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 |
| | a | b | c | d |
Vertex | (0, 0) | (5, 3) | (-8, 10) | (-3, 9) |
Opening direction | right | up | left | down |
Axis of Symmetry | y = 0 | x = 5 | y = 10 | x = -3 |
- Convert each standard equation into a general equation of the form Ax2 + By2 + Cx + Dy + F = 0.
a) y = 2(x + 4)2 2x2 + 16x - y + 32 = 0
b) x = (1/10)y2 (1/10)y2 - x = 0
- 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. | y = 5x2 |
b) Write the equation of this parabola in general form. |
5x2 + y = 0 |
c) Sketch the graph of this parabola.
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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.
x + 9 = (-3/4)(y - 4)2 |
b) Write the equation of this parabola in general form.
(3/4)y2 + x - 6y + 21 = 0 |
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
| Variable | The value of the variable decreases | The value of the variable increases | The value of the variable is 0. |
|---|
| h | the parabola shifts left | the parabola shifts right | the vertex of the parabola always lies on the y-axis |
| k | the parabola shifts down | the parabola shifts up | the vertex of the parabola always lies on the x-axis |
- 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
| Value of a | Effect on the Graph of the Parabola |
|---|
| |a| becomes farther from 0 |
the shape of the parabola becomes narrower
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| |a| approaches 0 |
the shape of the parabola becomes wider
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| |a| = 0 |
the result is a line
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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?
If the equation Ax2 + By2 + Cx + Dy + F = 0 defines a parabola, then either A or B must be zero and the other coefficients must be nonzero.
- A parabola is defined by the standard equation x - 7 = 12(y - 3)2 and by the general equation 12y2 - x - 72y + 115 = 0.
a) Show that these equations are equivalent.
One way is to expand the standard equation:
x - 7 = 12(y - 3)2
x = 12(y2 - 6y + 9) + 7
x = 12y2 - 72y + 108 + 7
12y2 - x - 72y + 115 = 0
This expanded equation is exactly equal to the general equation given in the question, therefore, we have just shown that the two equations given are equivalent.
Another way to show equivalency would be to complete the square on the general equation:
12y2- x - 72y + 115 = 0
12(y2 - 6y + 9) - (12)(9) + 115 = x
12(y - 3)2 = x - 7
x - 7 = 12(y - 3)2
This completed equation is exactly equal to the standard equation given in the question, therefore we have just shown that the two equations given 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.
Substitute the point (19, 4) into both equations:
x - 7 = 12(y - 3)2
19 - 7 = 12(4 - 3)2
12 = 12(1)2
12 = 12
12y2 - x - 72y + 115 = 0
12(4)2 - 19 - 72(4) + 115
= 12(16) - 19 - 288 + 115
= 192 - 19 - 288 + 115
= 0
- The standard equation x = 4y2 defines a parabola.
a) Sketch the parabola.
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?
(x + 5) = 4(y - 8)2
d) Explain how the equation in part (b) describes the translation of the parabola.
The expression (x + 5) describes that the parabola is translated 5 units left and the expression (y - 8) describes that the parabola is translated 8 units up.
- 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.
The parabola gets wider
b) The plane moves closer to the generator.
The parabola gets narrower
c) The plane intersects the generator.
The result is a line.
- 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.
The graph of the parabola is symmetrical on either side of the axis of symmetry. Therefore, if we plot points on one side of the axis of symmetry, we can easily graph the symmetrical points on the other side of the axis of symmetry. As an example: if we plot the point (3, 5) on the graph, and we know the x-axis is the axis of symmetry, then we know that the point (-3, 5) will also lie on the graph of the parabola.
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