algebra 2b - unit 3: a model family lessons 11-15
the table represents a ___ function. this can be identified because the ___ have a constant value of ___. x f(x) 0 0 1 4
0.67 of 1 blank 1: quadratic blank 2: second differences of the y-values blank 3: 12
what is the approximate value of log_13 (27)?
1.28
use the following graph of function f(x) = 2 log_2 x to answer the question.
[7, 8]
lesson 13
average rates of change
jenessa is competing in a race along a mountain path.
f(x) should be reflected across the y-axis
which option correctly defines period?
the horizontal length of one complete cycle of a function.
if the domain for a logarithmic function, f(x), is (1, ∞), what must be true?
the vertical asymptote is at x = 1
use the graph of the logarithmic function f(x) to answer the question. https://cdstools.flipswitch.com/asset/media/690431 what is the x-intercept?
(3, 0)
use the graph of logarithmic function f(x) to answer the question. https://cdstools.flipswitch.com/asset/media/689893 what is the range of f(x)?
(−∞, ∞)
what is the average rate of change for the function f(x) = x^3 - 6x^2 + 4x + 7 over the interval 0 ≤ x ≤ 5?
-1
use the following graph of the function f(x) = 4 cos (x), which , which passes through the approximate points a(-6, 3.84) and b(-4, -2.62), to answer the questions. what is the approximate slope of the secant line that passes through points a and b?
-3.23
if log_1/2 (16) = x, what is the value of x?
-4
which table represents a function whose average rate of change is -3 for the interval [-5, -3].
-5 = 23 -3 = 17
function f(x) passes through each point in the following table. use this information to complete part a and part b. (0.05, -2.97)...
0.25 of 1 a: f(x) is most likely the sum of a logarithmic function and a trigonometric function. b: https://cdstools.flipswitch.com/asset/media/690951
use the mapping statement f(x) -> 2 * f(x) to answer the questions. part a: which option gives the correct interpretation of the statement? part b: which graph correctly demonstrates this transformation, where f(x) is the preimage and g(x) is the image?
0.25 of 1 a: f(x) is vertically dilated by a factor of 2. b:https://cdstools.flipswitch.com/asset/media/696497
use the following graph of function f(x) = 3 sin x, which passes through....
0.25 of 1 a: the instantaneous rate of change increases from a to b to c. b: yes. the instantaneous rate of change is the same at all points in the graph of f(x).
the table represents a(n) ____ function. this can be identified because the ___ have a constant value of ___. x f(x) -5 49
0.33 of 1 blank 1: linear blank 2: second differences of the y-values blank 3: 8
select the graphs of the functions whose ranges are (−∞, ∞). select all correct answers.
0.5 of 1 https://cdstools.flipswitch.com/asset/media/690452 https://cdstools.flipswitch.com/asset/media/690455
the following graph shows the preimage of a transformation, f(x), and the image, g(x). which graph(s) demonstrate(s) a change in amplitude? select all that apply.
0.5 of 1 https://cdstools.flipswitch.com/asset/media/696654 https://cdstools.flipswitch.com/asset/media/696655
use the graph of logarithmic function f(x) to answer the question. https://cdstools.flipswitch.com/asset/media/698494 what is the base?
1/2
examine the graph of the logarithmic function f(x). https://cdstools.flipswitch.com/asset/media/690433 the function f(x) has a vertical asymptote at x = _____.
3
what is the average rate of change for the function f(x) = 3^x-9 over the interval 9 ≤ x ≤ 11?
4
use the table to calculate the average rate of change of f(x) over the interval [0, 4].
60
a small town recorded the number of car accidents that occurred there each year.
XXX a: [2009, 2010] b: the average rate of change of f(x) is constant over time. since the average rate of change between 2009 and 2010 is 6, the average rate of change between 2010 and 2011 will also be 6.
examine the radical function, f(x), and trigonometric function, g(x), shown in the graph. for which interval, if any, is the average rate of change of g(x) greater than the average rate of change of f(x)?
XXX [0, 3] XXX [1, 2] XXX the average rate of change of f(x) is always greater because it is a radical function.
alex is deciding between two investment options, option a and option b.
XXX idk this is annoying. a: 0.25 b: 0.11 c: alex should invest in option a. the corresponding function a(x) has a greater average rate of change for the three-year time interval than b(x), which means he will earn more money by investing in this option.
a function f(x) = cos(x) + 1 is transformed. the inverse of f(x) is then transformed by the mapping statement f^-1(x) --> f^-1(-x). part a: which mapping statement explains how f(x), the original function, was transformed? part b: which graph shows the transformation of f(x), where f(x) is the preimage and g(x) is the image?
XXXXX a: f(x) --> f(-x) b: https://cdstools.flipswitch.com/asset/media/696537
use the following graph of function f(x) = 2x+5/3x to answer the question.
[-1, 1]
over which of the following intervals does f(x) = −1/x have the greatest average rate of change?
[2, -1/4]
which option correctly defines the midline of a periodic function?
a line drawn at the average of the maximum and minimum values of a function.
what type of function is the result of adding a logarithmic function to a constant function?
a logarithmic function
derron wants to graph the function f(x) = log_10(x), , so he begins by sketching out some of the features of the function. his sketch includes the x-intercept of f(x), , the vertical asymptote, and the intersection of f(x) and the line y = 1. what does his sketch include?
a point at (10, 1) a point at (1, 0) an asymptote at x = 0
kalina is testing her new sprinkler system. the height of the water in inches after it is released from the sprinkler head is modeled by the function
a translation 5 units up
use the function f(x) = log_0.1(10x + 0.8) to answer the questions. a: what is the base of f(x)? b: what does the base indicate?
a: 0.1 b: the base is between 0 and 1, which indicates that the function is decreasing.
use the graph of the logarithmic function f(x) to answer the questions. https://cdstools.flipswitch.com/asset/media/690447 a: what is the base of the function? b: the base of the function can be identified as the x-coordinate of which ordered pair?
a: 2 b: (2, 1)
use the following graph of function f(x) = 6 4sqrtx, and the given information, to answer the questions.
a: 3/2 b: the slope of the tangent line at x = 1 is also the instantaneous rate of change at that point.
use the function f(x) = log_8/4 (16x) to answer the questions. part a: what is the base of f(x)? part b: what does the base indicate?
a: 8/4 b: the base is greater than 1, which indicates that the function is increasing.
a television station tracks its viewership during a 10-hour period.
a: [0, 2] b: [6, 8]
george invests a total of $5000 into various funds on the day his daughter is born.
a: [0, 5] b: george made moderately successful investment decisions. the investment balance grew over time, but at an increasingly slow rate.
use the table to calculate the average rate of change of f(x) over the intervals [0,1], [2, 3], [4, 5], [6, 7]. part a: over which interval does the function have the greatest average rate of change? part b: which answer correctly describes the general behavior of f(x)?
a: [6, 7] b: as x increases, the slope of f(x) gets steeper.
use the following function f(x) = 4 cos x, which passes through a(0, 4), b(pi/2, 0).....
a: [pi, 2pi] b: the average rate of change of f(x) over the interval (0, 2π) is 0 because the x-distance between a and e is also the length of one period of f(x). this means that f(x) has the same y-coordinate at points a and e. because the change in y for the interval is 0, the average rate of change is 0.
use the following functions to answer the questions. f(x) = 1/x and g(x) = x^2. part a: what type of function is the result of multiplying f(x) and g(x)? part b: what type of function is the result of dividing g(x) by f(x)?
a: a linear function b: a cubic function
use the graph of the two cosine functions, f(x) and g(x), to answer the question. https://cdstools.flipswitch.com/asset/media/698499 part a: which transformation(s) is/are applied to f(x) to obtain g(x)? part b: which mapping statement correctly maps f(x) to g(x)?
a: a vertical translation a: a horizontal translation b: f(x) --> f(x + π/2) + 2
use the following graph of function f(x) to answer the questions. https://cdstools.flipswitch.com/asset/media/690850 part a: which function family does f(x) belong to? part b: which function belongs to the same family to f(x)?
a: exponential b: g(x) = 3^x - 7
the following graph shows the preimage of a transformation, f(x), and the image, g(x). https://cdstools.flipswitch.com/asset/media/696674 part a: which mapping statement correctly describes the transformation? part b: which description correctly describes the transformation?
a: f(x) ---> f(-x) b: f(x) is reflected about the y-axis
a function, f(x), is transformed in such a way that the inverse of f(x) is transformed by the mapping statement f^-1(x) --> 7f^-1(x). part a: which mapping statement correctly explains how f(x), the original function, was transformed? part b: what transformation must have been done to f(x).
a: f(x) --> f(1/7x) b: f(x) is horizontally stretched by a factor of 7
use the following graph, where f(x) is the preimage of a transformation, and g(x) is the image, to answer the questions. https://cdstools.flipswitch.com/asset/media/696560 part a: which mapping statement correctly describes the transformation? part b: which function correctly defines f(x)?
a: f(x) --> f(x - 2) + 3 b: f(x) = log_5(x)
use the following graph, where f(x) is the preimage of a transformation and g(x) is the image, to answer the questions. https://cdstools.flipswitch.com/asset/media/696578 part a: which mapping statement correctly describes the transformation? part b: which description correctly describes the transformation?
a: f(x) --> f(x) + 4 b: f(x) is translated 4 units up
use the following mapping statement to answer the questions. f(x) -> f(x + 7) part a: which answer shows the correct interpretation of the statement? part b: the statement is an example of what type of transformation?
a: f(x) is moved 7 units to the left b: a translation
use the following information about two different functions, f(x) and g(x), to answer the questions.
a: f(x) is probably the product of a constant function and a rational function. b: no, g(x) is not likely a combination of the same function families as f(x). although g(x) has a vertical asymptote, it does not have a horizontal asymptote.
a logarithmic function is translated by the mapping statement f(x) --> f(x + 3). part a: which mapping statement correctly explains how f^−1(x), the inverse of f(x), will be transformed? part b: explain your response.
a: f^1(x) --> f^-1(x) - 3 b: when a function f(x) is transformed, its inverse is also transformed. Since an inverse function is generated by switching the x- and y-values of f(x), the transformations on the inverse function are also switched. thus, transformations that affect y will affect x.
frances knows that the function f(x) is the sum of two functions from different function families. she observes the graph and concludes that f(x) must be the sum of a trigonometric function and a constant function.
a: frances is probably incorrect. the general shape of f(x) resembles a trigonometric function, except that the y-values increase overall as x increases. however, a constant function would not cause this behavior in the y-values; it would translate the function only up or down. thus, f(x) is probably the combination of a trigonometric function and some other type of function. b: linear b: trigonometric
shawntea is going to dig a square garden in her backyard and install a fence around it. she will choose the garden size based on the cost of the fencing material.
a: l(x) = 4√x b: l(x) should be translated 10 units up
examine the following table, which shows the subset of values from the function f(x). then, answer part a and part b. part a: which function family does f(x) belong to? part b: the function family can be identified because the _[blank]_ remains constant.
a: linear function b: rate of change
part a: which of the given functions are decreasing? part b: justify your response.
a: log_1/4(36x) b: the function is decreasing because the coefficient is greater than 0 but less than 1.
marcus is a marine biologist who is studying the behavior of a species of carp. he kayaks out on a lake and observes a carp leaping out of the water.
a: no, marcus did not create a good model. the carp reaches a height of 6 feet, so f(x) should have a maximum y-value of 6. the fish leaves the water at x = 0 and reenters the water at x = 3, so f(x) should have an x-intercept at (0, 0) and another at (3, 0). b: https://cdstools.flipswitch.com/asset/media/690874
a scientist drops a small specimen of bacteria onto a petri dish and observes its cellular growth for several hours. as shown in the following graph, the bacterial cell count after f(x) = 1.2^x
a: point n has a greater instantaneous rate of change than point m. the bacteria specimen is growing more and more rapidly from m to n. b: the slope of line d is steeper than the slope of line c.
andrea is selling flowers at a farmers market. one day, she sells flowers to 8 out of the 20 people who walk by her flower stand.
a: s(x) describes the number of people who purchased flowers versus the total number of people who walked by andrea's flower stand. the fraction 8/20 need to be multiplied by x, so s(x) should be a linear function. b: daniel should multiply s(x) by the constant function f(x) = 2.
use the following graph of function f(x) to answer the questions. https://cdstools.flipswitch.com/asset/media/690804 part a: which function family does f(x) belong to? part b: what is the parent function of f(x)?
a: square-root functions b: g(x) = √x
examine the graph, which shows radical function f(x) = -x+1/x-2..
a: the average rate of change for f(x) in the given interval is 0.2, and it is less than the average rate of change of g(x), which is approximately 0.7. b: average rate of change depends on the given interval. the result from part a is specific only to these functions on this interval.
use the following graph of an exponential function, f(x), and a logarithmic function, g(x), to answer the questions. part a: which statement correctly explains which function has a greater average rate of change over the interval [1, 2]? part b: consider the families of functions to which f(x) and g(x) belong. would you always expect the result you found in part a? why?
a: the average rate of change for f(x) in the given interval is 2, and it is greater than the average rate of change of g(x), which is 1. b: average rate of change depends on the interval given. this means the result from part a is specific to only these functions on this interval.
a logarithmic function f(x) is transformed according to the mapping statement f(x) --> f(x) + 10. part a: what transformation is done to the inverse of f(x)? part b: if f(x) is a logarithmic function, what type of function is its inverse?
a: the inverse function is translated 10 units to the right. b: exponential
jarrod is pitching a business idea to potential investors.
a: this is the same thing as asking when profit will equal $0, which is represented by the x-intercept of v(x). the x-intercept occurs at (2, 0), so the business will break even after 2 years. b: the investors should invest, because v(x) is an increasing function.
use the following graph of function f(x) to answer the questions. https://cdstools.flipswitch.com/asset/media/690803 part a: which function family does f(x) belong to? part b: which feature of f(x) justifies the correct response to part a?
a: trigonometric functions b: f(x) is periodic and bounded by extrema.
roxana and paki are working in a lab to compare the average growth rates of two types of mold, type i and type ii.
a: type i has a higher average rate of change of 1.56 over the first 20 hours of the experiment. this is greater than the average rate of change of type ii, which is 0.84. b: roxana's results conflicted with paki's because the average rate of change for a function shows the function's behavior over only a specific interval. one function may grow faster for one interval, and a different function may grow faster in a different interval.
otto wants to graph the function f(x) = log_2(x). he begins by analyzing the function and recording the following items. part a: which answer correctly shows what otto should conclude about f(x)? part b: which graph is the correct graph of f(x)?
a: x-intercept: (1, 0), vertical asymptote: x = 0, intersection of f(x) and y = 1: (2, 1) b: https://cdstools.flipswitch.com/asset/media/690463
phoebe spends her weekends competing in chess matches. so far, she has won 10 out of 20 matches that she has played.
a: yes, phoebe chose the most appropriate function type for her model. the situation describes a ratio of two expressions involving x. since rational functions include the ratio of two polynomials, phoebe was correct to choose a rational function to model her situation. b: rational
silvio is an air-traffic controller who must calculate the altitude of an incoming plane. he sees the plane at an angle of elevation of 40° as it flies over a signal tower. silvio knows the signal tower is 3 miles away, so he creates a trigonometric model to calculate the altitude of the plane relative to his own position. part a: did silvio choose the most appropriate function family for his model? part b: which function family correctly models this situation?
a: yes, silvio chose the most appropriate function family for his model. the text describes a right-triangle problem, so a trigonometric model is appropriate. b: trigonometric
which statement correctly compares the average rates of change of f(x) and g(x) over the interval [−2, 1]?
although f(x) is generally decreasing and g(x) is increasing over the interval, f(x) is changing at a faster rate because the line secant to f(x) is steeper than the line secant to g(x).
use the graph of the function f(x) = log_b x + c. to answer the question. https://cdstools.flipswitch.com/asset/media/689919
b = 5 c = 0 f(x) has a vertical asymptote at x = 0
use the function f(x) = log_2 (x) - 10 to fill in the blanks. f(x) must be ____ because ____.
blank 1: increasing blank 2: the base is greater than 1
lesson 15
building models
cato has two beehives in his backyard. in june, the first hive had 4 pounds of honey stored in its honeycombs, and this amount continued to grow by 30% every month thereafter.
cato should use an exponential function to model the activity of the first hive, and a linear function to model the activity of the second hive. then, he should add the functions together to find the total honey produced.
dione is designing a water fountain for the lobby of a new hospital. she knows that the fountain will have a circular pool with water projected upward at a constant rate from the center of the pool. she needs to determine the maximum height, so she uses a linear model to represent the situation. which statement correctly explains whether dione chose a good model to represent the situation?
dione did not choose a good model. because the water is a projectile with a maximum height, she should use a quadratic model.
lesson 14
exploring function families
on monday, magda goes on a walk in her neighborhood.
f(h) should be vertically stretched by a factor of 1.5
the following graph shows the preimage of a transformation, f(x), and the image, g(x). https://cdstools.flipswitch.com/asset/media/696668 which mapping statement correctly represents the transformation?
f(x) --> -f(x)
use the graph of the transformation, where a radical function f(x) is the preimage and its image is g(x), to answer the question. https://cdstools.flipswitch.com/asset/media/696506 what is the mapping statement for this transformation?
f(x) --> f(5x)
use the following graph, where rational function f(x) is transformed to produce image g(x), to answer the question. https://cdstools.flipswitch.com/asset/media/696571 which mapping statement correctly describes the transformation?
f(x) --> f(x) + 3
examine the following graph, where radical function f(x) is the preimage of a transformation, and g(x) is the image. https://cdstools.flipswitch.com/asset/media/696490 which mapping statement describes the transformation?
f(x) -> f(x + 3) - 2
which function could result from multiplying a quadratic function and a linear function?
f(x) = x^3
corey records the water volume in a reserve tank as it is being filled. he writes his observations in the following table, modeling the water volume with the function f(x), where xx is time in hours.
f(x) has the greater average rate of change over the interval. f(x) is an increasing linear function, which means f(x) increases at a constant rate over its entire domain. g(x) is a trigonometric function, and it is periodic; its y-coordinates rise and fall within a range of [2, 4].
which statement correctly compares the rate of change of f(x) and g(x)?
f(x) is an exponential function, and it generally increases more rapidly than g(x), which is a logarithmic function.
valentina is trying to identify the two function families that were combined to create f(x). she studies the table and then concludes that f(x) is a combination of a quadratic function and a constant function. which analysis of valentina's conclusion is most accurate?
f(x) may be the combination of a quadratic function and a constant function. the y-values of f(x) have constant second differences, which indicates a quadratic function. combining a quadratic function with a constant function would not affect this characteristic.
wanda is researching the growth of an invasive species of ivy after it is introduced to a foreign environment.
f(x) should be reflected across the x-axis
if f(x) maps to g(x) by the mapping statement f(x) --> 2 * f(x + 5), which mapping statement correctly maps f^-1(x) to g^-1(x)?
f^-(x) --> f^-1(1/2x) - 5
lesson 11
graphing logarithms
which graphs represent logarithmic functions that are decreasing? select all that apply.
https://cdstools.flipswitch.com/asset/media/689856 https://cdstools.flipswitch.com/asset/media/689857 https://cdstools.flipswitch.com/asset/media/689858
which graph correctly illustrates the midline of a function, f(x), where f(x) = sin (x) + 1?
https://cdstools.flipswitch.com/asset/media/690005
consider the function,f(x) = log_4 (x). the function g(x) is the image of a transformation of f(x) by the mapping statement f(x) -> f(x) + 1. which graph correctly illustrates this transformation?
https://cdstools.flipswitch.com/asset/media/696482
a function, f(x), is transformed by the mapping statement f(x) --> -f(x). the following graphs show the preimage of the transformation, f(x), and the image, g(x). which graph illustrates this transformation?
https://cdstools.flipswitch.com/asset/media/696634
which logarithmic functions have an x-intercept at 2 and a vertical asymptote at x = 0?
https://cdstools.flipswitch.com/asset/media/698490 https://cdstools.flipswitch.com/asset/media/698491
which graph correctly shows the function f(x) = log_2 x + 3?
https://cdstools.flipswitch.com/asset/media/698495
if the logarithm function f(x) maps to g(x) by the mapping statement f(x) --> -1/2f(x), which graph correctly shows these two functions?
https://cdstools.flipswitch.com/asset/media/698502
match each graph with the type of function represented in the graph.
https://cdstools.flipswitch.com/asset/media/701966 : trigonometric function https://cdstools.flipswitch.com/asset/media/701968 : logarithmic function https://cdstools.flipswitch.com/asset/media/701970 : square root function https://cdstools.flipswitch.com/asset/media/701971 : rational function
use the graph of the logarithmic function f(x) to answer the questions. https://cdstools.flipswitch.com/asset/media/690451 what is the domain of the function?
i do not know.
use the graph of the logarithmic function f(x) to answer the question. https://cdstools.flipswitch.com/asset/media/690439 if f(x) increasing or decreasing?
increasing
match each logarithmic expression with the equivalent single logarithm.
log_2(3) + 2 log_2(6) : log_2(108) 2 log_2(6) - log_2(3) : log_2(12) log_2(3) - log_2(6) + log_2(4) : log_2(2)
a scientist wants to model the decay of a viral cell population after antibiotics are introduced to its host. the virus population begins at 1.2 million cells.
no, the scientist did not choose the best model for the situation. the problem says that the population shrinks by one-third every hour, and this is a description of exponential decay. this situation would be best described by an exponential model.
percy is in charge of a fund-raising event for a local charity. he knows that the donations the charity will receive depends on the number of people who attend the event. he creates a function, d(x), to model the total donations the event will earn, where x is the number of people who attend the event. in which situation is it appropriate for percy to vertically dilate the function d(x) by a factor of 2?
one of the donors offers to double the total donation amount at the end of the night.
the function p(x) = 2/x describes the probability that karl or his sister, kelly, will win a school raffle based on x, the number of other contestants participating. on the day of the drawing, 10 additional contestants enter. how should p(x) be transformed, if at all, to demonstrate this change?
p(x) does not need to be transformed; it is still an appropriate model.
use the following graph of the function f(x) to answer the question. https://cdstools.flipswitch.com/asset/media/690849 which of the following combinations could create the function f(x)? justify your response.
subtracting a linear function from an exponential function could result in the function f(x). the graph of f(x) resembles the graph of a positive exponential function, except that the asymptote of an exponential function is horizontal. subtracting a linear function could cause this change in the orientation of the asymptote.
which two function families were most likely combined to create f(x)? how can the correct response be justified? table: x f(x) -2 -4 -1 -7
the function has constant second differences, which indicates a quadratic function. there is no simple test to determine a constant function for this scenario, but it is a good candidate, because a constant function would not interfere with the second differences test. thus, f(x) is probably the combination of a quadratic function and a constant function.
which statement correctly compares the instantaneous rates of change of f(x) at x = 12 and x = 4?
the instantaneous rate of change of f(x) is greater at x = 12, because the tangent line at x = 12 is steeper than the tangent line at x = 4.
which option correctly defines amplitude?
the positive distance between the midline of a function and its extrema (minima or maxima).
a subset of values of the function g(x) is shown in the following table. what type of function is represented in the table?
the table represents a quadratic function because the second differences are all the same value.
lesson 12
transformations of graphs
which statement(s) correctly describe(s) how to find the slope of a line?
using two points on the graph of the line, find the ratio of the rise to the run between the two points. using the coordinates of two points on the line, find the difference of the y-coordinates and divide by the difference of the corresponding x-coordinates.
use the graph of logarithmic function f(x) to fill in the blank. https://cdstools.flipswitch.com/asset/media/689845 the line x = −1 is the _[blank]_ of the function.
vertical asymptote
the function f(x)log_4(x) intersects the line _[blank]_ when x = 4. which equation correctly fills in the blank in the previous sentence?
y = 1