Transforming Exponential Functions - Module 14.5 (Part 1)

Summary

In Module 14.5 Part 1, Mrmathblog takes viewers on a journey through transforming exponential functions. The lesson not only revisits the basics of exponential functions but expands on them by exploring transformations such as stretching, compressing, reflecting over the x-axis, and translating them away from the origin. Through practical examples and step-by-step calculations, Mrmathblog breaks down how changes in the constants a, b, and k in the function \( f(x) = a \cdot b^x + k \) affect the graph's shape and position. Additionally, viewers are guided through homework questions and are encouraged to attempt these exercises on their calculators, ensuring understanding of controlling the exponent feature and observing function behavior changes when parameters are adjusted.

Highlights

• Mrmathblog explores transforming exponential functions by altering constants a, b, and k. 🔍
• The 'a' value changes the graph's steepness or flips it when negative. 🤸
• Bigger 'b' powers speed up growth or, if a fraction, slow it down. ⏩⏪
• Adjusting 'k' moves the graph up or down on the coordinate plane. ⬆️⬇️
• Visual graphs and examples help in comprehending these adjustments. 📊

Key Takeaways

• Transforming exponential functions involves stretching, compressing, and shifting them. 🌀
• The 'a' value affects vertical stretch/compression, and reflects the graph if negative. 🤸‍♂️
• Exponent 'b' influences growth rate; greater than 1 = faster growth, between 0 and 1 = decay. ⬆️⬇️
• Adding or subtracting 'k' shifts the graph vertically. 📉📈
• Understanding transformations aids in graphing and real-world application of functions. 🌍

Overview

Mrmathblog dives into the transformative world of exponential functions, providing insights into how traditional exponential graphs can be modified. By tweaking the constants in the function, he demonstrates how graphs are stretched, compressed, or shifted, allowing a deeper understanding of function manipulation.

In this lesson, the focus is on changing the constants 'a', 'b', and 'k' in equations of the form \( f(x) = a \cdot b^x + k \). Mrmathblog uses examples to show how these adjustments impact the graph visually, from making the function grow faster to causing it to decay more gradually, each transformation adding a layer of complexity.

The lesson equips students with the knowledge necessary to tackle homework questions, bolstering comprehension through interactive calculation and visualization. Mrmathblog ensures that each viewer leaves with the tools to manipulate and understand these essential mathematical concepts, hinting at more to explore in the upcoming Part 2.

Chapters

• 00:00 - 00:30: Introduction to Lesson This chapter introduces transformations of exponential functions. It covers stretching, compressing, flipping over the x-axis, and translating these functions away from the origin. Additionally, it mentions that students can find related lessons on transmigrated math website. The chapter is part of Integrated Math 1, Module 14.5, and it hints at a subsequent part that will address homework questions.
• 00:30 - 03:00: Graphing Exponential Functions This chapter focuses on understanding how changes in the values of 'a', 'b', and 'k' affect the graph of the exponential function f(x) = a * b^x + k. The discussion emphasizes the behavior of the graph as 'a', 'b', and 'k' are altered individually. It is highlighted that 'b' cannot equal 1 while exploring its effects; however, 'b' can be any other positive number greater than one, a fraction, or a negative number. Additionally, the chapter notes the utility of using a calculator to visualize these changes effectively.
• 03:00 - 07:00: Comparison of Functions The chapter discusses how to complete a table for different functions, specifically comparing the functions f(x) = 1.2^x and g(x) = 1.5^x. It emphasizes the importance of correctly using a calculator's exponent feature, which may appear as the Y^X button, to determine values for given x inputs, such as x = -2.
• 07:00 - 10:00: Exploring Vertical Stretches and Compressions The chapter titled 'Exploring Vertical Stretches and Compressions' explains how to use the exponent feature on a calculator, represented by a caret button, to perform calculations involving base numbers and exponents. The example given uses a base of 1.2, and with the exponent of -2, demonstrates how to calculate the outcome using the calculator. The chapter briefly discusses verifying results and understanding the relationship between values, indicating some exercises for the reader to complete independently.
• 10:00 - 14:00: Effect of Negative Coefficients This chapter discusses the effects of negative coefficients in functions, focusing on the behavior as the variable 'x' changes. It highlights how different functions respond when 'x' increases or decreases. Specifically, the chapter points out that f(2) grows faster than f(1) as 'x' increases, and discusses which function approaches zero more rapidly as 'x' decreases.
• 14:00 - 18:00: Impact of Constants and Translations The chapter discusses the impact of constants and translations on functions. It starts by examining how certain functions decrease at varying rates. Specifically, it notes that when evaluating the function at x=2, certain functions decrease in value more rapidly than others. The chapter further explores the concept of y-intercepts, pointing out that these are the points where x equals zero and showing that in some cases, both y-intercepts are equal to 1. It emphasizes the relationship between decimals, constants, and variables, reminding that B, a constant in this context, cannot equal 1 but can be a decimal between zero and one. The text indicates practical examples where these values are plugged into functions to better understand their behavior.

Transforming Exponential Functions - Module 14.5 (Part 1) Transcription

• 00:00 - 00:30 hey everybody this lesson is going to take the last lesson exponential functions and we're gonna stretch them we're going to compress them we're gonna flip them over the x-axis and move them away from the origin so that's what this lessons about and don't forget all your lessons can be found at mr. math bug comm and this is integrated Math one so this is module 14.5 okay and then the second part of this I'll have a part two it'll help you go over some of the homework questions you guys so take a
• 00:30 - 01:00 look at that one also you guys so here's our question so how does some the graph of f of X equal a times B to the X change when a and B are changed and we're also gonna add a plus K on here constant and it's just gonna move it all around stretch it and compress it so we're gonna change the value of B and f of X equals B to the X and I remember B can't equal one so it can equal anything else it just can't equal 1 it could be negative it can be a fraction it can be something greater than one just can't equal one you're gonna need a calculator
• 01:00 - 01:30 for this you guys so so complete the table for the functions F of one of X is one point two to the X and then the other one is one point 5 to the X we're gonna see what changes here okay so they've given us a few of the values here they plugged in x equals negative 2 right here so you have to plug in one point two to the negative two power and and it depending on which calculator you have you guys so your exponent feature will either be this Y to the X button or
• 01:30 - 02:00 this little I call it a caret button that's your exponent feature right there so so let's try this one right here so if we plug in this base right here one point two and then we hit our exponent button the Y to the X and then we plug in negative 2 and then equals it should give us that right there okay so I'm gonna do the rest of that for those ones right there you guys can do that on your own right there and so there's the values right there and now we're gonna explore the relationship with those values here and we're going to underline which is true so as let's see does f of
• 02:00 - 02:30 1 or f of 2 of x increase more quickly as x increases okay so here's X X is getting bigger bigger bigger which one of these are getting bigger bigger bigger looks like f of 2 is Oh bigger than f of one okay so definitely f of two will be get bigger and then this one says which one approaches zero more quickly as X decreases so X decreasing this means they're getting smaller smaller smaller which one's
• 02:30 - 03:00 getting smaller faster this one's getting smaller faster so f of two again okay and then it's asking what are the y-intercepts while the y-intercepts are when x equals zero so both y-intercepts are equal one right there okay all right let's do that again with this one here okay here we have decimals there are numbers that are less than one remember remember B can't be one but it can be a decimal between zero and one so let's try this okay so they've given us a couple over here and we just got to plug them in I'm gonna save some time I've
• 03:00 - 03:30 done those already so there those are let's answer these questions again so which is true does f of three or F of four increase more quickly as X in as I'm sorry as X decreases okay so which one increases faster looks like this one is bigger as X decreases right there so f of three is bigger and then which one approaches zero more quickly as x increases so as x increases look at these get bigger which ones smaller this
• 03:30 - 04:00 one is so f of three again right there okay so those are easy enough and the y-intercepts are when x equals zero so the y-intercepts are one on both of those right there okay so now let's consider the function of where B is greater than one okay so we're gonna consider the function y equals one point three to the X and how will the graph compare with the first two that we did F of one and F of two and discuss the end behavior okay so here's F of 1 and here's F of 2 so now we're gonna do F of
• 04:00 - 04:30 3 right here so all three graphs will have the same y-intercept okay now we can make a table over here and plug in F of 3 but we should be able to generalize right here F of 3 is going to be one point three is somewhere in between one point two to the X and one point 5 to the X so it's going to be somewhere in between these two guys so it's going to fall between the other two crafts it's going to increase more crit quickly F of 1 but less quickly than F of 2 as X
• 04:30 - 05:00 goes to the right or as x increases right here and the graph Falls more quickly than this graphed and it does from this graph as as X decreases okay so as X decreases the numbers are getting smaller can you see which one's getting closer to 0 this one's getting closer to 0 faster so f of 3 will be getting closer to 0 faster then that guy right there it's gonna be somewhere in between that so now we're going to change the value in when B is greater than 1 okay so when B is greater than 1
• 05:00 - 05:30 then we get what's called a vertical stretch or vertical compression it's formed by whatever the absolute value of a is okay now this is when B is greater than 1 we did some of those already so now we're gonna change a so if the absolute value of a is greater than 1 then the graph shoots up faster it grows it goes by a factor of the absolute value of a ok and if it's less than 1 then the graph did compresses I'll show
• 05:30 - 06:00 you graphs of those on those in just a second row back right here so make a table of values for the function given that then graph it on the same graph as y equals 1 point 5 to the X we already did that one and describe the end behavior and find the y intercept of each graph ok so here we're gonna do 0.3 times 1.5 to the X okay so there's 1.5 to the X we already did that before right there and there's that graph right there sorry I had to do this with my finger on my little laptop so it's a little shaky in there I wasn't in school
• 06:00 - 06:30 at the time otherwise I'd have done it on my whiteboard pin on my Promethean Board actually all right so I'm gonna plug in all these values right here so I'm gonna do 1.5 to the negative 2 power and whatever that is times 0.3 okay so we get all of those values right there and when we graph them they're compressed they're smaller so so it's it doesn't it still rises but it rises that less of a rate right there still goes in the same direction so in behavior as X
• 06:30 - 07:00 goes to positive infinity you can either look at the graph the graph is going up so f of X would go to positive infinity and as X goes to negative infinity this graph gets down close to the x-axis which is f of X equals zero right there okay or you can look at this right here as X gets smaller f of X gets close to zero as X gets bigger f of X gets larger okay alright so to infinity right there
• 07:00 - 07:30 alright so now let's try it with where a is negative two okay all right so we already have that graph right there we already did that one right there so let's do so we're gonna go one point five to the negative 4 power and whatever that is we're going to multiply that times negative two and then here one point five to the negative 3 power whatever that is times negative 2 so when we do all of that that gives us those points right there and see this negative reflected it over the x-axis
• 07:30 - 08:00 right there okay so as as as X gets goes to infinity this direction f of X shoots down it goes to negative infinity and as X goes to negative infinity this way remember X is a left and right movement so when we're going to negative infinity this pink graph is going up towards zero right there and the y-intercept is at negative 2 it's when x equals zero so it's going to give us at negative 2 right there alright so
• 08:00 - 08:30 what can we say about the common behavior of the form of f of X equals a times B to the X when B is greater than 1 and what is the difference when the sign changes okay so all graphs of the form of a times B to the X when B is greater than 1 approach 0 as X approaches negative infinity see how they're approaching 0 right here and the sign determines the end behavior as X approaches this way so for when a is positive then we're looking at this top graph as then when X goes to infinity
• 08:30 - 09:00 then it goes to infinity over here when X goes to infinity when a is negative then it goes to negative infinity right there that's what that's saying right there okay all right so now let's uh explore when B is a fraction somewhere between 0 & 1 okay so we're going to make a table of values in the function and we're going to then graph it on the same coordinate plane with the graph of y equals point six to the X okay so we're going to explore with a being some number positive and
• 09:00 - 09:30 negative and describe the end behavior okay so here's a this graph right here when we here's a point six to the X right here gets us this graph right here look as X goes to infinity then f of X goes down to zero and as X goes to negative infinity then f of X goes to positive infinity that's this graph right here this negative is going to make it flip upside down right here okay so when we put those in we're gonna get those right there so in the calculator
• 09:30 - 10:00 we push point six and then our exponent feature and then we plug in say negative one and then we multiply that times times negative three and that gives us that negative five right there okay all right so there's that graph right there and as X goes to infinity f of X goes to zero and as X goes to negative infinity then f of X is shooting down to negative infinity right there okay that's what that says right there and the y-intercept of course is when x equals zero so remember anything to the zero
• 10:00 - 10:30 equals one so this to the zero equals one times negative three is negative three all right remember I'm doing a homework lesson in this all right let's in the next lesson you guys so don't don't feel like you're totally lost I'll help you with your homework and part two of this okay so when we plug in that one here okay so so that point five which is less than point six just makes it rise up over here a little bit less okay so our end behavior is described right there okay and then what can we say about the
• 10:30 - 11:00 common behavior of the graphs of f of X equals a times B to the X 1 B is somewhere between 0 and 1 and when the sign changes right there okay so all graphs approach 0 as X approaches positive infinity right here so if I go this way then they approach 0 right there and when a is positive then when we approach negative infinity f of X goes to positive infinity and when a is negative that's this graph right here
• 11:00 - 11:30 when we go X goes to negative infinity f of X goes to negative infinity remember the trick I said in the last lesson that will help you this also now we're going to add a constant okay so make a table of values for the function and put them together on the same coordinate plane find the y-intercepts okay so now we're going to do f of X equals 2 to the X and we're gonna do 2 to the X plus 2 that's just gonna make this go up by 2 it's gonna take this graph and just shift it up by 2 that's all this graph does right here okay
• 11:30 - 12:00 so here we go we're gonna plug in y equal 2 to the X there it is right there there's 2 to the X right there okay and then when we do +2 it's just gonna shift that graph up 2 units up right there so the y intercept of f of x equals 1 y intercept of G of X is 3 because it's 1 plus 2 okay it's always everything plus 2 and so it's a vertical translation up by 2 units right there okay so let's do that with this one where we have B is a fraction between 0 & 1 this minus 3 is
• 12:00 - 12:30 going to take this graph and just shift it down 3 so that's all that does right there so there's that graph of f of X right there so when we shift it down by 3 it would give us that graph right there it just went down 3 it's the same graph it's just down by 3 so the y intercept of f of X is 1 the y intercept of of G of X is 1 minus 3 which is negative 2 okay so the y intercept of G of X is is less than that of f of X
• 12:30 - 13:00 because it's down a vertical translation down by 3 okay I know it's going fast I'll help you in the homework in the next lesson you guys so how do we determine the y-intercept of the exponential function a times B to the X plus K when it's both stretched and translated okay well the y-intercept of all parent exponential functions is o is 1 because you just plug in 0 right there 0 anything to the 0 equals 1 okay so then we first multiply that one
• 13:00 - 13:30 time so this is going to be 1 1 times whatever that is and then add that and that's going to be our y-intercept so we go so it's gonna this is 1 because x equals 0 anything to the 0 equals 1 so B to the 0 equals 1 we take that multiply it times a and then we add K afterwards okay so describe the end behavior of the translated exponential function when P is greater than one as X approaches negative infinity well that's to the left since all points are shifted by K the
• 13:30 - 14:00 function approaches K instead of zero remember it approaches zero but it's approaching K on this one because we shift it by K as X approaches negative infinity all right and then when when when B is changed right here if B is greater than one then it then it increases B and makes the graph rise faster so when a when a is a positive number right there and when B is a
• 14:00 - 14:30 fraction somewhere between or decimal between zero and one that increases B by making the graph fall more gradually as x increases all right you guys now I'm gonna sign this don't forget part two of this it's gonna be it's gonna help you through this homework I know that was a lot right there but you'll see just get the hang of it don't give up on yourself take care you guys