graphs of quadratic function grade 9.pptx

graphs of quadratic function grade 9.pptx

• 1.
  Forms of QuadraticFunctions Give the different forms of quadratic function, determine the axis of symmetry, opening of the parabola and the vertex; and Identify the different forms of quadratic functions.
• 2.
  The vertex is(h,k), and the axis of symmetry is x = h which divides the graph of quadratic function into two equal parts. The value of a indicates the opening of the graph. If the value of a is positive, the graphs opens upward. If the value of a is negative, the graph opens downward.
• 3.
  To find thevalue of the vertex use the formula of: 𝒉 = −𝒃 𝟐𝒂 and 𝒌 = 𝟒𝒂𝒄−𝒃𝟐 𝟒𝒂 and the axis of symmetry is x = h.
• 4.
  Examples: 1.y = 3x²- 4 2.y = - x² + 2x + 3 3.y = 5x² 4.y = 2x² + 8x + 16
• 5.
  Graphs of Quadratic Functions
• 6.
  Learning Targets  Drawthe graphs of the quadratic function;  Analyze the effect of changing the values of a, h and k in the equation y = a(x-h)²+k of the quadratic function on its graph.
• 7.
  GRAPHS OF QUADRATICFUNCTION All quadratic functions have graphs similar to this graph. This U-shaped is called a parabola. 𝑷𝒂𝒓𝒂𝒃𝒐𝒍𝒂 𝒗𝒆𝒓𝒕𝒆𝒙 ( 0, -1 ) The “turning points” of the graph is called the vertex of the parabola. The parabola is symmetric with respect to a line that passes through the vertex. This line is called the axis of symmetry. It divides the parabola into two parts so that one part is a reflection of the other part. 𝒂𝒙𝒊𝒔 𝒐𝒇 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒚
• 8.
  EXAMPLE 1. Constructa table of values for the quadratic function and identify at least five points of the function. 𝑓 𝑥 = 1 − 2𝑥 − 𝑥2 𝒌 = 𝟒𝒂𝒄 − 𝒃𝟐 𝟒𝒂 𝒉 = −𝒃 𝟐𝒂 𝒂 = −𝟏 𝒃 = −𝟐 𝒄 = 𝟏 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒗𝒆𝒓𝒕𝒆𝒙. 𝒉 = −(−𝟐) 𝟐(−𝟏) 𝒉 = 𝟐 −𝟐 𝒉 = −𝟏 𝒌 = −𝟒 − 𝟒 −𝟒 𝒌 = 𝟒(−𝟏)(𝟏) − (−𝟐)𝟐 𝟒(−𝟏) 𝒌 = −𝟖 −𝟒 𝒌 = 𝟐 𝑽𝒆𝒓𝒕𝒆𝒙: (−1, 2)
• 9.
  x y −𝟏 𝟐 −𝟐 −𝟑 0 𝟏 1 −𝟐1 −𝟐 𝑓 𝑥 = 1 − 2(−2) − (−2)2 𝒙 = −𝟐 𝑓 𝑥 = 1 − 2𝑥 − 𝑥2 𝑓 𝑥 = 1 𝑓 𝑥 = 1 + 4 − 4 𝑓 𝑥 = 1 − 2(−3) − (−3)2 𝒙 = −𝟑 𝑓 𝑥 = −2 𝑓 𝑥 = 1 + 6 − 9 𝑓 𝑥 = 1 − 2(0) − (0)2 𝒙 = 𝟎 𝑓 𝑥 = 1 𝑓 𝑥 = 1 − 0 − 0 𝑓 𝑥 = 1 − 2(1) − (1)2 𝒙 = 𝟏 𝑓 𝑥 = −2 𝑓 𝑥 = 1 − 2 − 1 𝑽𝒆𝒓𝒕𝒆𝒙: (−1, 2)
• 10.
  x y −𝟏 (−𝟏, 𝟐) −𝟐 −𝟑 0𝟏 1 −𝟐 1 −𝟐 𝑓 𝑥 = 1 − 2𝑥 − 𝑥2 𝑣𝑒𝑟𝑡𝑒𝑥: (−1, 2) 𝟐 (−𝟐, 𝟏) (−𝟑, −𝟐) (𝟏, −𝟐) (𝟎, 𝟏) 𝐴𝑥𝑖𝑠 𝑜𝑓 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦: (−1) 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 1 𝑂𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: −2.5, 0.5 𝑅𝑎𝑛𝑔𝑒: 𝑦 𝑦 ≤ 2 𝐷𝑜𝑚𝑎𝑖𝑛: 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
• 11.
  EXAMPLE 2. Drawthe graph of the quadratic function. 𝑦 = −𝑥2 − 2𝑥 + 3 𝒌 = 𝟒𝒂𝒄 − 𝒃𝟐 𝟒𝒂 𝒉 = −𝒃 𝟐𝒂 𝒂 = −𝟏 𝒃 = −𝟐 𝒄 = 𝟑 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒗𝒆𝒓𝒕𝒆𝒙. 𝒉 = −(−𝟐) 𝟐(−𝟏) 𝒉 = 𝟐 −𝟐 𝒉 = −𝟏 𝒌 = −𝟏𝟐 − 𝟒 −𝟒 𝒌 = 𝟒(−𝟏)(𝟑) − (−𝟐)𝟐 𝟒(−𝟏) 𝒌 = −𝟏𝟔 −𝟒 𝒌 = 𝟒 𝑽𝒆𝒓𝒕𝒆𝒙: (−1, 4)
• 12.
  x y −𝟏 𝟒 −𝟐 −𝟑 0 𝟏 3 𝟎3 𝟎 𝑓 𝑥 = −(0)2 − 2 0 + 3 𝒙 = −𝟐 𝑦 = −𝑥2 − 2𝑥 + 3 𝑓 𝑥 = 3 𝒙 = −𝟑 𝑓 𝑥 = 0 𝒙 = 𝟎 𝑓 𝑥 = 3 𝒙 = 𝟏 𝑓 𝑥 = 0 𝑓 𝑥 = 0 − 0 + 3 𝑓 𝑥 = −(−3)2 − 2 −3 + 3 𝑓 𝑥 = −9 + 6 + 3 𝑓 𝑥 = −(−2)2 − 2 −2 + 3 𝑓 𝑥 = −4 + 4 + 3 𝑓 𝑥 = −(1)2 − 2 1 + 3 𝑓 𝑥 = −1 − 2 + 3
• 13.
  x y −𝟏 (−𝟏, 𝟒) −𝟐 −𝟑 0𝟏 3 𝟎 3 𝟎 𝑦 = −𝑥2 − 2𝑥 + 3 𝑣𝑒𝑟𝑡𝑒𝑥: (−1, 4) 𝟒 (−𝟐, 𝟑) (−𝟑, 𝟎) (𝟏, 𝟎) (𝟎, 𝟑) 𝐴𝑥𝑖𝑠 𝑜𝑓 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦: (−1) 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 3 𝑂𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: −3 𝑎𝑛𝑑 1 𝑅𝑎𝑛𝑔𝑒: 𝑦 𝑦 ≤ 4 𝐷𝑜𝑚𝑎𝑖𝑛: 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
• 14.
  EXAMPLE 3. Drawthe graph of the quadratic function. 𝑦 = (𝑥 − 2)2 − 4 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒗𝒆𝒓𝒕𝒆𝒙. 𝒉 = 𝟐 𝒌 = −𝟒 𝑽𝒆𝒓𝒕𝒆𝒙: (2, −4) 𝒇 𝒙 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌 𝑦 = (𝑥 − 2)2 − 4
• 15.
  x y 2 −𝟒 1 0 3 𝟒 −3 𝟎−3 𝟎 𝒙 = 𝟏 𝑦 = (−1)2 − 4 𝑓 𝑥 = −3 𝒙 = 𝟎 𝑓 𝑥 = 0 𝒙 = 𝟑 𝑓 𝑥 = −3 𝒙 = 𝟒 𝑓 𝑥 = 0 𝑦 = (𝑥 − 2)2 − 4 𝑦 = (1 − 2)2 − 4 𝑦 = (−2)2 − 4 𝑦 = (0 − 2)2 − 4 𝑦 = (1)2 − 4 𝑦 = (3 − 2)2 − 4 𝑦 = (2)2 − 4 𝑦 = (4 − 2)2 − 4
• 16.
  (𝟒, 𝟎) 𝑦 =(𝑥 − 2)2 − 4 𝑣𝑒𝑟𝑡𝑒𝑥: (2, −4) (𝟏, −𝟑) (𝟎, 𝟎) (𝟑, −𝟑) (𝟐, −𝟒) 𝐴𝑥𝑖𝑠 𝑜𝑓 𝑆𝑦𝑚𝑚𝑒𝑡𝑟𝑦: (2) 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 0 𝑂𝑝𝑒𝑛𝑠 𝑢𝑝𝑤𝑎𝑟𝑑 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 0 𝑎𝑛𝑑 4 𝑅𝑎𝑛𝑔𝑒: 𝑦 𝑦 ≥ −4 𝐷𝑜𝑚𝑎𝑖𝑛: 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 x y 2 −𝟒 1 0 3 𝟒 −3 𝟎 −3 𝟎
• 17.
  Generate a tableof values and graph each quadratic functions and determine the vertex, axis of symmetry, y- and x- intercepts, maximum or minimum value, opening of parabola, range, and domain. NOTE: Use GRAPHING PAPER for your graph and other paper for the table of values. 1. 𝒚 = 𝒙𝟐 + 𝒙 − 𝟒 2. 𝒚 = 𝒙𝟐 − 𝟑𝒙 − 𝟒 3. 𝒚 = 𝟐(𝒙 − 𝟏)𝟐 + 𝟑 4. 𝒚 = (𝒙 − 𝟑)𝟐 + 𝟐