Instructions:Answer all the following questions in the space provided. Simplify all answers.
- Identify the vertex, opening direction and axis of symmetry of the parabola defined by:
| a) x = 6y2 |
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c) x + 8 = -4(y - 10)2 |
b) y - 3 = 7(x - 5)2 |
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d) y = -(1/2)(x - 3)2 + 9 |
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a |
b |
c |
d |
Vertex |
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Opening direction |
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Axis of Symmetry |
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- Convert each standard equation into a general equation of the form Ax2 + By2 + Cx + Dy + F = 0.
a) y = 2(x + 4)2 ___________________________________
b) x = (1/10)y2 ___________________________________
- A parabola opens upward and its vertex is located at the origin. The shape of this parabola can be described by a = 5.
| a) Write the equation of this parabola in standard form. | ______________________________ |
b) Write the equation of this parabola in general form. |
______________________________ |
c) Sketch the graph of this parabola on the provided graph paper.
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- A parabola opens left and its vertex is located at (-9, 4). The shape of the parabola can be described by a = -3/4.
| a) Write the equation of this parabola in standard form. | ______________________________ |
b) Write the equation of this parabola in general form. |
______________________________ |
c) Sketch the graph of this parabola on the provided graph paper.
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- Describe the effect that varying h, k and a in the standard equations y - k = a(x - h)2 and x - h = a(y - k)2 has on the graph of a parabola by completing the following chart.
The Effect of h and k on the graph of y - k = a(x - h)2 and x - h = a(y - k)2
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Variable |
The value of the variable decreases |
The value of the variable increases |
The value of the variable is 0. |
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| h |
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| k |
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- Describe the effect of varying a in the standard equations x - h = a(y - k)2 and y - k = a(x - h)2 has on the graph of a parabola by completing the following chart.
The Effect of a on the graph of x - h = a(y - k)2 and y - k = a(x - h)2
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Value of a |
Effect on the Graph of the Parabola |
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| |a| becomes farther from 0 |
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| |a| approaches 0 |
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| |a| = 0 |
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- If the equation Ax2 + By2 + Cx + Dy + F = 0 defines a parabola, then what must be true about the values of the coefficients A and B?
- A parabola is defined by the standard equation x = 12(y - 3)2 and by the general equation 12y2 - x - 72y + 115 = 0.
a) Show that these equations are equivalent.
b) When two equations are equivalent they have identical solution sets. Verify that the point (19, 4) is a solution of both equations.
- The standard equation x = 4y2 defines a parabola.
a) Sketch the parabola on the provided graph paper.
b) What will the graph of the parabola in part (a) look like if it is translated so that its vertex is located at (-5, 8)? Sketch the graph of this parabola.
c) What will be the standard equation of the translated parabola?
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d) Explain how the equation in part (b) describes the translation of the parabola.
- A parabola is formed when a double-napped cone is cut by a plane that is parallel to the generator of the cone.
Describe what happens to the parabola when:
a) The plane moves farther away from the generator.
b) The plane moves closer to the generator.
c) The plane intersects the generator.
- A parabola has one axis of symmetry. Explain why (or show how) this axis of symmetry is useful when sketching the graph of a parabola.
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