Alg II Unit 4-2 Standard Form of a Quadratic Function


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Alg II Unit 4-2 Standard Form of a Quadratic Function

  1. 1. 4-2 STANDARD FORM OF AQUADRATIC FUNCTIONChapter 4 Quadratic Functions and Equations©Tentinger
  2. 2. ESSENTIAL UNDERSTANDING ANDOBJECTIVES Essential Understanding: y = ax2 + bx + c, a, b, and c provide key information about its graph Objectives: Students will be able to:  Graph quadratic equations  Identify the vertex, axis of symmetry, minimum and maximum from standard form.
  3. 3. IOWA CORE CURRICULUM Algebra A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F.IF.7. Graph functions expressed symbolically, and show features of the graph, by hand in simple cases and using technology for more complicated cases. F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F.IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) F.BF.1. Write a function that describes a relationship between two quantities.
  4. 4. REVIEW What is vertex form for a quadratic equation? Standard Form: y = ax2 + bx + c How do you think we switch from vertex form to standard form? Put each of the following equations in standard form. y = (x – 3)2 + 2 y = (x + 4)2 – 1 y = -(x – 1)2 + 5 y = 3(x + 2)2 + 7 How do you think we determine the vertex of an equation in standard form?
  5. 5. PROPERTIES Change the vertex form to standard form: a(x-h)2 + k a=a b = -2ah c = ah2 +k Solve the above for h and k
  6. 6. STANDARD TO VERTEX FORM Convert from Standard form to vertex form y = 2x2 + 10x + 7 y = -x2 + 4x – 5
  7. 7. PROPERTIES Without a calculator: The graph f(x) = ax2 + bx + c is parabola If a > 0, it opens up If a < 0, it opens down Axis of symmetry: x = -b/(2a) Vertex  X = -b/(2a)  Y = f(-b/(2a)) Y – intercept (0, C)
  8. 8. WITHOUT THE CALCULATOR Without graphing find the vertex, axis of symmetry, min/max value, y intercept, using the properties of quadratic functions. Then Graph the function by hand y = x2 + 2x + 3 y = -2x2 + 2x – 5 y= 2x2 + 5
  9. 9. EXAMPLES Using the calculator, graph y = 2x2 + 8x – 2 Identify the vertex, minimum/max, and the axis of symmetry, the domain, and the range Using the calculator, graph y = -3x2 – 4x +6 Identify the vertex, minimum/max, and the axis of symmetry, the domain, and the range
  10. 10. INTERPRETING A QUADRATIC GRAPH Where in real life do you see parabolas? The Zhaozhou Bridge in China is the oldest arch bridge, dating to A.D. 605. You can model the arch with the function f(x) = -0.001075x2 + 0.131148x, where x and y are in feet. How high is the bridge above its supports? Why does the model not have a constant term?
  11. 11. HOMEWORK Pg 206 – 207 # 9 – 30 (3s), 32, 38, 41, 45, 46 14 problems