Algebra 2 Tutorial
Preparing for Algebra 2
All too often, students begin classes in algebra II without a clear understanding of what is required to have a smooth and fulfilling learning experience. Obviously, a good grasp of concepts, processes and procedures introduced in algebra I is of paramount importance. Beyond this, the student should be aware that algebra II delves heavily into various types of equations and the various graph-types that correspond to them. In short, there are ten essential types of equations to be aware of, each of which yields its own unique kind of graph on the Cartesian plane. Listing them two at a time, these are referenced as linear and quadratic, cubic and quartic, radical and rational, exponential and logarithmic, and (finally) absolute value and circle equations.
Without proper understanding of the types of equations to be presented in algebra II and the fact that each type yields its own peculiar graph, students often find themselves swimming upstream, so to speak. Students need to gain familiarity with the forms and appearance of equations they will meet in algebra II. From algebra I they know how to identify linear and quadratic equations and the graph-type of each. For example, = graphs a straight line, and – graphs a parabola. It would be very beneficial to have familiarity with the other types of equations and their respective graphs.
Beyond linear and quadratic equations are the following:
The general form of a cubic equation is where letters a, b and c are coefficients of the variable x and d is a constant. Typically, the cubic equation will yield a graph with two up/down turns of a curved line (as opposed to the single, U-shaped turn of the curved line related to a quadratic equation).
The general form of a quartic equation is . The graph related to a quartic equation will typically show 3 up/down turns of the curved line.
The general form of a radical equation is where, as is evident, the operation is under a radical. The curved line associated with a radical equation typically shows an abrupt stop on a graph, indicating areas where values are forbidden. These would be values of that yield a negative answer–in this example, or less than . A negative number under a radical is known as an imaginary number.
The general form of a rational equation is
Because the denominator of a fraction cannot be, or be equal to, zero, there are limits to values the variable can take, for a given value. This typically results in a graph with two curved lines that have a very noticeable gap in between. The gap indicates where values of make the denominator equal zero.
The general forms, respectively, of exponential and logarithmic equations are and . The graphs of these equations show a line that falls or rises abruptly due to the extreme effect on values, when exponents are in the equation, as well as when variables increase or decrease logarithmically.
The general form of absolute value equations is (read “y equals the absolute value of ax + b”). The graph of this kind of equation typically yields two straight lines that meet to form a V shape with vertex on the x-axis. The indication is that will only have positive values for any value of . Finally, the general form of a circle equation is for circles with their center at the origin of the Cartesian plane. Otherwise the equation for a circle graph takes the form .
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