Michelle E. Chung

* EMAT6680 Assignment 3: Quadratic Equations

Let's think about quadratic equations.  Consider graphs in the xc plane and xb plane.

1. When we vary 'b' :

 Graph of Graph of Graph of The parabola opens down. The vertex of the parabola is (-0.5, 0.25). The axis of symmetry is x=-0.5. The parabola opens down. The vertex of the parabola is (0.5, 0.25). The axis of symmetry is x= 0.5. The parabola opens down. The vertex of the parabola is (-2.5, 6.25). The axis of symmetry is x=-2.5. The parabola opens up. The vertex of the parabola is (-50, -25). The axis of symmetry is x=-50. So, the graph of has following characteristics: The line of symmetry is The vertex of the parabola moves along Since this graph is passing the origin (0, 0), is always a tangent of the graph at (0, 0).

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2. When we vary 'a' :

Graph of Graph of Graph of The parabola opens down. The vertice of the parabolsaares(-0.5, 0.25) and (-0.1.5, 0.0625). The axixes of symmetry arex=-0.5 and x=-0.125. The parabolas opes up. The vertices of the parabolas are (0.5, -0.25) and (0.125, -0.0625). The axixes of symmetry are x=0.5 and x=0.125. When a=0, the graph of the equation would be the straight line, y=-x. So, the graph of has following characteristics: The line of symmetry of the graph is . When a=0, the parabola would be the line, y=-x. Since this graph is passing the origin (0, 0), y=-x would be a tangent of the graph at (0, 0).

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Consider the graphs of the cubic equation . Explore the pattern of roots in the xb, xc, or xd planes. (Okay, maybe look at the xa plane too!)

1. in the xb plane:

 Graph of Graph of Graph of The graph of this equation is NOT a conic. It is just a ALGEBRAIC CURVE. As we see, y=-x is the slant asymptote of the equation. It is under aymptote for the top graph (we can say this by checking the values of a ordered pair on the graph - an absolute value of x-coordinate is always less than an absolute value of y-coordinate) and above asymptote for the bottom graph (similarly, we can say this because an absolute value of x-coordinate is always less than an absolute value of y-coordinate). The SKYBLUE is the graph of the derivative of the given equation. As we see, it is a hyperbola. When the constant term ('d') is -1, the graph of the equation is the reflection the RED GRAPH over the origin, (0,0). It also has y=-x as its slant asymptote. Again, the PURPLE is the graph of the derivative of the given equation. As we see, it is a hyperbola. As we see, the relative maxima and minima seem to be on the y=-2x. (maybe not...???) When the value of the constant term increases, the value of the relative maxima decreases. When the value of the constant term decreases, the value of the relative minima increases. When n=-0.1   When n=0.1   When n=0 When n=0, i.e. the value of constant term is 0, the graph is a hyperbola. The graph of this kind of equation is NOT a conic. It is just a ALGEBRAIC CURVE. It has three different shapes - when the constant term is positive, the graph is a ALGEBRAIC CURVE, not a conic, and most part of the graph is under y=-x - when the constant term is 0, the graph is a CONIC, actually a HYPERBOLA, and y=-x and y-axis are the aymptotes of the hyperbola - when the constant term is negative, the graph is a ALGEBRAIC CURVE, not a conic, and most part of the graph is above y=-x y=-x is the slant asymptote of the equation. When the value of the constant term increases, the value of the relative maxima decreases, and when the value of the constant term decreases, the value of the relative minima increases.

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