# Michelle E. Chung

* EMAT6680 Assignment 2: Parabolas

 Graph w/Two Fixed Value Graph of

1. Construct graphs for the parabola

for different values of a, b, and c. (a, b, c can be any rational numbers.)

 When Graph of a=b=c=1 (a,b,c>0) As we know, this is the simplest parabola graph, which is    . The parabola opens up. The vertex of the parabola is (0,0). The axis of symmetry is x=0. When a=b=c=1, The parabola opens up. The vertex of the parabola is (-1/2, 3/4). The axis of symmetry is x=- 1/2. When a=c=1 and b=3, The parabola opens up. The vertex of the parabola is (-3/2, -5/4). The axis of symmetry is x=- 3/2. When a=c=1 and b=-3, The parabola opens up. The vertex of the parabola is (3/2, -5/4). The axis of symmetry is x= 3/2. When a=1, b=3 and c=-4, The parabola opens up. The vertex of the parabola is (-5/2, -41/4). The axis of symmetry is x=- 5/2. When a=-0.3, b=5 and c=5.3, The parabola opens down. The vertex of the parabola is (25/3, 339/5). The axis of symmetry is x= 25/3. When a=-0.3, b=-7.8 and c=-2, The parabola opens down. The vertex of the parabola is (-13, 48.7). The axis of symmetry is x= -13.

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2. Fix two of the values for a, b, and c. Make at least 5 graphs on the same axes as you vary the third value.

How do you change the equations to explore other graph?

Graph of Graph of Graph of when c varies: when b varies: when a varies: As you see,        'c' is y-intercept of the graph. As c increases, the graph moves up. As c decreases, the graph moves down. As you see,        'b' determines x-coordinate of the vertex of the graph with a. As b increases, the graph moves to the right. As b decreases, the graph moves to the left. As you see,        'a' determines whether the graph opens up or opens down and also determines the width of the graph. If a is positive, the graph opens up. If a is negative, the graph opens down. As |a| increases, the graph is getting thinner. As |a| decreases, the graph is getting wider. 'a' determines whether the graph opens up or opens down and also determines the width of the graph. - If a is positive, the graph opens up, and if a is negative, the graph opens down. - As |a| increases, the graph is getting thinner, and as |a| decreases, the graph is getting wider. 'b' determines x-coordinate of the vertex of the graph with 'a'. - As b increases, the graph moves to the right, and as b decreases, the graph moves to the left. 'c' is y-intercept of the graph. - As c increases, the graph moves up, and as c decreases, the graph moves down.

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4. Interpret your graphs. What happens to

(i.e. the case where b=1 and c=2) as a is varied?
Is there a common point to all graphs? What is it?
What is the significance of the graph where a=0?

Do the similar interpretations for other sets of graphs.
How does the shape change?
How does the position change?

Graphs of when n is in [-10, 10] These are the graphs of and when a=-4. The vertex of the graph of is (0.125, 2.0625). Since |a| is 4 and is greater than 1, it is thinner than . Both graphs pass (0, 2).     These are the graphs of and when a=-0.05. The vertex of the graph of is (10, 7). Since |a| is 0.05 and is less than 1, it is wider than . Both graphs pass (0, 2).     These are the graphs of and when a=0. There is no vertex because the graph of is a straight line. Both graphs pass (0, 2).     These are the graphs of and when a=0.08. The vertex of the graph of is (-6.25, -1.125). Since |a| is 0.08 and is less than 1, it is wider than . Both graphs pass (0, 2).     These are the graphs of and when a=8. The vertex of the graph of is (-0.0625, 1.96875). Since |a| is 8 and is greater than 1, it is thinner than . Both graphs pass (0, 2). These are the graphs of and when a=0. When a=0, the graph of is a straight line because the equation would be y=bx+c, which is a linear equation. So, to be a parabola, 'a' should not be '0'; otherwise, it would be a straight line. Even though the graph is a straight line, the graph of when a=0 still passes (0, 2). 'a' determines whether the graph opens up or opens down and also determines the width of the graph. - If 'a' is positive, the graph opens up. - If 'a' is negative, the graph opens down. - When |a| increases, the graph is getting thinner. - When |a| decreases, the graph is getting wider. When a=0, the graph is a straight line because y=bx+c, which is a linear equation. Since c is y-intercept, the graph always passes (0, 2). So, (0, 2) is the common point of the graphs.

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