ELF.01.2 – Graphing ELF.01.2 – Graphing Exponential FunctionsExponential Functions

MCB4U - SantowskiMCB4U - Santowski

(A) Review(A) Review at this stage of the course, you should be adept at at this stage of the course, you should be adept at

graphing and analyzing graphs graphing and analyzing graphs in particular, the following graphs should be masteredin particular, the following graphs should be mastered

(i) polynomial functions(i) polynomial functions (ii) rational functions(ii) rational functions

you should be able to algebraically and graphically you should be able to algebraically and graphically identify the following features on graphs:identify the following features on graphs: Domain, range, x- and y-interceptsDomain, range, x- and y-intercepts Where in its domain a function increases or decreasesWhere in its domain a function increases or decreases Maximum and minimum points (turning points)Maximum and minimum points (turning points) ““End” behaviourEnd” behaviour ConcavityConcavity asymptotesasymptotes

(B) Exponential Functions(B) Exponential Functions exponential functions have the general formula y = aexponential functions have the general formula y = axx where the variable is now the where the variable is now the

exponentexponent so to graph exponential functions, once again, we can use a table of values and find so to graph exponential functions, once again, we can use a table of values and find

pointspoints ex. Graph y = 2ex. Graph y = 2xx

x yx y -5.00000 0.03125-5.00000 0.03125 -4.00000 0.06250-4.00000 0.06250 -3.00000 0.12500-3.00000 0.12500 -2.00000 0.25000-2.00000 0.25000 -1.00000 0.50000-1.00000 0.50000 0.00000 1.000000.00000 1.00000 1.00000 2.000001.00000 2.00000 2.00000 4.000002.00000 4.00000 3.00000 8.000003.00000 8.00000 4.00000 16.000004.00000 16.00000 5.00000 32.000005.00000 32.00000

(B) Features of y = 2(B) Features of y = 2xx

(i) no x-intercept and the y-intercept is 1(i) no x-intercept and the y-intercept is 1 (ii) the x axis is an asymptote - horizontal (ii) the x axis is an asymptote - horizontal

asymptote at y = 0+asymptote at y = 0+ (iii) range { y > 0}(iii) range { y > 0} (iv) domain {xER}(iv) domain {xER} (v) the function always increases(v) the function always increases (vi) the function is always concave up(vi) the function is always concave up (vii) the function has no turning points, max or (vii) the function has no turning points, max or

min pointsmin points

(C) Graphs of Various Exponential (C) Graphs of Various Exponential FunctionsFunctions

(C) Transformed Exponential (C) Transformed Exponential GraphsGraphs

As seen in the previous slide, the graph As seen in the previous slide, the graph maintains the same “shape” or maintains the same “shape” or characteristics when transformedcharacteristics when transformed

Depending on the transformations, the Depending on the transformations, the various key features (domain, range, various key features (domain, range, intercepts, asymptotes) will changeintercepts, asymptotes) will change

(D) Investigating Exponential (D) Investigating Exponential Functions: f(x) = aBFunctions: f(x) = aBb(x+c)b(x+c) + d + d

We will use a GDC (or WINPLOT) and investigate:We will use a GDC (or WINPLOT) and investigate:

(i) compare and contrast the following: y = {5,3,2}(i) compare and contrast the following: y = {5,3,2}xx and y and y = {½, 1/3, 1/5}= {½, 1/3, 1/5}xx

(ii) compare and contrast the following: y = 2(ii) compare and contrast the following: y = 2xx, y = 2, y = 2x-3x-3, , and y = 2and y = 2x+3x+3

(iii) compare and contrast the following: y = (1/3)(iii) compare and contrast the following: y = (1/3)xx, and , and y = (1/3)y = (1/3)x+3x+3 and y = (1/3) and y = (1/3)x-3x-3

(iv) compare and contrast the following: y = 8(2(iv) compare and contrast the following: y = 8(2xx) and ) and y = 2y = 2x+3x+3

(E) Further Investigation – Internet (E) Further Investigation – Internet TutorialTutorial

Go to this link Go to this link from from AnalyzeMathAnalyzeMath and and work through the tutorial on transformed work through the tutorial on transformed exponential functionsexponential functions

Consider how y = aConsider how y = axx changes changes i.e. the i.e. the range, asymptotes, increasing/decreasing range, asymptotes, increasing/decreasing nature of the function, shifting and nature of the function, shifting and reflectingreflecting

(G) Homework(G) HomeworkNelson text, p103-105, Q1-5, 8, 9,11,13Nelson text, p103-105, Q1-5, 8, 9,11,13