Presentation on theme: "Sullivan Algebra and Trigonometry: Section 4.3 Quadratic Functions/Models Objectives Graph a Quadratic Function Using Transformations Identify the Vertex."— Presentation transcript:
Sullivan Algebra and Trigonometry: Section 4.3 Quadratic Functions/Models Objectives Graph a Quadratic Function Using Transformations Identify the Vertex and Axis of Symmetry of a Quadratic Function Graph a Quadratic Function Using the Vertex, Axis of Symmetry, and Intercepts Use a Graphing Utility to Find the Quadratic Function of Best Fit
A quadratic function is a function of the form: where a, b, and c are real numbers and a 0. The domain of a quadratic function consists of all real numbers. The graph of a quadratic function is called a parabola.
Graphs of a quadratic function f(x) = ax 2 + bx + c, a 0 a > 0 Opens up Vertex is lowest point Axis of symmetry a < 0 Opens down Vertex is highest point Axis of symmetry
Graph the function fxxx() Find the vertex and axis of symmetry. First, rewrite the function by completing the square
Now, graph the function using the transformations discussed in Chapter 3. (0,0) (2,4) (0,0) (2, -8)
(2, 0) (4, -8) (2, 7) (4, -1) Vertex : (2,7)
Properties of the Quadratic Function Parabola opens up if a > 0. Parabola opens down if a < 0.
Given the function, determine whether the graph opens upward or downward. Find the vertex, axis of symmetry, the x- intercepts, and the y-intercept. fxxx() x-coord. of vertex: y-coord. of vertex: Axis of symmetry: x b a 2 3 Vertex: (-3, -13)
fxxx() y-intercepts: f(0) = 5; so the y-intercept is (0,5) x-intercepts: Solve the equation = 0 fxxx() The x-intercepts are approximately (-5.6,0) and (-.45,0) Summary: Parabola opens up Vertex (-3, -13) y-intercept: (0,5) x-intercepts: (-0.45, 0) and (-5.55,0)
Now, graph the function using the information found in the previous steps. Vertex: (-3, -13) (-5.55, 0)(-0.45, 0) (0, 5)
If a mathematical model of a real world situation leads to a quadratic function, the properties of the function can be applied to the model. Example: The John Deere Company has found that the revenue from sales of heavy duty tractors is a function of the unit price p that it charges. If the revenue R is what unit price p should be charged to maximize revenue? What is the maximum revenue?
Note that the revenue function is a quadratic function that opens downward, since a < 0. So, the maximum revenue will be achieved at the vertex of the function. x-coord. of vertex: y-coord. of vertex: So, the maximum revenue occurs when they charge $1900. The maximum revenue is $1,805,000.
Suppose you throw a ball straight up and record the height of the ball in 0.5 second intervals and obtain the following data:
Use a graphing utility to complete the following: a.) Draw a scatter diagram of the data. b.) Find the quadratic function of best fit. c.) Draw the quadratic function of best fit on the scatter diagram.