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Quadratic Functions and Their Graphs

Quadratic Functions and Their Graphs. More in Sec. 2.1b. 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

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Quadratic Functions and Their Graphs

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  1. Quadratic Functions and Their Graphs More in Sec. 2.1b

  2. 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............

  3. 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

  4. 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

  5. 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

  6. 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.

  7. Guided Practice 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

  8. 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:

  9. 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?

  10. 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

  11. 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.

  12. 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:

  13. Guided Practice 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.

  14. Guided Practice 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!!!

  15. Guided Practice 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!!!

  16. Guided Practice 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!!!

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