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.
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.
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?
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
STANDARD TO VERTEX FORM Convert from Standard form to vertex form y = 2x2 + 10x + 7 y = -x2 + 4x – 5
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)
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
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
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?