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Section 10.2

Section 10.2. Parabolas. Analyze the Equation of a Parabola. Write y = – x 2 – 2 x + 3 in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. y = – x 2 – 2 x + 3 Original equation y = –1( x 2 + 2 x ) + 3 Factor –1 from the x -terms.

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Section 10.2

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  1. Section 10.2 Parabolas

  2. Analyze the Equation of a Parabola Write y = –x2 – 2x + 3 in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. y = –x2 – 2x + 3 Original equation y = –1(x2 + 2x) + 3 Factor –1 from the x-terms. y = –1(x2 + 2x + ■) + 3 – (–1)(■) Complete the square on the right side. y = –1(x2 + 2x + 1) + 3 + 1(1) The 1 added when you completed the square is multiplied by –1. y = –1(x + 1)2 + 4 (h, k) = (–1,4) Answer: The vertex of this parabola is located at (–1, 4) and the equation of the axis of symmetry is x = –1. The parabola opens downward.

  3. Example: What is the equation y = 2x2 + 4x + 5 written in standard form? What is the vertex, direction of the opening and the axis of symmetry? A.y = 2(x + 2)2 + 5 B.y = 2(x + 1)2 + 5 C.y = 2(x + 1)2 + 3 D.y = 2(x + 2)2 + 3

  4. Example:Graph y = 2x2. For this equation, h = 0 and k = 0. The vertex is at the origin. Since the equation of the axis of symmetry is x = 0, substitute some small positive integers for x and find the corresponding y-values. Since the graph is symmetric about the y-axis, the points at (–1, 2), (–2, 8), and (–3, 18) are also on the parabola. Use all of these points to draw the graph.

  5. Example:Graph y = 2(x – 1)2 – 5. The equation is of the form y = a(x – h)2 + k, where h = 1 and k = –5. The graph of this equation is the graph of y = 2x2 in part A translated 1 unit right and 5 units down. The vertex is now at (1, –5). Answer:

  6. A. B. • C. D. B Example: What is the graph of y = –3(x + 1)2 + 3? • A • B • C • D

  7. directrix: length of latus rectum: or 1 unit. focus: Graph an Equation Not in Standard Form Graph x + y2 = 4y – 1. First write the equation in the form x = a(y –k)2 + h. Then use the following information to draw the graph based on the parent graph x = y2. vertex: (3, 2) axis of symmetry: y = 2 direction of opening: left, since a < 0

  8. A. B. C.D. Which graph is x – y2 = 6y + 2?

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