Algebra 2B / Precal B

Please read and reply.

  1. Graph the following two functions using a graphing calculator (if you use the graphing calculator ap in your i-phone, you will have a better resolution).                                                                                                                 f(x)= log(x times x) and f(x) = 2logx.                                                                                                                                                 Are the graphs equivalent? What might account for any differences in the two equations?
  2. Write, in your own words, a brief explanation comparing the algebra of complex numbers and the algebra of vectors.


13 thoughts on “Algebra 2B / Precal B

  1. 1. The graphs are not equivalent. They run the same on the positive x-axis but the graph of 2logx doesn’t extend into the negative x-axis. The differences might be because you are squaring a value and then taking it’s log and for the second one, doubling the value obtained after taking a number’s log.

    2. The algebra involved with complex numbers is there to provide roots for every quadratic equation so that any equation to the n degree has exactly n roots, both real and possibly imaginary.
    The algebra involved with vectors is there to provide distance measurements and as a whole, measurements used in order to find things such as areas formed by shapes that intersecting vectors create.
    Both concepts use algebra for calculations but complex numbers use it to find roots while vectors use it to find things such as areas.

    These are my thoughts on your proposed query and some of my explanations may not be 100% correct but at the present, this is what I think it should be.

  2. 1.Yeah. The graphs are equivalent. When f(x)= log(x times x) is expanded, the x times x, which equals x^2 would need to move the 2 in front of the log, which makes the graphs equal.

  3. 2. The two are similar because if you were to plot 3+2i on a graph as a complex alebraic, the point (3,2) would represent the expression and the vector going from the origin to that point would represent the vector.

  4. 1. The graphs are not equivalent. The first graph is a line curving out of the X axis in two different directions, while the second graph is a line curving out of the X axis in only one direction. This is likely because in one equation, X is being doubled while in the other it’s being squared, resulting in different values and lines.
    2. The algebra of complex numbers works on the principle that real numbers can be put together in equations with imaginary numbers. Those numbers can be graphed on a scale where X and Y represent real and imaginary numbers. Imaginary numbers are numbers that do not exist because there have been no mathematical equivalents that have been set to them.

    Vectors are simply made up of points arranged mathematically in an order where they form a line on a coordinate plane. They can be added together to more complex things such as planes and shapes, but all vectors are essentially are points that change formulaically.

    Sorry if it’s bad. I tried.

  5. 1. The graphs are not equivalent. f(x) log(x*x) has a reflection across the y-axis which is not present in f(x) = 2logx. The former take up all four quadrants while the latter crosses through two (I and IV). I assume that squaring a number and then taking its’ log accounts for this difference since the other graph instead involves multiplying a number by two.

    2. I want to say that complex numbers are only arithmetic based, while vectors include a bit of geometry, although that’s all I can say.

  6. 1. The graphs are not equivalent. The first one goes along the positive and negative side of the x-axis. The second one only curves toward the positive side. This change occurs because it one equation you are squaring it, and in the other you are doubling it. I assume that squaring the equation causes it to curve out towards the positive and negative side.

    2. The algebra in complex numbers and vectors are similar because they can both be used to represent a location on a graph. However, complex numbers can be used as roots of a quadratic and vectors represent the direction.

  7. 1.) Both logs are not equivalent because they both go in opposite directions. In the first log, it curves to the negative axis, and the second log curves to the positive axis. The reason for that is because the first one, the two x’s were squared, but on the second one the 2 is be multiplied to the x.

    2. The algebra in complex numbers and in vectors are similar because they both can be graphed on the graph. However, the alebra complex numbers can have square roots and imaginaries and the vectors is more like the distance from the each point to the other.

  8. 1)The graphs are not equivalent; the first one goes on the negative and positive x-axis and the second curves toward the positive side. They differ most likely because one equation is squared and the other is being doubled.
    2)Algebra in complex numbers and in vectors have similar factors;they can both be graphed. Though complex numbers can have imaginary numbers and square roots; where vectors use formulas and are based more on points on a graph.

  9. I did not have access to a graphing calculator but log of x times x is the same thing as 2log of x because: a number times log x is equal to log x to the power of that number. Therefore, log of x times x equals log of x squared.

  10. Complex numbers and vectors are similar because they have similar components. In complex number a + bi , a stands for the real axis (real number) and b stands for the imaginary axis. In the vector the similar situation occurs where vector (ai + bj) i stands for magnitude (length) and j stands for the direction of the vector.

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