In this picture you see that we had to create a number with a repeating decimal. You must remember that in order to get the sum, it has to be rational. First we get the first portion of the decimal as the a sub one. Then we place that into the summation formula. It is critical to remeber how to properly add, multiply and divide fractions.
The trickiest part of doing a partial decoposition problem with repeating factors is distinguishing the difference between one problem with distinct factors. We always have to start of by getting the least common denominator and multiplying the same to both top and bottom for each different part. After combining like terms and simplifying, remember to use those systems and plug them into a matrix. Matrices can be very tricky because although it can be easy, the steps are very easy to mess up. At the end it is crucial to go back and check your problems answer.
The trickiest part of this problem would be to follow the appropriate steps accordingly. Make sure that whatever is multiplied to the bottom of a fraction must always be multiplied to the top. Also, it is easy to forget or confuse coefficients. Go back and check all distributions and simplifying.
Matrices can be very confusing to solve. First we have to make sure we simplyify the equations before moving on. Also, make sure that you copy down the correct COEFFICIENTS because if one value is wrong, the entire solution is. The other tricky part is canceling out to get zeroes and ones because it although they are simple steps, it's easy to make a mistake.
The trickiest part of graphing a logarithmic equation is remembering how to find the components of these equations. For a logarithmic the asymptote is always x = h, which in this case is -3. The domain is always dependent on the asymptote. Another tricky part is plugging in the equation correctly into my calculator because we have to use the change of base formula.
ps. The end of the video showing the graph and the plotted points were cut off. The ordered pairs I used were: (-2, -2), (1, -1), (3, -.708), (5, -.5). The graph goes near the asymptote, almost touching it but never crossing it.
The trickiest part to graphing the equation was getting my intercepts. When we get our asymptote, it is better to then figure out whether we have x-intercepts which could be really confusing. We will only have x-intercepts if it crosses the x axis either below or above the asymptote (depending on the sign of a).When finding an x-intercept, we usually will have to divide logs so remembering to use parenthesis is critical.
In this video the trickiest part is to realize that any log can be broken up in different ways. A log can be expanded in several different ways correctly. Also, it is easy to forget that regardless of the amount of clue already given to us we always have two "hidden" clues. In this case, ours were log base 8 of 8 which if we were refer back to concept eight, we know that if the base and the value are the same number, it simplifies to one. The second given clue is log 8 of 1 equalling zero because any log base of 1 equals zero. Always remember that the numbers MUST be broken down using only using the clues to then expand the log completely.
In this video we discuss how to get all needed plots in order to graph a rational function. First, we need a rational function equation and a rational function equation must be a fraction with both the numerator and the denominator being polynomials. Then we decide whether our equation has a horizontal asymptote or a slant because a function can NEVER have both. If we refer to our in-class-chant we know that if the degree on Top is One bigger than the degree on bottom, we have a slant! And to get that slant equation, we need to use long division but ignore the reminder. Next is to find vertical asymptotes and holes. If after factoring both the top and the bottom we notice that a set in parenthesis can be canceled from top and bottom, then that becomes our hole(s) and we can now use the "simple" equation to set the denominator to zero and get our vertical asymptotes. Rememeber, Domain is always the vertical asymptotes and the holes (only x-intercept). To get the x and y intercepts, the simple equation must always be used. For x, set the numerator equal to zero; for the y, plug in 0 for every x value. After getting all of that, we must find three different plots for each side and only then may we solve. Check your graphing to get your plot ordered pairs.
The trickiest part in solving this equation was actually figuring out how to graph it. When plugging into my graphing calculator, it wouldn't give me an accurate graph and I didn't know why until I remembered that *parenthesis must always be put seperately around both the numerator and the denominator. After that it was then easy to solve.
In this Student Video, I cover the process of how to find all zeroes of 4th/5th degree polynomials. There were four main things asked to look for: one, find possible positive and negative zeroes; two, find rational zeroes for each function; three, find all zeroes (most crucial step); and four, writing the complete factorization of the zeroes. When finding all of these things, it is helpful to write everything down. I will go over each step as thouroughly as I can in the video.
While watching this video, Isuggest to pay attention to the side notes which tell what are steps will bne. This will minimize confusion and could clarify any question about any section of the mathematical process. I do try to speak slowly to keep from confusing the viewer but if you feel like you understand, feel free to skip any section of my explanation. Thank you for watching!
In this Student Problem, students were to create their own Quadratic with our four selected zeroes (one with a multiplicity of two and two more each with the multiplicity of one). With the zeroes, we needed to complete the square in order to make the equation "graphable." But before graphing, the end behavior must be identified. For example, my own Quadratic has an End Behavior with both pointing up because it was an Even Positive. To find End Behavior, look at the Leading Coefficient and Exponent. Finally, we graph! All five features (Equation, Factored Equation, End Behavior, X-intercepts- with multiplicities, and Y-intercept.
The trickiest part for this Student Problem was finding four zeroes that would work smoothly together. After that, the entire problem ran generally well.
In this Student Problem, we were to create our own Quadratic Equation and identify its components. The first step was to complete the square to make graphing the equation easier by setting it in its parent form. But instead of leaving it in parent graph form, we must find the x-intercepts. After the work is done: the vertex, max or min, y-intercept, axis of symmetry, and x-intercepts must be identified. Finally, all five points and the line of symmetry must be sketched.
The trickiest part to identifying all parts of this quadratic is identifying everything correctly. The x-intercepts especially could be found confusing at times. Also, it is easy to forget to identify the parabola as having a "max" (maximum extrema) or a "min" (minimum extrema).
