Quadratic Functions and Their Graphs More in Sec. 2.1b Homework: p. 176 19-41 odd

What are they??? Quadratic Function – a polynomial function of degree 2 Recall the basic squaring function?  Any quadratic function can be obtained via a sequence of transformations of this basic function…………observe............

reflection across x-axis, Quick Examples Describe how to transform the basic squaring function into the graph of the given function. Sketch its graph by hand. Vertical shrink by 1/2, reflection across x-axis, translation up 3 units

Translation left 2 units, Quick Examples Describe how to transform the basic squaring function into the graph of the given function. Sketch its graph by hand. Translation left 2 units, vertical stretch by 3, translation down 1 unit

More Generalizations… Consider the graph of If , the parabola opens downward If , the parabola opens upward Axis of Symmetry (axis for short) – line of symmetry Vertex – point where the parabola intersects the axis

Definition: Vertex Form of a Quadratic Function (Standard Quadratic Form) Any quadratic function , , can be written in the vertex form The graph of f is a parabola with vertex (h, k ) and axis x = h, where and . If a > 0, the parabola opens upward, and if a < 0, it opens downward.

Guided Practice Vertex: (–2, 5) Axis: x = –2 Vertex: (3/2, –1) Find the vertex and axis of the graph of the given functions. Vertex: (–2, 5) Axis: x = –2 Vertex: (3/2, –1) Axis: x = 3/2

Guided Practice Use vertex form of a quadratic function to find the vertex and axis of the given function. Rewrite the equation in vertex form. Standard form: So, a = –3, b = 6, and c = –5 Coordinates of the vertex:

How about a graph to support these answers? Guided Practice Use vertex form of a quadratic function to find the vertex and axis of the given function. Rewrite the equation in vertex form. Vertex: Axis: Vertex form of f : How about a graph to support these answers?

First, let’s make sure we remember how to complete the square… Solve by completing the square: Get x terms by themselves Complete the square!!! Factor Take square root of both sides Solve for x

We can complete a similar process when changing forms of quadratics: Use completing the square to describe the graph of the given function. Support your answer graphically. The graph of f is a upward-opening parabola with vertex (–2, –1), axis x = –2, and intersects the x-axis at about –2.577 and –1.423.

Characterizing the Nature of a Quadratic Function Point of View Characterization Verbal Polynomial of degree 2 Algebraic or Graphical Parabola with vertex (h, k), axis x = h; opens upward if a > 0, opens downward if a < 0; initial value = y-int = f(0) = c; x-intercepts:

Guided Practice Vertex: (5/2, –77/4), Axis: x = 5/2, Opens Use completing the square to describe the graph of the given function. Support your answer graphically. Vertex: (5/2, –77/4), Axis: x = 5/2, Opens upward, intersects the x-axis at about 0.538 and 4.462, Vertically stretched by 5.

Guided Practice Check with a calculator graph!!! Write an equation for the parabola shown, using the fact that one of the given points is the vertex. Plug in (3, –2) for (h, k): (6, 1) Plug in (6, 1) for (x, y), solve for a: (3, –2) Check with a calculator graph!!!

Guided Practice Check with a calculator graph!!! Write an equation for the parabola shown, using the fact that one of the given points is the vertex. Plug in (–1, 5) for (h, k): (–1, 5) Plug in (2,–13) for (x, y), solve for a: (2,–13) Check with a calculator graph!!!

Guided Practice Check with a calculator graph!!! Write an equation for the quadratic function whose graph contains the vertex (–2, –5) and the point (–4, –27). Plug in the vertex: Plug in the point: Check with a calculator graph!!!