People often confuse obstacles, setbacks, challenges and screw ups as factors in their success. Every great person who ever achieved the success they wanted had a tonne setbacks, obstacles, challenges and screw ups block their path at some stage. So, are you willing and ready to get out of your own way and stop being the obstacle to the results you want to create in all areas of your life?
Tip: as with mindset, get out there and research and learn the habits and behavioual traits of successful people have that unsuccessful people don’t.
Learning how to not be the obstacle to your own success is powerful and liberating beyond words. If you missed any of the other days in this 5 day mini-series you can find them in the links below. So true “People often confuse circumstances and situations as obstacles and setbacks. Get my FREE workbook - How to set yourself up to succeed with achieving your personal and professional goals. Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising.
This lesson covers finding the minimum(s), maximum(s), end behavior, domain, and range of a polynomial graph.
Save or share your relevant files like activites, homework and worksheet.To add resources, you must be the owner of the Modality. Most prominent or noticeable coefficient in the polynomial is defined as that noteworthy figure when compared to additional coefficients in the function for having either a very huge or very insignificant value. Note that examples of transformations of Rationals can be found here, and more examples of transformations of Exponential and Log Functions can be found here.
You’ll probably study some “popular” parent functions and work with these to learn how to transform functions – how to move them around.   We call these basic functions “parent” functions since they are the simplest form of that type of function, meaning they are as close as they can get to the origin (0, 0). The chart below provides some basic parent functions that you should be familiar with.  It also includes the domain and range of each function, and if they are even, odd, or neither, which we learned here in the Compositions of Functions, Even and Odd, Increasing and Decreasing Section. You’ll learn about the trigonometric parent functions and transformations here in the Graphs of Trig Functions and Transformations of Trig Functions sections, respectively. We haven’t really talked about end behavior, but we will in the Asymptotes and Graphing Rational Functions and Graphing Polynomials sections. Again, the “parent functions” assume that we have the simplest form of the function; in other words, the function either goes through the origin (0, 0), or if it doesn’t go through the origin, it isn’t shifted in any way. When functions are transformed on the outside of the  part, you move the function up and down and do the “regular” math, as we’ll see in the examples below.  These are vertical transformations or translations.
Here are the rules and examples of when functions are transformed on the “outside” (notice that the y values are affected). The t-charts include the points (ordered pairs) of the original parent functions, and also the transformed or shifted points. Notice that the first two transformations are translations, the third is a dilation, and the last is a reflection. I didn’t include the absolute value function for the horizontal flip, since it will just be the same function! You may be asked to perform a rotation transformation on a function (you usually see these in Geometry class).  A rotation of 90° counterclockwise involves replacing (x, y) with (–y, x), a rotation of 180° counterclockwise involves replacing (x, y) with (–x, –y), and a rotation of 270° counterclockwise involves replacing (x, y) with (y, –x). Most of the problems you’ll get will involve mixed transformations, or multiple transformations, and we do need to worry about the order in which we perform the transformations.
You might see mixed transformations in the form  ,  where a is the vertical stretch, b is the horizontal stretch (remember you do the opposite math), h is the horizontal shift to the right (again, do opposite math), and k is the vertical shift upwards. Now, what we need to do is to look at what’s done on the “outside” (for the y’s) and make all the moves at once, by following the exact math.  Then we can look on the “inside” (for the x’s) and make all the moves at once, but do the opposite math. So we’re starting with the function  .  If we look at what we’re doing on the outside of what is being squared, which is the ,  we’re flipping it (the minus sign), stretching it by a factor of 3, and adding 10 (shifting up 10).
Now if we look at what we are doing on the inside of what we’re squaring, we’re multiplying it by 2, which means we have to divide by 2  (horizontal compression by ?), and we’re adding 4, which means we have to subtract 4 (a left shift of 4).  Remember that we do the opposite when we’re dealing with the x. Also remember that we always have to do the multiplication or division first with our points, and then the adding and subtracting. Note that this transformation take the original function, flips it around the y axis, performs a horizontal stretch by 2, moves it right by 1, and then down by 3.


Just remember if you’re having trouble drawing the graph from the transformed ordered pairs, just take more points from the original graph to map to the new one! Let’s say we want to use a “function notation” transformation to transform a parent or non-parent equation.   We can do this without using a t-chart, but by using substitution and algebra. For example, if we want to transform    using the transformation  , we can just substitute “x – 1” for “x” in the original equation, multiply by –2, and then add 3.
The second example was found here in the Solving Quadratics by Factoring and Completing the Square section; the last will be shown here in the Exponential Functions section.
Note that when figuring out the transformations from a graph, it’s difficult to know whether you have an “a” or a “b” in the equation   .    Since a vertical stretch (“a”) is really the same thing as a horizontal compression (), for simpler graphs like cubics and quadratics below, we can just look for an “a” (vertical stretch), and ignore the “b” part (by pulling the “a” outside of the function). If a cubic function is vertically stretched by a factor of 3, reflected over the y axis, and shifted down 2 units, what transformations are done to its inverse function? We need to do transformations on the opposite variable.  So the inverse of this function will be horizontally stretched by a factor of 3, reflected over the x axis, and shifted to the left 2 units. Now let’s look at taking the absolute value of functions, both on the outside (affecting the y’s) and the inside (affecting the x’s).
Let’s look at a function of points, and see what happens when we take the absolute value of the function “on the outside” and then “on the inside”.  Then we’ll show absolute value transformations using parent functions. Note that with the absolute value on the outside (affecting the y’s), we just take all negative y values and make them positive, and with absolute value on the inside (affecting the x’s), we take all the 1st and 4th quadrant points and reflect them over the y axis, so that the new graph is symmetric to the y axis.
Definition: Rational functions are functions which can be written as a ratio of two polynomials. Technically, a polynomial is also a rational function just as an integer is also a rational number with a denominator of 1. As stated before, a rational function is a function which can be written as a ratio of two polynomials. Divide out all common factors between the numerator and denominator before finding zeros or asymptotes or graphing the function. ZEROS: When discussing polynomial functions we were often interested in the zeros of the function. When a rational function is equal to zero (that is, its output is equal to zero) then its NUMERATOR is equal to zero. EXAMPLE: The zeros of the function h(x) described above would be found by setting the NUMERATOR equal to zero.
A line is an asymptote if the distance between the curve and the line approaches zero as we move out farther and farther on the line. When a vertical line is an asymptote then the graph gets closer and closer to the vertical line. Remember, you have already divided out all common factors between the numerator and denominator before finding zeros or asymptotes or graphing the function. As the name suggests, end behavior asymptotes model the behavior of the function at the left and right ends of the graph. While there are short-cuts to find the end behavior asymptote in two of these three cases, they all derive from the same procedure. The end behavior asymptote will allow us to approximate the behavior of the function at the ends of the graph.
The end behavior asymptote (the equation that approximates the behavior of the original function at the ends of the graph) will simply be y = quotient. Once again, to find the end behavior asymptote we will divide the denominator, (6 + x2) into the numberator, 3x. Because the denominator has degree LARGER than the degree of the numerator, the denominator will divide into the numerator 0 times.
Look at the graph on your graphing calculator (or look at the graph of the function shown above) to confirm that this is so.
The reason it is great news is because learning the skills to do this are totally within your control. In it he talks about how life is, by its nature, pull of problems and that our growth and development comes from both taking responsibility for and finding ways through them. We decide end behavior on the basis of degree of function and also the most prominent coefficient in the polynomial function.


Judge the end behavior of the polynomial function given as follows: u4 – 4 u3 + 3 u + 25? You might recall that a polynomial is an algebraic expression in which the exponents of all variables are whole numbers and no variables appear in the denominator.
We usually talk about non-polynomial rational functions AFTER we have discussed polynomial functions so that we can build on some of the shared characteristics. If you divide out a common factor, then you must also state that the domaindoes not include the number that would have made the denominator zero. That is, we wanted to know the values in the Domain (input) of the function that would make the Range (output) equal to zero. The graph becomes vertical so the vertical line is a model of what the graph looks like as the graph gets closer to the line.
The distance between the curve and the line approaches zero as we move out further and further out on the line. A function describes a relationship between two sets such that for every input there is exactly one output. I will first summarize the procedure, will then explain why it works, and will then give several examples based on the function defined at the beginning.
This will always be the case when the degree of the denominator is larger than the degree of the numerator. On the intervals where the sign change number line is + the graph of the rational function will be above the x-axis. You can’t expect to get the results you really want unless you are behaving, making decisions, and thinking in ways that deliver success. You can do all the other tasks and activities we have talked about but if you don’t step up to this one you will always be capping the level of success you achieve. That is a very interesting perspective your throw out about your business being a source of personal development.
At this point, graphs of functions typically show the domain on the horizontal axis - this means that the zeros would show up as the values on the horizontal axis where the graph touches or crosses.
The graph of the function is a visual representation of the relation decribed by the function rule. On the intervals where the sign change number line is - the graph will be below the x-axis.
This video not only reviews some of the steps listed here but will gives some examples of rational functions used to model events. They are just things, circumstances and situations to be dealt with on the path to success.
You can view what you see happening in your business as a mirror reflecting where you need to upgrade your mindset and who you are being as a business owner to get better outcomes.
Suppose we have a polynomial function F (X), its end behavior can be explained by considering the values of F (x) i.e. When the denominator of the function gets closer to zero the magnitude of the range (output) of the function gets closer to infinity which is what the vertical line indicates. In some cases the rule that describes the function relationship can be quite complicated and time consuming to evaluate but if a linear asymptote exists, then it can be used to approximate the output of the function without going through the complicated function definition. This is why I always say that it isn't just the business skills and processes you need to master to be successful it is also who you are BEING in the business and all areas of your life.
The graph never actually touches the vertical line so the denominator is never actually zero.
In the case of the vertical asymptote we are just describing the line that the graph approaches but never touches.



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