<< Chapter < Page Chapter >> Page >

Use a graphing utility to find an exponential regression formula f ( x ) and a logarithmic regression formula g ( x ) for the points ( 1.5 , 1.5 ) and ( 8.5 ,  8 .5 ) . Round all numbers to 6 decimal places. Graph the points and both formulas along with the line y = x on the same axis. Make a conjecture about the relationship of the regression formulas.

Got questions? Get instant answers now!

Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary.

First rewrite the exponential with base e : f ( x ) = 1.034341 e 0 .247800x . Then test to verify that f ( g ( x ) ) = x , taking rounding error into consideration:

g ( f ( x ) ) = 4.035510 ln ( 1.034341 e 0 .247800x ) 0.136259 = 4.03551 ( ln ( 1.034341 ) + ln ( e 0 .2478 x ) ) 0.136259 = 4.03551 ( ln ( 1.034341 ) + 0 .2478 x ) 0.136259 = 0.136257 + 0.999999 x 0.136259 = 0.000002 + 0.999999 x 0 + x = x

Got questions? Get instant answers now!

Find the inverse function f 1 ( x ) for the logistic function f ( x ) = c 1 + a e b x . Show all steps.

Got questions? Get instant answers now!

Use the result from the previous exercise to graph the logistic model P ( t ) = 20 1 + 4 e 0.5 t along with its inverse on the same axis. What are the intercepts and asymptotes of each function?

Graph of P(t)=20/(1+40.5e^(-0.5t)) and P(t)=(ln(4)-ln((20/t)-1)/0.5.

The graph of P ( t ) has a y -intercept at (0, 4) and horizontal asymptotes at y = 0 and y = 20. The graph of P 1 ( t ) has an x - intercept at (4, 0) and vertical asymptotes at x = 0 and x = 20.

Got questions? Get instant answers now!

Chapter review exercises

Exponential Functions

Determine whether the function y = 156 ( 0.825 ) t represents exponential growth, exponential decay, or neither. Explain

exponential decay; The growth factor, 0.825 , is between 0 and 1.

Got questions? Get instant answers now!

The population of a herd of deer is represented by the function A ( t ) = 205 ( 1.13 ) t , where t is given in years. To the nearest whole number, what will the herd population be after 6 years?

Got questions? Get instant answers now!

Find an exponential equation that passes through the points (2, 2 .25) and ( 5 , 60.75 ) .

y = 0.25 ( 3 ) x

Got questions? Get instant answers now!

Determine whether [link] could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

x 1 2 3 4
f(x) 3 0.9 0.27 0.081
Got questions? Get instant answers now!

A retirement account is opened with an initial deposit of $8,500 and earns 8.12 % interest compounded monthly. What will the account be worth in 20 years?

$ 42 , 888.18

Got questions? Get instant answers now!

Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with 7.5 % APR, compounded daily, in order to reach her goal in 3 years?

Got questions? Get instant answers now!

Does the equation y = 2.294 e 0.654 t represent continuous growth, continuous decay, or neither? Explain.

continuous decay; the growth rate is negative.

Got questions? Get instant answers now!

Suppose an investment account is opened with an initial deposit of $10,500 earning 6.25 % interest, compounded continuously. How much will the account be worth after 25 years?

Got questions? Get instant answers now!

Graphs of Exponential Functions

Graph the function f ( x ) = 3.5 ( 2 ) x . State the domain and range and give the y -intercept.

domain: all real numbers; range: all real numbers strictly greater than zero; y -intercept: (0, 3.5);

Graph of f(x)=3.5(2^x)
Got questions? Get instant answers now!

Graph the function f ( x ) = 4 ( 1 8 ) x and its reflection about the y -axis on the same axes, and give the y -intercept.

Got questions? Get instant answers now!

The graph of f ( x ) = 6.5 x is reflected about the y -axis and stretched vertically by a factor of 7. What is the equation of the new function, g ( x ) ? State its y -intercept, domain, and range.

g ( x ) = 7 ( 6.5 ) x ; y -intercept: ( 0 ,  7 ) ; Domain: all real numbers; Range: all real numbers greater than 0.

Got questions? Get instant answers now!

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask