Lec 46 - Exponential Functions: Graphing
Summary
TLDRThis educational video script explores the properties of exponential functions, particularly focusing on functions of the form f(x) = a^x. It discusses the domain, range, intercepts, and end behavior of these functions, highlighting that for a > 1, the function is always increasing with no x-intercept, and for 0 < a < 1, it is decreasing. The script also emphasizes the importance of understanding the graphical behavior of exponential functions without relying solely on visual aids, and concludes with a summary table differentiating the characteristics of exponential functions based on the value of 'a'.
Takeaways
- đ The script discusses the properties of exponential functions, specifically focusing on functions of the form f(x) = a^x.
- đ When 'a' is greater than 1, the domain of the function is the entire real line, and the range is from 0 to infinity.
- đ For a > 1, the function is always increasing and never touches the x-axis, meaning there is no x-intercept.
- đ When 0 < a < 1, the function is decreasing, and the end behavior is that as x approaches infinity, f(x) approaches 0, and as x approaches negative infinity, f(x) approaches infinity.
- đ The y-intercept for both cases (a > 1 and 0 < a < 1) is (0, 1), as a^0 equals 1 for any a.
- đ« There are no roots for exponential functions where a > 0, as the function never crosses or touches the x-axis.
- đĄ The horizontal asymptote for all such exponential functions is y = 0, indicating the behavior as x approaches infinity or negative infinity.
- đ The script introduces the concept of graphing exponential functions without the need for a graphing tool by understanding their properties.
- đ€ The behavior of the function changes based on the value of 'a', and understanding this allows for predicting the shape and behavior of the graph.
- đ The script uses specific examples, such as f(x) = 2^x and g(x) = 5^(-x), to illustrate the general properties of exponential functions.
- đđ The reflection of the graph across the y-axis (by replacing x with -x) results in a change from an increasing to a decreasing function and vice versa.
Q & A
What is the domain of the function f(x) = 2^x?
-The domain of the function f(x) = 2^x is the entire real line, as it is defined for all real values of x.
What is the range of the function f(x) = 2^x?
-The range of the function f(x) = 2^x is from 0 to infinity (0 to â), since 2^x is always positive and never reaches 0.
Does the function f(x) = 2^x have any x-intercepts?
-No, the function f(x) = 2^x does not have any x-intercepts because it never touches the x-axis, as it is always greater than 0.
What is the y-intercept of the function f(x) = 2^x?
-The y-intercept of the function f(x) = 2^x is at the point (0, 1), which occurs when x = 0.
What is the end behavior of the function f(x) = 2^x as x approaches infinity?
-As x approaches infinity, the function f(x) = 2^x also tends to infinity because the exponential growth rate is faster than linear growth.
What is the end behavior of the function f(x) = 2^x as x approaches negative infinity?
-As x approaches negative infinity, the function f(x) = 2^x approaches 0, but never actually reaches it, due to the horizontal asymptote at y = 0.
Are there any roots for the function f(x) = 2^x?
-No, there are no roots for the function f(x) = 2^x because it never crosses or touches the x-axis.
Is the function f(x) = 2^x increasing or decreasing throughout its domain?
-The function f(x) = 2^x is increasing throughout its domain because for any x1 < x2, 2^x1 < 2^x2.
What happens when the base 'a' of the exponential function f(x) = a^x is greater than 1?
-When the base 'a' is greater than 1, the function f(x) = a^x has similar properties to f(x) = 2^x: it is increasing, has no x-intercepts, and has a horizontal asymptote at y = 0.
What is the effect on the graph of the function when x is replaced with -x in an exponential function?
-Replacing x with -x in an exponential function reflects the graph across the y-axis, changing the increasing function to a decreasing one and vice versa.
What is the general behavior of the graph of f(x) = a^x when 0 < a < 1?
-When 0 < a < 1, the graph of f(x) = a^x is a decreasing function with the same domain and range as when a > 1, but with a horizontal asymptote at y = 0 and tending to infinity as x approaches negative infinity.
Outlines
đ Introduction to Exponential Functions
This paragraph introduces the topic of exponential functions, emphasizing the importance of understanding their properties before analyzing specific examples. The domain of the function f(x) = 2^x is discussed, highlighting that it covers the entire real line. The range is identified as starting from 0 to infinity, with the function never reaching negative values. The concept of the y-intercept at (0, 1) is introduced, and the behavior of the function as x approaches positive and negative infinity is explored, noting that it does not touch the x-axis and has a horizontal asymptote at y=0.
đ Analyzing the Graph and Behavior of Exponential Functions
The paragraph delves into the graphical representation and behavior of the exponential function f(x) = 2^x. It discusses the function's end behavior, noting that as x approaches infinity, the function also tends towards infinity, and as x approaches negative infinity, the function approaches 0. The absence of roots is established, and the function's monotonic increase across the entire real line is emphasized. Key points on the graph, such as the y-intercept and the point (1, 2), are identified, and the overall increasing nature of the function is reiterated.
đ Generalizing Exponential Functions with Bases Greater Than 1
This section generalizes the properties of exponential functions where the base 'a' is greater than 1. It asserts that all such functions share the same characteristics as 2^x, including the domain being the entire real line, the range from 0 to infinity, the absence of x-intercepts, and the horizontal asymptote at y=0. The paragraph also explains that the function's values change with different bases but the overall shape and behavior remain consistent, simplifying the process of graphing these functions.
đ Reflecting on Exponential Functions with Bases Between 0 and 1
The final paragraph examines the behavior of exponential functions where the base 'a' is between 0 and 1, using 1/5^x as an example. It discusses the reflection of the graph across the y-axis, resulting in a decreasing function. The domain remains the real line, and the range is still from 0 to infinity. The y-intercept remains at (0, 1), but the end behavior is reversed, with the function approaching 0 as x approaches infinity and tending towards infinity as x approaches negative infinity. The paragraph concludes by summarizing the properties of these functions and introducing the concept of natural exponential functions for the next video.
Mindmap
Keywords
đĄExponential Function
đĄDomain
đĄRange
đĄY-Intercept
đĄHorizontal Asymptote
đĄEnd Behavior
đĄRoots
đĄIncreasing Function
đĄDecreasing Function
đĄReflection Across Y-Axis
đĄNatural Exponential Function
Highlights
Introduction to the concept of exponential functions and the importance of understanding their properties.
Explanation of the domain of an exponential function, emphasizing the entire real line.
Analysis of the range of the function 2^x, highlighting it is always greater than 0 and approaches infinity.
Identification of the y-intercept at (0,1) for the function 2^x and its significance.
Discussion on the horizontal asymptote of the function 2^x, which is y=0.
Clarification that the function 2^x never touches the x-axis, indicating no x-intercept.
End behavior analysis of the function 2^x as x approaches both positive and negative infinity.
Investigation of the roots of the function 2^x, concluding there are none due to the function never touching zero.
Explanation of the function's monotonicity, asserting that 2^x is strictly increasing.
Graphical representation and identification of special points for the function 2^x.
Generalization of the properties for exponential functions where the base a is greater than 1.
Introduction to the reflection property of exponential functions when the base a is between 0 and 1.
Analysis of the function 1/5^x, discussing its reflection across the y-axis and its end behavior.
Identification of the y-intercept and the behavior of the function 1/5^x at x=1.
Summary of the properties of exponential functions based on the value of the base a.
Conclusion and transition to the topic of natural exponential functions in the next video.
Transcripts
Welcome back. So, I hope you must have done your exercises and you must have developed
some understanding about the exponential functions. Let us try to collect recollect that understanding
through 2 examples given here. So, let us first take 1 a which is f of x
is equal to 2 raised to x. If you have used DESMOS, you must have got the figure of the
function. But prior to receiving the figure of the function, let us see what should be
the domain of a function. We have already discussed in greater detail that the domain
of a function can be a real line, entire real line ok.
Now, if you look at this function which is 2 raised to x, this 2 is greater than 1 and
the 2 raised to x will always be greater than 2 raised to 0 which is equal to 1, 2 raised
to x is always greater than 2 raised to 0 whenever x is positive correct.
Now, because I x greater than 0, then 2 raised to x is always greater than 2 raised ok. So,
if x is less than 0, what will happen? 2 raised to x, when x is less than 0 will always be
less than 1. This is also possible. hm But when this 2 raised to; can this 2 raised to
x become negative? No. So, it is always greater than 0. So, if
you have this understanding, then you can easily write the function has a range which
is 0 to infinity. So, there is a split from when you consider a point 1, there is something
happening at point 0 comma 1 right. What is 0 comma 1? 0 comma 1 actually is an y intercept
ok, something is happening at 0 comma 1 because I have put 0 here for then it is I am getting
1. So, 0 comma 1 is also y intercept and there
is something happening which is going below 0. Is going below 1, your graph is going below
1 and therefore, this particular thing is going down, but it never goes below 0. This
is an interesting fact because if you consider 2 raised to x, it never goes below 0. It cannot
go to a negative number. Therefore, will it touch the x axis? It will
not touch x axis. In fact,x intercept is nil ok, but it is approaching 0. So, the something
that is approaching 0, so x intercept is actually it will never touch it; but it will actually
go along that line. So, this y is equal to 0, it will touch at infinity ok. So, such
a thing, we call as horizontal asymptote ok. oh
So, such a thing you call as horizontal asymptote. So, with this understanding, these these are
the things that I can make out directly without looking at the graph. So, let us now look
at the graph ok,before going to that, let us see what happens to the end behavior.
End behavior of a function as x tending to infinity. So, as 2 raised to x,you consider
a function 2 raised to x as x increases, this also increases. In fact it increases at a
rapid rate than x . So, this also should tend to infinity ok and as x tends to minus infinity,
we have already figured out y is equal to 0 is the horizontal asymptote. So, 2 raised
to x will actually go to 0 ok. Then, the question that we used to quantify
while consideringthe function, what are the roots of this function. So, do they have any
roots? In fact, using graphical method, it is very clear that it never touches 0. So,
there are no roots ok and the functionsincrease and decrease.
So, the domains of increase and decrease like polynomials, we studied domains of increase
and decrease; but here, I think my claim is no need to identify the domains of increase
and decrease. Why? Because you look at a function 2 raised to x, let us take x 1 not equal to
x 2 or x 1 lesser than x 2, without loss of generality, we can take this. Then, what what
can you say about 2 raised to x 1 and 2 raised to x 2?
See x 1 is less than x 2, so naturally ifit is raised to the power 2 power; 2 is raised
to power of x 1 and 2 is raised to power of x 2, this relation should hold. So, what I
am saying is the function is actually an increasing function and increasing functions are 1 to
1 . Therefore, I do not have any doubt that the increase and decrease, it is only increasing;
throughout the real line, the function is only increasing.
So, let us look at the graph of a function f of x is equal to 2 raised to x. Let us identify
the points. So, here you can identify a point right. So, this point we have seen as y intercept
and that point was 0 comma 1 right. Then, the one in this case, let us look at this
point which is 1 and where will it go? It will actually tell you 2. So,the point is
1 comma 2, the second point ok. So, these 2 points are very special points,
they tell you something. So, in particular, had it not been 2 raised to x, but a raise
to x, then that point would have been 1 comma a and if you mimic this graph over here y
x graph of y x is over here ok, this is a point 1, this is a point 0 and this is a point
which is a. So, that says a greater than 1; a greater
than 1, this relation is there, is greater than 0 yeah and therefore, the graph was a
point which lies here, which is here right. As x tends to infinity, this graph actually
goes to infinity; as x tends to minus infinity, this graph goes to 0.
These two points is these two point and this is an increasing function. As afrom left as
you come from left to right, it increases. So, this is an increasing function, y is equal
to 0 is the horizontal asymptote, that is very clear ok. The range of a function is
0 to infinity, that is also very clear. The domain of a function is entire real line .
So, we have got all the details necessary for finding this . Now, what it so special
about 2 raised to x, if I replace this 2 with 3, still I will have y intercept to be 0,
1 because 3 raised to 0 is also 1 and I will again have domain of f to be equal to R; range
of f to be equal to 0 infinity; no x intercept; y is equal to 0 will be horizontal asymptote;
x tends to infinity a raised to x tends to infinity, x tends to minus infinity, a raised
to x tends to 0. There are no roots. The function is only increasing.
And therefore, I will state this as a fact that every f of x is equal to a raised to
x, for a greater than 1 will have same properties as 2 raised to x. So, I do not there is no
need to draw different different values. The behavior is same only the values will
change. For example, in this case, where you have seen the graph of this 1, 2 is a point;
1, 2 is a point, suppose I consider 3 raised to x, 1 3 will be the point. So, only the
values are changing; but the shape, the behavior, everything else that is listed here remains
the same. Therefore, you do not have to draw a graph every time, only thing is you need
to evaluate the values in general. So, what is the graph of f of x equal to a
raised to x in general? It is this way for a greater than 1. So, remember that line that
we have drawn which is that the line for a, where we have eliminated these 2 points such
as 0, this is 1, we have identified what is the case for a greater than 1. You have also
identified the case, where a is less than 1 and greater than 0 . So, let us go back
and see what happens when 0 is less than a less than 1. So, if a lies here how is the
behavior? So, you have already analyzed. And let us take this function as g of x andtake
it to be g of x is equal to a raised to x and this is 1 by 5 raised to x. hm Now, you
do not really have to draw this graph, what you can do is ok. So, g of x is equal to 5
raised to minus x. So, here x is replaced by minus x.
So, what will be the change in the behavior? So, when x is replaced by minus x, you know
its reflection across y axis, you have solved many examples in the assignments. This y axis,
this is x; then when I put it as minus x, it will be simply reflected along y axis.
So, if you look at this graph and try to draw a graph of this function, then it should be
something like coming from here going here, it should be something like this, it should
actually look like a reflection along y axis. So, let us try to show it as reflection ok.
This will actually go very close, but never touch. So, let me erase this ok. So, this
is how it will look like. hm So, without actually thinking about anything else, you can simply
draw a graph of 1 by 5 raised to x; but still let us try to do it inregular set up.
So, what will be the domain of this function? The domain of this function is very clear
becausewe have used it several times, the domain of this function will be real line.
Range, nothing changes; 0, infinity because it is a reflection across y axis. So, let
us look at this function. So, the domain will be R; range will be 0 to infinity. What will
be the y intercept? Because it is a reflection, so y intercept would not change, so it will
be 0, 1 only. x intercept will be nil, there would not be any x intercept.
And therefore, no roots and what is what about the end behavior? End behavior is like x tending
to infinity, x tending to minus infinity . So, when x tends to infinity, the end behavior
will be because it is a reflection you see. So, when x was tending toinfinity there, it
was going to infinity. So, and x tending to minus infinity, function
5 raised to x would have behaved, it will go to 0. So, that reflection will make this
a raised to x or 5, 1 by 5 raised to x whatever is the function 1 by5 raised to x, let me
do it properly. So, this will make 1 by 5 raised to x to go
to 0 and this function 1 by 5 raised to x will go to infinity ok. Good. Then, because
it is a reflection, the increasing thing will become decreasing. So, there is nointelligence
here. So, this will be in fact a decreasing function wonderful. So, we have analyzed everything
without taking much efforts. This is the beauty of once you understand the functions on graphical
plane. So, here is the graph of a function which
is given to us 1 by 5 raised to x, you also might have plotted and naturally, the we will
analyze whether it coincides with our thing. So, this is a point 0 comma 1, now it is 1
by 5. So, your point will be somewhere here, sorrythis is 5. So, the point 1 is here and
this point is 1 by 5. So, 1 comma 1 by 5, this is done. Then, as
x tends to infinity, as x tends to infinity, this function goes to 0. As x tends to minus
infinity that is this way, this function actually goes to infinity ok and this function is decreasing.
From left to right if you come, you are actually coming down. So, it is a decreasing function.
So, this completely gives us an understanding of what the graph of a function will look
like. Also, the same fact is true that every f x
a raised to x, where 0 is less than a is less than 1 has same properties as 1 by 5 raised
to x. Therefore, it is a representative class. So, you do not have to worry about the because
it is a representative class, you have to worry about all other functions. All other
functions will have a similar behavior. So, we have done a lot, let us summarize these
things in a neat table which is this. So, this is the summary ofthe table . So, if I
have been given a function f of x is equal to a raised to x, thento be more precise,
let me draw a line here. This is a line; it does not look like a line, but assume that
this is a line. This is the point 1, then I am talking about 0 less than a less than
1 that this zone. In this zone, the domain of a function is
R; range of a function is 0 to infinity. There are no x intercepts, no; y intercept is 0,
1. Horizontal asymptote y is equal to 0 is there. The function is decreasing. The end
behavior as x tends to infinity, f of x tends to 0; as x tends to minus infinity, f of x
tends to infinity correct. Then,you look at the function which is a greater
than 1, domain is real line, range is 0 infinity, nil; 0, 1, y intercept is 0, 1. Horizontal
asymptote is y is equal to 0. The only distinguishing feature is the function is increasing here
and a function is decreasing here and because it is increasing and decreasing, the end behavior
changes that is because it is decreasing, it will decrease to 0 because it is bounded
below by 0 and because this is increasing, it will increase to infinity, but here it
will go to 0 ok. Then finally, you see the prototypes, just
look at the graphs of these two functions ok. This ends our topic on exponential functions.
Now, we will introduce something which is called natural exponential function in the
next video.
5.0 / 5 (0 votes)