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Published byMartin Atkins Modified over 8 years ago
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Parabolas Our Fourth Conic Form (9.5)
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POD What other conic forms have we looked at? Why do we call them conic forms? What’s the primary skill we’ve used to manipulate equations?
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Parabolas we’ve seen Rewrite this equation into vertex form. (What’s the first step?) y = x 2 – 4x + 3
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Parabolas we’ve seen Rewrite this equation into vertex form. (What’s the first step? Completing the square.) y = x 2 – 4x + 3 y – 3 = x 2 – 4x y – 3 + 4 = x 2 – 4x + 4 y + 1 = (x – 2) 2 What is the vertex of this parabola? How does it open?
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Parabolas we’ve seen For y + 1 = (x – 2) 2, vertex (2, -1) opens in the positive vertical direction What are the intercepts? The axis of symmetry?
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Parabolas we’ve seen For y + 1 = (x – 2) 2, vertex (2, -1) opens up What are the intercepts? y-intercept (0, 3) x-intercepts (3, 0) and (1, 0) The axis of symmetry? x = 2
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New parabola What would x = y 2 – 4y + 3 look like? Try rewriting it into vertex form.
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New parabola What would x = y 2 – 4y + 3 look like? Try rewriting it into vertex form. x = y 2 – 4y + 3 x – 3 = y 2 – 4y x – 3 + 4 = y 2 – 4y + 4 x + 1 = (y – 2) 2 What’s the vertex? Want to make a guess as to what it looks like?
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New parabola What would x = y 2 – 4y + 3 look like? What’s the vertex? Want to make a guess as to what it looks like? How does it compare to the first parabola with an equation of y = x 2 – 4x + 3 ?
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New parabola What would x = y 2 – 4y + 3 look like? This graph is the inverse of y = x 2 – 4x + 3. That means the x and y values switch. The vertex that was (2, -1) becomes (-1, 2). The axis of symmetry is now y = 2. It opens in the positive horizontal direction.
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New parabola What would x = y 2 – 4y + 3 look like? This graph is the inverse of y = x 2 – 4x + 3. That means the x and y values switch. The y-intercept that was (0, 3) becomes the x-intercept at (3,0).
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New parabola What would x = y 2 – 4y + 3 look like? This graph is the inverse of y = x 2 – 4x + 3. Remember, that means the x and y values switch. The x-intercepts that were (1, 0) and (3,0) become y- intercepts at (0, 1) and (0, 3).
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New parabola What would x = y 2 – 4y + 3 look like? A key point is that parabolas with y 2 terms open sideways.
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Start with the horizontal parabola Without having the inverse to start with, find the vertex, intercepts, and axis of symmetry for this parabola. x = -2y 2 + 12y – 10
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Start with the horizontal parabola Without having the inverse to start with, find the vertex, intercepts, and axis of symmetry for this parabola. Complete the square. x = -2y 2 + 12y – 10 x + 10 = -2(y 2 – 6y) x + 10 – 18 = -2(y 2 – 6y + 9) x – 8 = -2(y – 3) 2
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Start with the horizontal parabola For x – 8 = -2(y – 3) 2 vertex x-intercept y-intercepts (Use Quadratic Formula.) axis of symmetry
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Start with the horizontal parabola For x – 8 = -2(y – 3) 2 vertex (8, 3) x-intercept (-10, 0) y-intercepts (0, 5) and (0, 1) axis of symmetry y = 3 Graph it.
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Start with the horizontal parabola For x – 8 = -2(y – 3) 2 vertex (8, 3) x-intercept (-10, 0) y-intercepts (0, 5) and (0, 1) axis of symmetry y = 3
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