Grade 11 Pro Maths Online Lesson (Maths Exponential Functions)

MATHEMATICS GRADE 11 LESSON TOPIC : EXPONENTIAL FUNCTIONS 20 MAY 2021

Focus : Functions Lesson Duration : Saturday: Test 2 2hr Tuesday: Thursday: Consolidate Paper 1 Exponential Functions Topics Friday: Saturday : Consolidate Exponential Hyperbola Functions & Start Hyperbola 2

Grade 11 : Exponential Function Exponential Function !!!!!

Exponential Functions Learners need to be able to recognize the A function f will decrease when a > 0 & 0 < b < 1 equation of an exponential function and the where the graph is above the horizontal asymptote) sketched exponential graph itself. y = ab x + p + q where b > 0 ; b ≠ 1 ➢ Intercepts must be labelled Learners must be able to sketch/draw the The asymptote must be clearly indicated with a Exponential graph and pay focus to the dashed line and the values where it intercepts following: the y - axis must be given. ➢ Shape is important and remember a function Remember the exponential function only has f will increase when a > 0 & b > 1 and the one asymptote (Horizontal: y = q, the value of q) graph will be above the horizontal and only the required information on the asymptote. A function f will also increase Cartesian Plane must be included. when a < 0 & 0 < b < 1 where the graph lies below horizontal asymptote) 4

The importance and real life applications of exponential functions. • Exponential functions are mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies (managing money), risk analysis and risk management (calculation of actuarial age/life expectancy), construction and keeping airplanes in the air.

What careers use exponential functions to solve real life problems? • The business managers, financial analysts , computer programmers, research scientists, professional engineers, actuarial scientists and health care professionals apply the knowledge of the exponential functions to solve real life problems on a daily basis respectively.

Select a =2 and utilize at least the following five ordered pairs: y — x |—2]-1/0}1]2 y=?’ 1 }1 4 yl—- |= ]1 {2/4} 3 2 4 |2 Plot these points on =Ppomain —R Cartesian Plane. Range = (0; oD) Draw best fit graph. <q. BEEBE EE EEE BEEBE EER ER EBER EE Horizontal Asymptote: -2.| -1 1 2 Defined by vy =0 =1

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Properties of the exponential functions: 𝑥+𝑝 y = f(x) = a.𝑏 +q ✓We will explore the effect of the parameters a and q and investigate the effect of p on the graphs of the exponential function defined by ✓ y = f(x) = a.𝑏 𝑥+𝑝 + q , where b > 0 and b ≠ 1. ✓Where a determines the shape of the graph as well as on which side of the asymptote the exponential function lies. Together with a , the value of b determines whether function is increasing or decreasing. ✓The value of q resulted in a vertical translation (up or down) and the line y = q is the horizontal asymptote. The value of p resulted in a horizontal translation.

Properties of the exponential functions: 𝑥+𝑝 y = f(x) = a.𝑏 +q • Domain: (the other notation of can be provided) • Range: y > q for a > 0

Properties of the exponential functions: 𝑥+𝑝 y = f(x) = a.𝑏 +q • Domain: ( the other notation of can be provided) • Range: y < q for a < 0 (the other notation of can be provided)

Properties of the exponential functions: 𝑥+𝑝 y = f(x) = a.𝑏 +q • Domain: (the other notation of can be provided) • Range: y > q for a > 0

Properties of the exponential functions: 𝑥+𝑝 y = f(x) = a.𝑏 +q • Domain: (the other notation of can be provided) • Range: y < q for a < 0 (the other notation of can be provided)

Asymptotes : Discussion & clarify Asymptotes using a function

Domain & Range Discussion & clarify : Domain & Range

Activity : Sketching and properties

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: Solution promaths . 1)" Given: f(x)= ; +4 Discussion Area 1 | y=4

Continuation of the solution Discussion Area •

Discuss the Reflection along x axis (y = 0) & y axis (x = 0)

Activity-Sketching and properties

Discussion Area [aauleuus | For y-intercept/Vir y-afsnit substitution x = 0: = y=4.2°41 =§ H(0;5) 2 | For x-intercept! Vir y-afsnit y=0 ie/di, 4.2" +1=0 427 =-1 "= ~ which is impossible, since 2” >0 for xe R

Promaths Continuation of the solution Discussion Area y=l pe

Continuation of the solution ay Discussion Area 4 a(i)=40" +2] =47" +48 The graph of h is translated 7 units upwards to form g’

Activity - Finding the equation and properties. •

Activity- Finding the equation and properties

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Solution Promaths 1 q=-6 3 , Discussion Area 3 xint:2°'=2 2.x=2 +.(2:0) 1 1 int: y=3.27-6=-4— +.) 0:-4— pint: y 5 ( :| Average Gradient ovat 2 2-0 9 =e or ot 4 4 4 y=3.277 -6

Activity on Exponential Functions 1. Consider the function f defined by f (x) = 4 - x - 2 1.1 Calculate the coordinates of the intercepts of f with the axes. 1.2 Write down the equation of the asymptote of f . 1.3 Sketch the graph of f . 1.4 Write down the equation of g if g is the graph of f shifted 2 units upwards. 29

Asymptotes & Intercepts of an Exponential Function 1.1 Calculate the coordinates of the intercepts o f f (x) = 4− x − 2 with the axes. y-intercept : x = 0  y = 40 − 2 = −1  y − intercept at (0; −1) y = 0  y = 4− x − 2 = 0 x-intercept :  4− x = 2  2−2 x = 21 Asymptote of f: −2x = 1 y = −2  x − intercept at (− 12; 0 ) (1.2 ) Write down the equation of the asymptote of f . 30

Graphing an Exponential Function 1.3 Sketch the graph of f . y x f (x) = 4− x − 2 31

Transforming an Exponential Function 1.4 Write down the equation of g if g is the graph of f shifted 2 units upwards g(x) = 4− x 2 f (x) = 4− x − 2 32

Concluding Remarks Following our today lesson, I Repeat this procedure until you are want you to do the to: confident. Read through what the learner need to Do not forget: Practice makes understand and master in your learner perfect! material. Complete the activities Attempt as many as possible other similar examples on your own from the Text-Book and the past exam papers. 33

o Thank you Promaths y