Lesson 1.3 Exponential Functions

7Lesson 1.3

Exponential Functions

Part I HW: page 26: 1-20For 19, just use the calculator to determine an exponential model, rather than completing a and b.

Drill: Solve each equation

x3 + 9 = 17

2y2 + 2 = 10

½z3 - 8 = 24

Exploration on the graphing calculator:

You have 15 minutes to complete this on a separate

piece of paper. Graph the function f(x) = ax for a = 2, 3, 5.

Use the window [-5, 5] by [-2, 5]. For what values of x is it true that 2x < 3x <

5x ? For what values of x is it true that 2x > 3x >

5x ? For what values of x is it true that 2x = 3x =

5x ? Graph the function y = (1/a)x for a = 2, 3, 5 Repeat parts 2-4 for the function in part 5.

Exponent Rules Product of Powers Postulate

ax ● ay = ax+y

Power of a Power Postulate (ax)y = axy

Power of a Product Postulate (ab)x = ax bx

Quotient of Powers Postulate (ax /ay ) = ax-y

Power of a Quotient Postulate (a/b)x = ax / bx

Zero Exponent Theorem a0 = 1

Exponential Function Let ‘a’ by a positive real number

other than 1. The function f(x) = ax is the exponential function with base a.

Graph the function f(x) = 2(3x) – 4. State domain and range.X Y

-2-1012

-3.8-3.3-22

14-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-5

0

5

10

15

Domain : a l l r ea l

numbers( -∞ ,∞)

Range : y > 4( -4 , ∞ )

Finding Zeros (x-intercepts)

Find the zeros of f(x) = 5x – 2.5 Let y1 = f(x) Let y2 = 0 Graph (standard window is fine…ZOOM

6) 2nd TRACE, 5, ENTER, ENTER, ENTER In the case of multiple zeros, you will

need to move the cursor towards the other zero(s) before hitting ENTER, ENTER, ENTER

Rewriting Bases Rewrite 4x with a base of 2

4 = 22

So 4x = (22)x

Leaving 4x = 22x

Rewrite (1/64)x with a base of 4 64 = 43

So (1/64) = 4 -3

(1/64)x = (4-3)x

= 4 -3x

Exponential Growth vs. Decay: y = k(a)x, k>0

Growth a>1 Domain: (-∞,∞) Range: (0,∞) y-intercept is (0,k) As x increases, for a

> 1, f(x) also increases without bound

The x-axis (y = 0) is the asymptote

Decay 0<a<1 Domain: (-∞,∞) Range: (0,∞) y-intercept is (0,k) As x increases, for

0<a<1, f(x) decreases, approaching zero

The x-axis (y = 0) is the asymptote

Predicting Population In 1995, the US population was

estimated at 264,000,000 people and was predicted to grow about 0.9% a year for the near future.

A. With these assumptions, state a formula for the US population x years after 1995.

B. From the formula, estimate the population in 2010.

Cost of a Penn State Education/Semester (Tuition

Only) for PA Residents

x = years after 1991 and y = cost of tuition

Pick any two points on your curve. Step 1: Using the formula y = kax, form a

system: Step 2: Divide the equations (higher

power/lower power) to find a: Step 3: Substitute a into EITHER equation

to find k: 4: Rewrite, substituting a and k:

Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003Tuition 2166 2274 2376 2483 2594 2717 2816 2920 3081 3273 3527 4004 4648

Using the calculator…

1) Enter data into STAT, 1:edit, L1 = x, L2 = y2) Plot on the calculator: STATPLOT, Type:

scatterplot (1st one), Xlist = L1, YList = L23) ZOOM, 9:Zoomstat4) Exponential regression: STAT, →Calc,

0:ExpReg5) To put into Y=: VARS, →Y-Vars, 1:Function,

1:Y1

6) 9:ZoomStat to see line with points

DrillYear Populatio

n for Virginia (thousan

ds)1998 6901

1999 7000

2000 7078

2001 7193

2002 7288

2003 7386

1. Find the exponential regression by hand using any two points.

2. (Let x = year after 1990)

3. Find the exponential regression on the calculator.

4. Predict the population for Virginia in 2011

Half-Life Formula: A = A0(.5) t/h

A = final amount after t years. h = half life time period A0 = original amount

A certain substance has a half-life of 24 years. If a sample of 80 grams is being observed, how much will remain in 50 years? A = final amount after 50 years. h = 24 years A0 = 80 grams A = 80(.5)50/24 = 18.88 grams

compound interest formula: A = P (1 +r/n)nt

A = final amount P = original

amount R = interest rate N = number of

compounding periods

T = time

A bank is currently offering a certificate of deposit paying 5.25% interest compounded quarterly. Find the value of the CD after two years if $1000 is invested. A = final amount P = 1000 R = 5.25 = .0525 N = 4 T = 2 A = 1000 (1 +.0525/4)4*2

= $1109.95

If the interest were compounded continuously, the amount would approach the irrational number e » 2.718281828….

Continuously Compounded Interest

A = Pert

P = principal, r = rate, t = years

e

Suppose you invest $100 at 4.5% interest, compounded continuously, for 5 years. Calculate how much will be in the account.

)5045(.100 e23.125$

Compare this to an account compounded monthly.

)125(12045.1100 18.125$

Doubling Time Determine how long it will take for

an investment of $P to triple if you compound continuously at a rate of 3.7% A = Pert

3P = Pe .037t

3 = e .037t

Let y1 = 3 and y2 = e .037t

Intersect (You will need to change windows): t = 29.69 years

Group activity: copy and complete the tables below. Hand in at the end of class!

x y= 2x-3

Change ∆y

1 -1 ---2 1 23 3 24 5 2

x y = x2 Change ∆y

1 ---234

x y = -3x + 4

Change ∆y

1 ---234

x y = 3ex

Ratio (yi / yi-

1)1 ---234

example

Hw: p. 27: 21-32p. 28: 39-46

p. 2-: 1-4