7.1 Graph Exponential Growth Functions

Use a calculator to graph the following:

Exponential Function f(x) = bx where the base b is a positive number other than one. Graph f(x) = 2x Note the end behavior x→∞ f(x)→∞ x→-∞ f(x)→0 y=0 is horizontal asymptote

Asymptote A horizontal line that a graph approaches as you move away from the origin The graph gets closer and closer to the line y = 0 ……. As values Of x become larger And larger negative Negative numbers… But NEVER reaches it 2 raised to any power Will NEVER be zero!! y = 0

Look at the activity on p. 465 (OLD page) Graph y= a * 2x (first with a = 1/3 then a= 3) Now let a = -1/5 then let a = -5….compare these graphs with those graphed above. What is the effect of a on the graph of y? Passes thru the point (0,a) (the y intercept is a) The x-axis is the asymptote of the graph D is all reals (the Domain) R is y>0 if a>0 and y<0 if a<0 (the Range)

Consider: y = abx If a>0 & b>1 ……… The function is an Exponential Growth Function

Example 1 Graph y = (½) 3x Plot (0, ½) and (1, 3/2) Then, from left to right, draw a curve that begins just above the x-axis, passes thru the 2 points, and moves up to the right

D= all reals R= all reals>0 y = 0 Always mark asymptote!!

Example 2 Graph y = - (3/2)x Plot (0, -1) and (1, -3/2) Connect with a curve Mark asymptote D=?? All reals R=??? All reals < 0 y = 0

To graph a general Exponential Function: y = a bx-h + k Sketch y = a bx h= ??? k= ??? Move your 2 points h units left or right …and k units up or down Then sketch the graph with the 2 new points.

Example 3 Graph y = 3·2x-1-4 Lightly sketch y=3·2x Passes thru (0,3) & (1,6) h=1, k=-4 Move your 2 points to the right 1 and down 4 AND your asymptote k units (4 units down in this case)

D= all reals R= all reals >-4 y = -4

Now…you try one! Graph y= 2·3x-2 +1 State the Domain and Range! D= all reals R= all reals >1 y=1

Exponential Growth Models When a real life quantity increases by fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by: y = a(1+r)t Where a is the initial amount and r is the percent increase expressed as a decimal. The quantity 1+r is called the growth factor

A=P(1+r/n)nt Compound Interest P - Initial principal r – annual rate expressed as a decimal n – compounded n times a year t – number of years A – amount in account after t years

Compound interest example A=P(1+r/n)nt You deposit $1000 in an account that pays 8% annual interest. Find the balance after I year if the interest is compounded with the given frequency. a) annually b) quarterly c) daily A=1000(1+ .08/1)1x1 = 1000(1.08)1 ≈ $1080 A=1000(1+.08/4)4x1 =1000(1.02)4 ≈ $1082.43 A=1000(1+.08/365)365x1 ≈1000(1.000219)365 ≈ $1083.28