EXPONENTIAL AND LOGARITHMIC FUNCTIONS IT LESSON
1 - GRAPHS OF EXPONENTIAL FUNCTIONS (SEC 4 ADDITIONAL MATHEMATICS)
Software: LiveMath
(previously known as MathView)
Thinking Skills: Induction
and Deduction At the end of the lesson, the students should be able to:
(1) sketch the graphs of exponential functions,
(2) describe the characteristics of the graphs of exponential functions.
INTRODUCTION
Why study the graphs of exponential functions? Click on the Powerpoint
Presentation IT4AMExpGraph.ppt
to look at two real-life applications: radioactive decay and Moore's Law.
Dr. Gordon Moore is the co-founder of Intel.
The first document on the left (IT4AMExpGraph1.thp) shows
the graphs of the exponential functions
y1 = ax, y2= bx
and y3 = cx. Change the value of a
to 4, the value of b to 3
and the value of c to 2.
Question 1:
What is the y-intercept for all the 3 curves?
Question 2:
What is the asymptote for all the 3 curves?
Question 3:
When x > 0, which curve has the largest y-values (i.e. which curve
is on top)?
y = ______ has the largest y-values,
i.e. 4x > _____ > _____ .
Question 4:
When x < 0, which curve has the largest y-values (i.e. which curve
is on top)?
y = ______ has the largest y-values,
i.e. 4x < _____ < _____ .
Question 5:
y = ex is called the natural exponential
function where e
2.71828. Where do you expect the curve y = ex to lie?
y = ex will lie between y = ___ and y = ___ .
Now change the value of a to e
to check your answer in Q5.
2. Set c = 1.
Question 6:
What is the shape of the curve y = 1x? Is this
an exponential function?
3. Set a = 2 and
b
= 1/2.
Question 7:
What do you observe between the curves y=(1/2)x
and y = 2x?
y = (1/2)x is the _____________ of y = 2x in
the __________ .
Question 8:
Convert the equation of the curve y = (1/2)x to the form
y = d–x. Therefore y = _____–x is the _____________
of y = 2x in the ________ .
4.
Click on the second graph to see the animation of the curves
y = ax from a = 0.5 to a = 4
in steps of 0.5.
Note:
The animation will stop at 3.5. It
will not include the last value.
B. GRAPHS OF EXPONENTIAL FUNCTIONS 2:
y = ex + c
1.
The 2nd document on the left (IT4AMExpGraph2.thp) shows
the graph of y = ex + c and its asymptote. Change the value
of c to 0 so that the graph shows the
curve y=ex.
Question 9:
For the curve y = ex, the y-intercept is _____
and the asymptote is y= ____ .
2.
Change the value of c from –2
to 4 in steps of 1. Record the y-intercepts and the asymptotes
in the table in the Worksheet (see below).
Curve
y-intercept
Asymptote
y = ex – 1
y =
y = ex
y = ex + 1
y = ex + 2
y = ex + 3
y = ex + 4
Question 10:
With reference to the curve y = ex, infer by
induction
the effect of c on the curve y = ex+c.
c will shift the curve y = ex _______ if c > 0 and ________
if c < 0.
3.
Click on the second graph to see the animation of the curves
y = ex + c from c = –2 to c = 4
in steps of 1.
Note:
The animation will stop at 3. It
will not include the last value.
C. GRAPHS OF EXPONENTIAL FUNCTIONS 3:
y = eax+b
1.
The third document on the left (IT4AMExpGraph3.thp) shows
the graph of y = eax+b. Change the value of a
to 2 and the value of b to 1.
Question 11:
For the curve y = e2x+1, the y-intercept is
_______ and the asymptote is y = _______ .
Hint: To find the y-intercept,
put x = 0 and use a calculator to evaluate y to 3 significant figures.
Or use the knife button (first button
on top right hand corner of graph) to draw a region enclosing the y-intercept
and then release. This will zoom in on the selected region. Repeat the
process until you get the y-intercept to 3 significant figures.
2.
Change the values of a and b to obtain the curves given
in the table below. Record the y-intercepts and the asymptotes.
Curve
y-intercept
(to 3 sig. fig.)
Asymptote
y = ex+2
e2 =
y =
y = e2x–1
y = e–x+3
y = e3x–2
Question 12:
What do you notice about the asymptotes?
3.
Set b = 1. Change the
value of a from –2 to 3 in steps of
1 except for a = 0 (why?). Record the y-intercepts and the asymptotes in
the table in the Worksheet (see below).
Curve
y-intercept
(to 3 sig. fig.)
Asymptote
y = e–2x+1
e1 =
y =
y = e–x+1
y = ex+1
y = e2x+1
y = e3x+1
Question 13:
What do you notice about the y-intercepts and the asymptotes?
4.
Click on the second graph to see the animation of the curves
y = eax+b from a = –2 to a = 4
in steps of 1 (when b = 1).
Note:
The animation will stop at 3. It
will not include the last value.
D. EXERCISE
1.
Do NOT use your computer to do
the exercise. You can check your answers after that.
2.
Apply what you have learnt to deduce
the shapes of the following curves and sketch them, showing the y-intercept
and the asymptote clearly.
(a) y = ex
(b) y = ex–2
(c) y = e–x
(d) y = e3x+2
(e) y = e–x+1
(f) y = e–2x+1
2.71828. Where do you expect the curve y = ex to lie?