Tutor profile: Abigail J.
What does it mean for there to be a "point of inflection" on a graph?
Recall that we previously talked about concavity: a graph, or section of a graph, can be either concave up (sometimes called convex) or concave down. Concave up/convex is when the graph looks like a bowl--imaging you dropped a marble onto the graph, it would roll down into the bowl and remain at rest. Concave down/concave is when the graph looks like a mountain--if you dropped a marble onto the graph, it would roll off the side of the mountain and out of sight. A "point of inflection" is when a graph switches from one type of concavity to another. There can be multiple points of inflection on a graph (especially for higher order polynomials) or there can be no points of inflection on a graph (for example, y = x^2). For practice finding points of inflection, try graphing y = (x)^3(x + 2)^4(x − 3)^5. If you're having trouble keeping track of all the inflection points, try tracing the graph with your finger from left to right--this will help you sense the points where the inflection changes. See how many inflection points you can find, and then we will check your answer together.
Why do some organisms undergo asexual reproduction and others use sexual reproduction?
There are different advantages and disadvantages to both sexual and asexual reproduction which help explain why both are seen in nature. Asexual reproduction is advantageous because (1) you don't need a mate and (2) it allows for rapid population growth. This would be helpful for certain types of fungus, for example, because it allows faster growth so that it can take over a particular niche environment. It is disadvantageous, though, because (1) it limits genetic variation--the only genetic variation you will get is from mutations from mistakes in DNA replication or environmental mutagens. Sexual reproduction, on the other hand, is advantageous because it allows for (1) greater genetic variability (hence, evolution!). However, it is disadvantageous because (1) organisms need to find a mate which is a slower process. At the end of the day, sexual reproduction offers way more genetic variation than asexual reproduction because of things like crossing over and independent assortment (we will cover these topics when we discuss meiosis!) and things like the random nature of fertilization and migratory gene flow.
When do I have to check to see if I have an extraneous solution?
Great question! As we know, sometimes its important to plug our answers back into our original equation to double check that they actually satisfy the equation we are trying to solve. The first thing to remember is that it will never hurt to plug in your answers and see if they satisfy the equation--when in doubt, plug them in and double check! However, there are only certain types of equations that you would actually expect to find "extraneous solutions" for. For example, equations that have variables in the denominator of a fraction or variables underneath a radical sign are often culprits for having extraneous solutions. If you plug in one of your solutions and it either (A) doesn't satisfy the equation or (B) causes you to divide by 0, you want to throw that solution out--only box as a final answer the solutions that perfectly satisfy your original equation.
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