Functions Video 1 Popper 04 Increasing decreasing graphs Basic Graphs Worksheets Linear Quadratic Cubic Cube Root Square Root Absolute Value Exponential Logarithmic Video 2 Popper 05 Rational Functions 1 Video 3 Popper 06 Polynomials End behavior x intercepts and behavior at them Working backwards Sequences Notation Domain and Range Bounded Sequences Appendices Interval Notation Factoring The Guaranteed Method Complete the Square Method Factoring by Grouping 2 In this module we will be looking more deeply at functions in particular the graphs of functions Starting off with 3 important properties which describe the behavior of the graph we ll define increasing decreasing and neither and see how certain graphs exhibit these properties Then we ll review of various families of graphs lots of types of functions all except rational functions and polynomials We ll look at only rational functions in the second video and only polynomials in the third video These types of functions are so important that they each get a separate video 3 Increasing and decreasing graphs We often talk of a turn around point a place where a graph changes direction so it is important to be able to talk about what is happening at a turnaround point Most graphs can be described as increasing decreasing or neither and knowing where the graph is exhibiting these properties helps us understand the behavior that the graph represents Let s look at a graph that has all three movements to get an intuitive sense of the descriptions Domain Range Intercepts Increasing Decreasing Neither 4 An application of the concept If you graph a revenue function for a retail item the most common shape is a cupped down parabola Now firms are always wanting to increase revenue Let s look at how increasing decreasing neither would work to alert a manager of what to do to keep the revenues flowing in We have months on the x axis and dollars on the y axis here First off why is revenue cupped down And what do the parts of the graph below the x axis mean Where is the graph incr decr and n Secondly When is the BEST time to have a sale 5 Identifying exactly WHICH x is the x of the turn around point is a focus of this class We will learn how to do this in the Derivatives section We can identify increasing and decreasing on a graph and note that neither occurs at a turn around point or if the graph is a flat horizontal line We always do this in a certain way First we look at the x s and we scan the axis from left to right Note that the smaller x s are on the left In fact if we number a first x and a second x moving left to right we have x1 x 2 Then we check the motion of the y values on the y axis as the x s get larger and larger So identifying the relevant property is a sort of scanning process If the y values are moving UP the y axis as they get used then the function is increasing If the y values are moving DOWN the y axis as they get used then the function is decreasing This has nothing to do with the sign of the y value a graph can be in Quadrant 3 with negative numbers for x and y and still be increasing see the first example below For example Here is the graph of y 0 5x 6 Domain Range Intercepts Increasing Decreasing Turn around points 6 This graph of a line is increasing everywhere on its domain even in Quadrant 3 and Quadrant 4 where the y s are negative numbers Another example Here is the graph of f x x 2 What are the domain range and intercepts Where is the graph increasing decreasing or neither Domain Range Intercepts Increasing Decreasing Turnaround point 7 Note that you always specify where a graph is increasing or decreasing on intervals of the x axis even though you re talking about the behavior of the y values Here is the graph of f x 10 x an exponential graph The domain is all real numbers and the range is 0 Note the leftward half asymptote too Are there any turn around points Let s trace out how we know it s increasing on its domain Here s a polynomial The domain and range are each all real numbers What is the formula for this polynomial and how would you know 8 Where is the graph increasing Decreasing Neither 9 Basic Graphs Worksheets Here we ll look at various families of graphs always starting with the basic or parent graph and then looking at a couple of shifted graphs We ll review the function facts like domain range and intercepts along with discussing increasing decreasing neither for each family For an excellent review of graph shifting check out the videos in the UH Math 1310 textbook for College Algebra http online math uh edu Math1310 Look for Chapter 3 Section 4 on transforming graphs While you re there you might check around for any other topics you want a review on the videos are just excellent 10 Basic Linear Function f x x Domain Range x intercept y intercept 0 0 0 0 Increasing everywhere This function is one to one f x y 3x 5 Domain Range x intercept y intercept 5 3 0 0 5 Increasing on its domain 11 Telling what s happening depends on the slope m of the line Let s see how f x 2 m 0 Intercepts Neither f x 2 x 8 m 0 Intercepts Increasing on its domain 12 f x 3 x 6 m 0 Intercepts Decreasing on its domain Graph below for demo Let s trace out the x and y movements on the axis to help see the decreasing description 13 Popper 04 Question 1 14 Basic Quadratic function f x x 2 Domain Range x intercept y intercept 0 0 0 0 0 Decreasing Increasing 0 0 Neither at 0 0 the turn around point This function is not one to one Example of a shifted graph f x x 2 4x 5 Domain Range x intercepts y intercept 9 5 0 and 1 0 0 5 vertex 2 9 the turn around point decreasing increasing 2 2 neither at the vertex 15 Let s work out a whole problem here reviewing as we go Here s the graph of a nice cupped up parabola y Where are the intercepts on the graph What are the point coordinates for the intercepts x intercepts x y intercept What do the x intercepts tell us What is the formula Let s multiply that formula out and then find the vertex using algebra and see if it looks right where the graph has it Now where is the graph increasing decreasing or doing neither Report your answer in interval notation 16 Popper 04 Question 2 17 Basic Cubic function f x x 3 Domain Range x intercept y intercept 0 0 0 0 Increasing everywhere This function is one …
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