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Understanding Parabola Translations in Quadratic Functions

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Learn how to apply vertical and horizontal translations to parabolas, shifting their position without changing shape. Explore vertex form and quadratic functions modeling.

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Understanding Parabola Translations in Quadratic Functions

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  1. Section 8.2 • Parabolas and Modeling

  2. Objectives Vertical and Horizontal Translations Vertex Form Modeling with Quadratic Functions (Optional)

  3. Vertical and Horizontal Translations • The graph of y = x2 is a parabola opening upward with vertex (0, 0). • All three graphs have the same shape. • y = x2 • y = x2 + 1 shifted upward 1 unit • y = x2 – 2 shifted downward 2 units • Such shifts are called translations because they do not change the shape of the graph only its position

  4. Vertical and Horizontal Translations • The graph of y = x2 is a parabola opening upward with vertex (0, 0). • y = x2 • y = (x – 1)2 • Horizontal shift to the right 1 unit

  5. Vertical and Horizontal Translations • The graph of y = x2 is a parabola opening upward with vertex (0, 0). • y = x2 • y = (x + 2)2 • Horizontal shift to the left 2 units

  6. Example Sketch the graph of the equation and identify the vertex. Solution The graph is similar to y = x2 except it has been translated 3 units down. The vertex is (0, 3).

  7. Example Sketch the graph of the equation and identify the vertex. Solution The graph is similar to y = x2 except it has been translated left 4 units. The vertex is (4, 0).

  8. Example Sketch the graph of the equation and identify the vertex. Solution The graph is similar to y = x2 except it has been translated down 2 units and right 1 unit. The vertex is (1, 2).

  9. Example Compare the graph of y = f(x) to the graph of y = x2. Then sketch a graph of y = f(x) and y = x2 in the same xy-plane. Solution The graph is translated to the right 2 units and upward 3 units. The vertex for f(x) is (2, 3) and the vertex of y = x2 is (0, 0). The graph opens upward and is wider.

  10. Example Write the vertex form of the parabola with a = 3 and vertex (2, 1). Then express the equation in the form y = ax2 + bx + c. Solution The vertex form of the parabola is where the vertex is (h, k). a = 3, h = 2 and k = 1 To write the equation in y = ax2 + bx + c, do the following:

  11. Example Write each equation in vertex form. Identify the vertex. a. b. Solution a. Because , add and subtract 16 on the right.

  12. Example (cont) b. This equation is slightly different because the leading coefficient is 2 rather than 1. Start by factoring 2 from the first two terms on the right side.

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