modeled by C = 5x2– 70x+ 700, where x is the number of workers hired to do the work. How many workers should be hired to minimize the cost?

  42. Question 11(E) • Jeff wants to build five identical rectangular pig pens, side by side, on his farm using 32m of fencing. The area that he will evaluate is given by the equation, A= -3w2 + 16w, where A is the total area in m2, and w is the width of the pig pen in m. Determine the dimensions (length and width) of the enclosure that would give his pigs the largest possible area. Calculate this area.

  43. Question 11(F) • Studies have shown that 500 people attend a high school basketball game when the admission price is $2.00. In the championship game admission prices will increase. For every 20¢ increase 20 fewer people will attend. The revenue for the game will be R= -4x2+ 60x + 100, where R is the revenue in dollars, x is the number of tickets sold. a) What price will maximize the venue? b) What is the maximum revenue? • = -4x2 + 60x + 100 • = -4(30)2+ 60(30) + 100 • =-1700

  44. X versus Yintercepts • (0, y) (x, 0)

  45. EXAMPLE: • Find the y intercept of y=5x2+ 2x – 4 • Solution: Substitute x=0 • y=5(0)2+ 2(0) – 4 • = - 4 ∴, the coordinates of the y-intercept are (0, -4) • (0, y) Calculate by substituting 0 for y First number is 0

  46. Y-intercept Where it touches the y-axis • To find the y intercept, the x coordinate is 0 and the y coordinate is found by substituting 0 for x in the relation and calculating the value for y. • (0, y) Calculate by substituting 0 for y First number is 0

  47. EXAMPLE: • Find the y intercept of y=5x2+ 2x – 4 • Solution: Substitute x=0 • y=5(0)2+ 2(0) – 4 • = - 4 ∴, the coordinates of the y-intercept are (0, -4) • (0, y) Calculate by substituting 0 for y First number is 0

  48. x-intercept (aka “zeros”) Where it touches the x-axis • To find the x intercept, the y coordinate is 0 and the x coordinate is found by substituting the values for a, b and c into the equation below, and then calculate x. QUADRATIC FORMULA • (x, 0) Calculate by using the above formula The second number is 0

  49. x-intercept (aka “zeros”) Where it touches the x-axis • So the quadratic formula is used to find the x-intercepts. • There can be 2, 1 or no x-intercepts. • (x, 0) Calculate by using the above formula The second number is 0

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