Name of School:
ROUTINE INFORMATION Inhlakanipho High School.
Student Surname and Name:
Maphumulo Mthokozisi
Student Number:
207503878
Grade:
10
Subject:
Mathematics
Topic:
Exponential graph
Content /Concept Area:
Characteristics of :- y=ax : a>0: a â‰
CAPS page number:
55
Duration of Lesson:
45 minutes
1
Specific Aims: To develop fluency in solving exponential form To provide opportunity to obtain skills of plotting points of exponential graph. To integrate exponential function to its independent and dependant variables. To explore systems used to derive exponential function into a function other than exponential expressions that everybody underestimate. Lesson Objectives:
Knowledge Learners should gain
Skills Learners should be able to do
Value /Attitude Learners should acquire
knowledge and
following :
values and attitudes
understanding of the
conducive to :
following : Explain the function of a
Sketch
a
graph
of
function of exponential graph.
exponential
graph
function
an
using grid pattern.
Review Exponential laws
Explain
the
formula
Produce
exponential
graph
through the points produced from a function.
for
calculating exponential graph.
Introduce the idea of domain and range of an exponential graph.
Analyse
the
concept
of
Recall the first set of numbers
Produce
exponential function. Describe two types
of
and the second set of numbers. Differentiate between
using few sets of ordered pairs. Acknowledge the formula of
exponential growth and decay.
exponential graph where a
exponential graphs
exponential
graph
Âż0 . Approaches / Teaching Strategies:
Role-plays. Case studies Class discussions. Questioning. Whole class-discussion Group Work
Resources:
Calculators Chalkboard Newspaper Worksheet Introduction:
During the introduction of this lesson, we make use of population growth increase to explain exponential growth. The aim of doing that is to draw learner’s attention words exponential growth and exponential decay. This is an accounting class, at least I have a clue of how should students be made to enjoy a lesson of an exponential graph within pure applied maths. Simple algebraic expressions would mean absolutely nothing to them; they are used to work with tangible things such as goods, capital, and infrastructure. Indeed population growth is the most common examples of observing population growth. For example if a population of a group increases by say 5% every year, we can make use of exponential graph to represent how the population explodes over a period of time. If the population grows by 5% each year, we can take the total population and multiply it by 105% to determine a number of population after one year.
i)
Therefore, after 1st year, the population is s 1.05 times from the original
ii) iii)
population. After the 2nd year, the population will be (1.05)2 times what it originally was. After 100 years, the population will be (1.05)100 times what it originally was. After x years, the population will be (1.05)
x
times what it originally was.
Imagine what would happen if times of disasters such as holocaust massacres or pandemic take place. The graph of exponential growth would drop down rapidly. Indeed that would be opposite of growth. Exponential decay is the decrease in a quantity according to the law. In the meantime, the function of an exponential graph is introduced in the chalkboard. Students are also questioned to explain what are exponents and their few basic rules. The main aim is to let them revise exponents required in the expression of exponential function to be able to complete the table of dependant factors. Successful completion for such table would enable students to plot the exponential graph. This is group work where students must sit into a group with six (6) members per group. Exponential function involves exponential laws on which, y=ax
a
x
POWER Base
The exponential laws are as follows: i)
If you want to simplify exponents, where bases have to be multiplied, see if bases are the same and add exponents. Therefore: am + an = am+ n: the bases are the same, we add exponents.
ii)
If you want to simplify exponents, where bases have to be divided, see if bases are the same and subtract exponents. 2
Therefore: iii)
(am)n= (a)m n
iv)
a-1=
1 a
b b1
= b2-1, the bases are the same we can equate exponents.
In the graph of y= ax: a>0, a≠1 Domain =R and Range = (0: ∞), for example: 2x ¿ 0 for all x ∈ R . There is no x-intercept and the y-intercept in the point (0: 1), since a0=1. The axis (y=0) is a horizontal asymptote. The function is one to one.
Development:
A class is divided into six groups where three groups must develop an exponential graph of y= ax for a>0, a≠1. Another group must deal with y = ax, for 0 ¿ a ¿ 1 Methods to be followed: i) ii)
Set up a table of x-values from -3 to 3 to be used to fill the ordered pairs In their different groups of six members learners have to complete the following
iii)
function of exponential graphs : Complete the exponential graph using a function of y= 2x and y= 2-x The other
iv) v) vi) vii)
group complete an exponential graph of y= 2-x Let y= 2x and y= 2-x be your function , calculate properly, do not skip methods Identify input and output clearly on your table Use Cartesian Plane to plot your graph Check ordered pairs to make sure their estimation are done correctly and plot
viii)
them on a graph. Join the co-ordinates to produce exponential graph.
In this exercise, students are dealing with an introduction. This classroom exercise aimed at training to be able to draw, interpret and analyse graph independent of grid pattern. As one should have noticed, the graphs are divided into two so that there is the one for exponential growth and exponential decay. Exponential growth graph: an exponential function graph
x Y=2x
-4 -3 -2 -1 0.0625 0.1250 0.2500 0.500
0 1
1 2
2 4
3 8
4 16
Y= 2x, Exponential Growth y=ax, for a ¿ 0
In the graph of y= ax, where the actual trigonometric function is y= 2x. Domain = R and range (0 : ∞ ¿ There is no x-intercept, but y-intercept is (0: 1). Axis of symmetry: it is reflected about y=2-x so Axis of Symmetry is y= 0. The axis if symmetry is subject to interpretations for a sketch maker since there are cases where y= -2x and y= -2-x. This is so curious, so marvellous to discuss because it offers professional insights about reflection of exponential graph. As x increases, y increases As a increases, the curve become steeper. Exponential decay graph: exponential function graph
x Y=2-x
-4 16
-3 8
-2 4
-1 2
0 1
1 0.5
2 3 0.2500 0.125 0
4 0.062 5
Y= 2-x, Exponential Decay. This is still a graph of f(x) = ax 0 ¿ a<¿ 1 ¿
This type of exponential graph: is a Decay Exponential Graph: As x increases, y decreases. As â&#x20AC;&#x2DC;aâ&#x20AC;&#x2122; decrease, the curve become steeper. This is a one to one relation. There are no x-intercept, y-intercept is (0: 1) Then thee two graphs have common symmetry if they are pulled together as shown below.
Take note of the following regarding exponential graphs: (responding to the caution needed for their drawing question):-
A) NB: y=ax and y= abx and y= 2x are used interchangeable. B) NB: y= a-x and y= ab-x and y= 2-x are also used interchangeable. Where b is concerned (of appear with a formula, it takes a y- intercept and also forms the
turning point of an exponential function graph). These conditions take place only and only if this is a trigonometric function. NB sometimes we ignore b to be used as a, so ‘a’ can act as both ‘a’ and b , this is thus far the most critical discovery we have found to be true . A grade 11 and 12 educator also agreed that sometimes a value is ignored and it can play two roles in the graph as explained above. Therefore it is always helpful to reflect the exponential function graph from its longest formula: y= abx+p + q. This revealed that exponential function graph is paraplegic in nature because it relies from other methods to derive its original function .Similarly hyperbolic function and parabolic functions were also found to be semi-permanent in nature. So they are also paraplegic. This need to be born in mind, when dealing with drawings of these graphs, full stop.
2x
2-x
Now express f(x) in terms of a and b twice as much as it is explained from the above tables Solution: f(x) = y = 2-x , or y = a-x where a = 2, but also represents coefficient of b that is used to determine the shape of a graph or y = ab -x, where ‘a’ value is ignored even though it is known for
its use in determining the shape of an exponential graph. Therefore in y=ab x+p =
p, b will always represents y- intercept as well as a component of a turning point. This rule applies only and only in the graph is an exponential function. Similarly f(x) = y = 2x or y= ax, where also’ ‘a’ is used interchangeable with b. Therefore ‘a’ and b replace one another, where b is presented ‘a’ is retrieved. The absence status of a does not mean it is not used, the syntax operators +ve and –ve reveals the power and influence of a in an exponential graph. (e.g. evaluate y = -2 x and y= +2x). It clear that the change or effect is brought by power of syntax operators applied to the value of a – the coefficient of ‘b’) so
where y= -2x it is same as y = -1 (2 x) or y = -1(2)x.. But a cannot be equal to 1, a ¿ 1 so
a is
integrated to b strategically. This is not a rare thing to in maths we often observe it happening with the rationalization of sides of the special angles where two radius differ in dimension but angles takes similar function.( e.g. sin 450 =
1 is same as sin 45 √2
0
=
√2 . Where r=1 and 2
or r= 2.) In all mathematics books this point is widely debated. (I will strongly recommend further research on this matter or input from senior university professors). These are brief problems that students must realise in exponential graph. Planned Questions:
a) Define briefly; what is a difference between exponential graph of decay and growth. b) List three main important things you will need when you have to draw an exponential graph. c) Describe using the following equations the effect of an exponential function -x -x i) Y= 2 , and y= -2 ii)
x
Y = 2 and y = -2
x
d) Draw these graphs on same set of axes. e) If these were set of hyperbola, where would you introduce asymptote or axis of symmetry? f) What do you think would be a complete expression of exponential graph method or formula? (See 124 leaners book). . Comment extensively. g) What caution and conversation would you lay down about asymptote and axis of symmetry in the exponential graph. ( hint look at value of a and integrated drawing where all graphs are drawn simultaneously in c i) and ii)
Consolidation and Conclusion: In an exponential graph, function has a bifurcated name because it stands as a heading and yet at the same time students must have seen difference between the two sets of exponential graphs. Therefore, there ae two main types of exponential graphs learnt so for (excluding logarithmic) exponential graph and decay graph. From all previous examples, we have explained that:
A) For a ¿ 1, the graph of f(x) = ax increases from left to right. The greater the value of a, the steeper the curve and this is called exponential growth. B) For, 0 ¿ x< 1,the graph of f ( x ) = ax decreases from left to right. For smaller values of a, the graph becomes less steep, this is called decay. (Refer to a graph of f(x) = 2 x and 2-x from the above section.). Think about the fact that a graph of y= 2 x could not grow the same as the graph of 3x.
Y= 3
Y=2
x
x
, steeper
, less steep
Assessment: Compare a graph
of
1 3 ¿
)x and (
1 2
) x and briefly explain which one is exponential growth and
decaying.
Memorandum:
a) - For a> 1, the graph of f(x) = ax increases from left to right. The greater the value of a, the steeper the curve and this is called exponential growth. Exponential growth present itself in many different forms. These forms are observable in a scientific
phenomenon. Exponential function could also reveal how rise presents itself in natural phenomena such as cells to the expansion of animal population. -
For, 0<x<1, the graph of f(x) = ax decreases from left to right. For smaller values of a, the graph becomes steeper, this is called decay. Similarly, decay exponential graph follows the very same natural phenomenal such as population growth of certain cells organisms. Decay may follow exponential growth and it may be related to competition of food, space and other related environmental factors. Therefore, a pure applied maths make use of exponential graph function is to depict, describe and analyse real.
b) - Draw a table to insert input variables from – 4 to 4. - Identify dependant and independent variable. - Use a function of an exponential graph to calculate your outputs. - Draw your vertical and horizontal line or simple construct your Cartesian plane. - Plot your ordered pairs (set of new co-ordinates) - Label your graph - Use scientific calculator - Knowledge of exponential expressions and logarithmic expressions prior to actual calculations, etc. c) i) Y= +2
-x
-x
and y= -2 :
In a graph where a
¿
0, for y= ax all point of a graph lie above y= 0 (the horizontal
asymptote like line). e.g.: y= 2x. Domain = R, a ¿ 0, a ≠ 1. There is no x-intercept and y-intercept is (0: 1). x
ii) Y = +2 and y = -2
x
:
The negative sign on the value of a, where ‘a’is less than zero (a ¿
0) makes the graph to
lie below the horizontal asymptote of y= zero (0). Bothe graphs where a is less than zero lies below a horizontal line. This clearly seem to reflect a graph of hyperbola with fixed values of asymptotes where the graph neither cuts nor touches a horizontal line of y, in this case y= 0). If so we could also argue that a value of a is sometimes ignored and write the equation of exponential graph as y= abx+p +q. (therefore the value of a and b is common, b = point where
a graph intersects y – axis. However when we examine a point b in conjunction with ‘a’, a takes +ve and –ve descriptive of the shape of a graph and a point where a graph intersects yaxis and sometimes turning point. \Please take notice that this applies only and only if a graph is an exponential graph. However if b is absent such as a simple graph of y= a x , we figure out positive (+ve) and negative (-ve) syntax operators to determine the shape of a graph. d) See Drawings Bellow.
Y= +2
-x
F(x) = -2
FIGURE C
Y= +2x
-x
F(x) =-2x
e) i)
y= 0, a horizontal line or x= 0 a vertical line.
ii) y= abx+p + q, implies q = 0 and p= 0 : so y=-x or y= x. Therefore, it is important to note the following: -Domain is the set of x-values of a function -Range â&#x20AC;&#x201C;is the set of y-values of a function -Symmetrical â&#x20AC;&#x201C; If a shape is symmetrical about the line, it has the same shape on either side of the line. (e.g. see y= 2x and y=2-x drawn together.) - Axis of symmetry: the line about which the function is symmetrical. -Turning point: - the maximum and minimum value of b, other graphs it is (p: q) -Asymptote: - a line, which the function never touches, e.g.: in y=ax , possibilities are that y=0 is an asymptote. NB in exponential graphs, one need to be curious about conditions of asymptotes. -Creativity is also required, if you have y= 2 x you could also produce y= 2-x and lay down an argument that graphs are have common asymptote of y= o. The conclusion that their symmetrical line is also x=0. Similarly if all graphs are drawn in one scale new forms of axis of symmetry could emerge as x=0, y= -x and y=x , and y=0 because a graph has been transformed to take a form of both parabola and hyperbola. The end