Module 5: Functions

Under which condition is the confirmed absolute maximum NOT also a local (or relative) maximum? it increases infinitely it is located at an endpoint the function is constant the function has only one point

it is located at an endpoint

If the average rate of change of a function is always the same and is not undefined, then the function is... constant. shifting. increasing. linear.

linear.

A positive horizontal stretch is... taller and thinner than the original function. shorter and wider than the original function. taller and wider than the original function. shorter and thinner than the original function.

shorter and wider than the original function.

A positive vertical compression is ... taller and thinner than the original function. shorter and wider than the original function. taller and wider than the original function. shorter and thinner than the original function.

shorter and wider than the original function.

The phrase, "for any given number of hours worked, Jacques completes two fewer assignments than Jamal" represents a... vertical shift up. vertical shift down. horizontal shift left. horizontal shift right.

vertical shift down.

The following graph represents which transformation? The black function is the original function and the green function is the transformed function. horizontal shift, g(x) = f(x+5) horizontal shift, g(x) = f(x-5) vertical shift, g(x) = f(x) + 5 vertical shift, g(x) = f(x) - 5

vertical shift, g(x) = f(x) - 5

The phrase, "at each point in the journey, it is expected that the new traveler will have earned four times as much as the original traveler" represents a... vertical stretch. vertical compression. horizontal stretch. horizontal compression.

vertical stretch.

Which graph best represents the following equation: y=2x+1 ?

A

For this section, you will provide feedback to a student. A student was asked to find the slope and y-intercept of the following equation: y = 2x + 1. Graph the line. Below, you will see the response they provided for the question. Using the rubrics below, provide the appropriate feedback for the student. Find the slope and y-intercept of the following equation: y = 2x + 1. a. slope = 2 y-intercept = (0,1) The answer key says the following for the equation: Find the slope and y-intercept of the following equation: y = 2x + 1. Graph the line. Slope intercept form is y = mx + b where m = slope and the y-intercept is (0,b). Therefore, the slope is 2, and the y-intercept is (0,1). What score should this student receive in Mathematical Concept? The answers to the question should demonstrate the correct problem-solving techniques and provide the correct answers. All answers are correct, and effective and efficient strategies were used to arrive at the correct answer. All answers are correct, but effective and/or efficient strategies were not always used to arrive at the correct answer. Most answers are correct, or the student did not use the most efficient and effective strategies to arriv

All answers are correct, and effective and efficient strategies were used to arrive at the correct answer.

In general, f°g and g°f are the same functions. True False

False

To add two functions f(x) and g(x) together, the operation is written as (f°g)(x) = f(g(x)). True False

False

Under which of the following cases would a function have no maximum value? point or interval of the function is undefined the highest value can be found on multiple points of the function at least one end point increases infinitely the function is a horizontal line that does not increase or decrease

at least one end point increases infinitely

One thing that distinguishes relative extrema from absolute extrema is that a relative extrema is... compared to other points within a function. compared to other points within an interval. a less significant tool for evaluating functions. relatively accurate, but not absolutely accurate.

compared to other points within an interval.

Writing a complicated function as a composition of two simpler functions is known as... evaluation. decomposition. finding the domain. an inverse function.

decomposition.

Given the x values {4,3,5} and the equation f(x)= 3x + 2, solve for f(x). f(4)=10, f(3)=11, f(5)=19 f(4)=14, f(3)=11, f(5)=17 f(4)=6, f(3)=8, f(5)=10 f(4)=17, f(3)=14, f(5)=11

f(4)=14, f(3)=11, f(5)=17

Use the table provided to evaluate f(g(1)). f(g(1))=6 f(g(1))=8 f(g(1))=3 f(g(1))=1

f(g(1))=3

Use the table provided to evaluate f(g(3)). f(g(3))=6 f(g(3))=8 f(g(3))=3 f(g(3))=1

f(g(3))=8

Use the graph to evaluate f(g(5)). f(g(5))=0 f(g(5))=2 f(g(5))=5 f(g(5))=6

f(g(5))=6

Use the graph to evaluate g(f(2)). g(f(2))=1 g(f(2))=2 g(f(2))=3 g(f(2))=4

g(f(2))=3

Use the table provided to evaluate g(f(3)). g(f(3))=3 g(f(3))=5 g(f(3))=2 g(f(3))=7

g(f(3))=2

Use the table provided to evaluate g(f(4)). g(f(4))=3 g(f(4))=5 g(f(4))=2 g(f(4))=7

g(f(4))=3

Use the table to evaluate g(f(4)). g(f(4))=3 g(f(4))=5 g(f(4))=2 g(f(4))=7

g(f(4))=3

Shift the function f(x) = x^2 four units to the right. g(x) = x^2+4 g(x) = x^2-4 g(x) = (x-4)^2 g(x) = (x+4)^2

g(x) = (x-4)^2

Which of the following equations represents the transformation from f(x) =x2 to g(x) as pictured in the graph below? The black function is the original function and the green function is the transformed function. g(x) = -1/3x^2 g(x) = -3x^2 g(x) = -(1/3x)^2 g(x) = -(3x)^2

g(x) = -1/3x^2

Reflect the function f(x) = x^2 across the x-axis. g(x) = x^2-x g(x) = -x^2 g(x) = -x^2-x g(x) = x^2

g(x) = -x^2

Which of the following equations represents the transformation from to g(x) as pictured in the graph below? The black function is the original function and the green function is the transformed function. g(x) = 1/4 square root x g(x) = 4 square root x g(x) = square root 1/4x g(x) = square root 4x

g(x) = square root 4x

Shift the function f(x) = x^2 down 3. g(x) = x^2 + 3 g(x) = x^2 - 3 g(x) = 3x^2 g(x) = -3x^2

g(x) = x^2 - 3

Shift the function f(x) = x^2 down 3, right 4, and then reflect the function across the x-axis. g(x) =-(x-4)^2-3 g(x) =x^2-3x g(x) =(x-3)^2+4 g(x) =-(x-3)^2-4

g(x) =-(x-4)^2-3

Which of the following equations represents the transformation from f(x) =x^2 to g(x) as pictured in the graph below? The black function is the original function and the green function is the transformed function. g(x) =(x+3)^2 g(x) =(x-3)^2 g(x) =x^2+ 3 g(x) =x^2- 3

g(x) =x^2+ 3

Which transformation does the following graph represent? The red function is the original function and the green function is the transformed function. horizontal stretch by a factor of horizontal compression by a factor of horizontal stretch by a factor of 2 horizontal compression by a factor of 2

horizontal compression by a factor of 2

The phrase, "it takes Jacques two more hours than Jamal to finish the same number of assignments the Jamal has completed" represents a... vertical shift up. vertical shift down. horizontal shift left. horizontal shift right.

horizontal shift left.

The following graph represents which transformation? The black function is the original function and the green function is the transformed function. horizontal shift, g(x) = f(x+3) horizontal shift, g(x) = f(x-3) vertical shift, g(x) = f(x) + 3 vertical shift, g(x) = f(x) - 3

horizontal shift, g(x) = f(x-3)

The phrase, "we expect the new traveler to get to each point in the journey half as fast as the original traveler" represents a... vertical stretch. vertical compression. horizontal stretch. horizontal compression.

horizontal stretch.

If the point (8, -14) is reflected over both the x-axis and the y-axis, where will the new point (x', y') be located? (8, 14) (-8, 14) (8, -14) (-8,- 14)

(-8, 14)

Solve the following system of equations: y=-3x+5 and 2x+4y=10. (2,3) (1,2) (-2,-3) no real solution

(1,2)

Solve the system of equations using elimination: 4x+2y=14 5x+2y=16 (3,4) (2,3) (-2,4) (-2,-3)

(2,3)

If the point (3, 2) is reflected over the x-axis, where will the new point (x', y') be located? (3, 2) (-3, 2) (3, -2) (-3,- 2)

(3, -2)

If the point (-4, -7) is reflected over the y-axis, where will the new point (x', y') be located? (4, 7) (-4, 7) (4, -7) (-4,- 7)

(4, -7)

Given the following data, find the rate of change between 2007 and 2009 for gasoline. -0.22 per year 0.22 per year -0.43 per year 0.43 per year

-0.22 per year

Given the equation y = - 2/3x + 6, what is the slope of the line? 2/3 -2/3 6 -6

-2/3

Find for f/gx the following equations: f(x)= -2x^2 g(x)=3x^2+5 x^2-5x 3x^2-4x 4x^2/3x^2 + 5x -2x^2/3x^2 + 5

-2x^2/3x^2 + 5

Find (f+g)(x) for the following equations: f(x)= -2x^2 g(x)= -3x^2+5x -5x^2+x+5 -5x^2-5x 5x^2+5 -5x^2+5x

-5x^2+5x

Use a graphing calculator to visualize the following function. If you don't have a graphing calculator, you can use online tools such as Mathway, Desmos, or Symolab. Identify the function's minimum point in the interval from x = 0 to x =8. 0 0.105 0.488 1.422

0.105

Use a graphing calculator to visualize the following function. If you don't have a graphing calculator, you can use online tools such as Mathway, Desmos, or Symolab. Identify the function's average rate of change in the interval from x=2 to x=7. 0.1698 0.2617 0.5583 0.9199

0.1698

Given the following data, find the rate of change between 2005 and 2011 for gasoline. -0.21 per year 0.21 per year -1.27 per year 1.27 per year

0.21 per year

Use the set below to answer questions: {(1,4),(2,8),(3,12),(4,16),(5,20)} Write the domain as a list of numbers separated by commas and no spaces 1,4,2,8,4 4,8,12,16,20 1,2,3,4,5 16,5,12,20

1,2,3,4,5

For functions f(x) = 7y^2-2y+9 and g(x) =4y^2-8y-7, find (f+g)(x). 11y^2-10y+2 3y^2+6y+16 3y^2-10y+16 11y^2+6y+2

11y^2-10y+2

The number of unwatched shows in Sylvia's DVR is 85. This number grows by 20 unwatched shows per week. The function N(t) = 85 + 20t represents the relation between the number of unwatched shows, N, and the time, t, measured in weeks. Find the value for N(4). 105 165 115 100

165

Use a graphing calculator to visualize the following function. If you don't have a graphing calculator, you can use online tools such as Mathway, Desmos, or Symolab. Which of the following is a relative maximum in the interval x = -1 to x = 8? 1.235 2.438 2.905 8

2.438

Given f(t)=t^2 - t and h(x)=3x + 2, evaluate h(f(-2)). 4 8 16 20

20

For functions f(x) = 6x^3 + 30x^2 and g(x) =3x2, find (f/g) (x). 2x + 10 3x + 13 5x + 3(x-1) 2x - 10

2x + 10

Use the set below to answer questions: {(1,4),(2,8),(3,12),(4,16),(5,20)} Write the range as a list of numbers separated by commas and no spaces 1,4,2,8,4 4,8,12,16,20 1,2,3,4,5 16,5,12,20

4,8,12,16,20

Given the equation y = - 2/3 + 6, what is the y-intercept of the line? 2/3 -2/3 6 -6

6

For functions f(x) = 6x^2 and g(x) =x + 5, find (g∘f)(x). 180 6x+5x^2 6x^2+5 6x^2+60x+150

6x^2+5

For functions f(x) = 6x^2 and g(x) =x + 5, find (f∘g)(x). 180 6x+5x^2 6x^2+5 6x^2+60x+150

6x^2+60x+150

Find (f×g)(x) for the following equations: f(x)= -2x^2 g(x)= -3x^2+5x 6x^4-10x^3-15x^2-3x 6x^2-10x 63-5x^3 6x^4-10x^3

6x^4-10x^3

Evaluate f(2) for the function f(x) = 5x - 3. 1 3 2 7

7

For functions f(x) = 9w^2 - 7w + 5 and g(x) =2w^2 - 4, find (f - g)(x). 11w^2 - 7w + 1 7w^2 - 5w - 9 11w^2 - 7w - 1 7w^2 - 7w + 9

7w^2 - 7w + 9

Given f(t)=t^2 - t and h(x)=3x + 2, evaluate h(f(2)). 4 8 16 20

8

For functions f(x) = 4y+3 and g(x) =2y-5, find (f • g)(x). 2y^2 - 7y - 8 8y^2 - 14y - 15 9y^2 + 14y + 4 8y^2 + 14y + 15

8y^2 - 14y - 15

Find the equation that is represented by this graph: y = -2/3x + 2 y = 1/2x - 2 y = 1/3x + 2 y = 2/3x - 2

y = 1/2x - 2

Find an equation of a line parallel to y = -2x − 3 that contains the point (2,3). Write the equation in slope-intercept form. y= -2x+7 y=2x-7 y= -4x+8 y=2x+5

y= -2x+7

Find the equation of a line that passes through (3,7) and (5,11). y=2x+1 y=2x+5 y=3x+4 y=3x-1

y=2x+1

Which of the following is correctly written in slope intercept form? slope=x+y+b y=mx+b y=x+mb y=xb+m

y=mx+b

Select the type of function depicted in the following graph. Linear Square Cube Square Root

Square

Determine the range of the following set of coordinates:{(1,0),(2,2),(3,5),(4,7),(5,12)} {1,2,3,4,5} {0,2,5,7,12} {(1,0),(2,2),(3,5),(4,7),(5,12)} {1,4,8,11,17}

{0,2,5,7,12}

Determine the domain of the following set of coordinates:{(1,0),(2,2),(3,5),(4,7),(5,12)} {1,2,3,4,5} {0,2,5,7,12} {(1,0),(2,2),(3,5),(4,7),(5,12)} {1,4,8,11,17}

{1,2,3,4,5}

Given the equation y = - 2/3 + 6, describe the slope of the line. It extends in a positive manner. It extends in a negative manner. It is a horizontal line. It is a vertical line.

It extends in a negative manner.

For this section, you will provide feedback to a student. A student was asked to find the slope and y-intercept of the following equation: y = 2x + 1. Graph the line. Below, you will see the response they provided for the question. Using the rubrics below, provide the appropriate feedback for the student. Find the slope and y-intercept of the following equation: y = 2x + 1. a. slope = 2 y-intercept = (0,1) The answer key says the following for the equation: Find the slope and y-intercept of the following equation: y = 2x + 1. Graph the line. Slope intercept form is y = mx + b where m = slope and the y-intercept is (0,b). Therefore, the slope is 2, and the y-intercept is (0,1). What score should this student receive in Diagram/Sketch/Graph - Clarity and Completion? Data should be clear, concise, and easy to understand. The scale of the graph should be regular and consistent with the equations presented. All of the following should be included: titles, labeled axes, a key or legend, and labels. The work submitted should not confuse the reviewer and should follow a logical order. All required elements are present and easy to understand. Almost all required elements are present and easy

Many required elements are missing, and the data is difficult to understand.

For this section, you will provide feedback to a student. A student was asked to find the slope and y-intercept of the following equation: y = 2x + 1. Graph the line. Below, you will see the response they provided for the question. Using the rubrics below, provide the appropriate feedback for the student. Find the slope and y-intercept of the following equation: y = 2x + 1. a. slope = 2 y-intercept = (0,1) The answer key says the following for the equation: Find the slope and y-intercept of the following equation: y = 2x + 1. Graph the line. Slope intercept form is y = mx + b where m = slope and the y-intercept is (0,b). Therefore, the slope is 2, and the y-intercept is (0,1). What score should this student receive in Diagram/Sketch/Graph - Accurate Information? The diagrams, sketches, and/or graphs should be accurate. All diagrams, sketches, or graphs are fully accurate. Most diagrams, sketches, or graphs are accurate, but a few errors may be present. One diagram, sketch, or graph contains multiple errors. More than one diagram, sketch, or graph contains multiple errors. All diagrams, sketches, or graphs contain multiple errors, or one or more answers are missing a diagram, sketch,

One diagram, sketch, or graph contains multiple errors.

Find the slope and y-intercept of the following equation: y=2x+1. The slope is 4, and the y-intercept is (2,1). The slope is -4, and the y-intercept is (2,1). The slope is 2, and the y-intercept is (0,1). The slope is -2, and the y-intercept is (0,1).

The slope is 2, and the y-intercept is (0,1).

One single average rate of change value can be calculated across an interval where the function both increases and decreases. True False

True

To do a composition, the output of the first function, g(x), becomes the input of the second function. True False

True

The number of unwatched shows in Sylvia's DVR is 85. This number grows by 20 unwatched shows per week. The function N(t) = 85 + 20t represents the relation between the number of unwatched shows, N, and the time, t, measured in weeks. What is the dependent variable? Sylvia Unwatched shows Time The DVR

Unwatched shows

If given the two functions, and f°g = g°f, then the composition of the functions is commutative. True False

True

Is the following graph a function? Yes No

Yes

Use the set below to answer questions: {(1,4),(2,8),(3,12),(4,16),(5,20)} Is the set a function? Yes No

Yes

A negative horizontal stretch or compression is equivalent to... a negative output. a positive output. a reflection over the x-axis. a reflection over the y-axis.

a reflection over the y-axis.

Which of the following best defines vertical stretch? - when the "x" (or input value) of an original function is multiplied by a number whose absolute value is greater than 1, which results in the new transformed version (g(x)) reaching each output value faster than the original function (f(x)). It is squeezing the graph toward the y-axis. - when the "x" (or input value) of an original function (f(x)) is multiplied by a number whose absolute value is less than 1, which results in the new transformed version (g(x)) reaching each output value later than the original function (f(x)). It is the stretching of the graph away from the y-axis. - when an original function f(x) is multiplied by a number whose absolute value is less than 1; when compared to the original function, the new transformed version will have a smaller output absolute value at every x (or at every input point). It is the squeezing of the graph toward the x-axis. - when an original function f(x) is multiplied by a number whose absolute value is greater than 1; when compared to the original function, the new transformed version will have a larger output absolute at every x (or at every input point). It is the stretching of th

when an original function f(x) is multiplied by a number whose absolute value is greater than 1; when compared to the original function, the new transformed version will have a larger output absolute at every x (or at every input point). It is the stretching of the graph away from the x-axis.

Which of the following best defines vertical compression? - when the "x" (or input value) of an original function is multiplied by a number whose absolute value is greater than 1, which results in the new transformed version (g(x)) reaching each output value faster than the original function (f(x)). It is squeezing the graph toward the y-axis. - when the "x" (or input value) of an original function (f(x)) is multiplied by a number whose absolute value is less than 1, which results in the new transformed version (g(x)) reaching each output value later than the original function (f(x)). It is the stretching of the graph away from the y-axis. - when an original function f(x) is multiplied by a number whose absolute value is less than 1; when compared to the original function, the new transformed version will have a smaller output absolute value at every x (or at every input point). It is the squeezing of the graph toward the x-axis. - when an original function f(x) is multiplied by a number whose absolute value is greater than 1; when compared to the original function, the new transformed version will have a larger output absolute at every x (or at every input point). It is the stretching o

when an original function f(x) is multiplied by a number whose absolute value is less than 1; when compared to the original function, the new transformed version will have a smaller output absolute value at every x (or at every input point). It is the squeezing of the graph toward the x-axis.

Which of the following best defines horizontal stretch? - when the "x" (or input value) of an original function is multiplied by a number whose absolute value is greater than 1, which results in the new transformed version (g(x)) reaching each output value faster than the original function (f(x)). It is squeezing the graph toward the y-axis. - when the "x" (or input value) of an original function (f(x)) is multiplied by a number whose absolute value is less than 1, which results in the new transformed version (g(x)) reaching each output value later than the original function (f(x)). It is the stretching of the graph away from the y-axis. - when an original function f(x) is multiplied by a number whose absolute value is less than 1; when compared to the original function, the new transformed version will have a smaller output absolute value at every x (or at every input point). It is the squeezing of the graph toward the x-axis. - when an original function f(x) is multiplied by a number whose absolute value is greater than 1; when compared to the original function, the new transformed version will have a larger output absolute at every x (or at every input point). It is the stretching of

when the "x" (or input value) of an original function (f(x)) is multiplied by a number whose absolute value is less than 1, which results in the new transformed version (g(x)) reaching each output value later than the original function (f(x)). It is the stretching of the graph away from the y-axis.

Which of the following best defines horizontal compression? - when the "x" (or input value) of an original function is multiplied by a number whose absolute value is greater than 1, which results in the new transformed version (g(x)) reaching each output value faster than the original function (f(x)). It is squeezing the graph toward the y-axis. - when the "x" (or input value) of an original function (f(x)) is multiplied by a number whose absolute value is less than 1, which results in the new transformed version (g(x)) reaching each output value later than the original function (f(x)). It is the stretching of the graph away from the y-axis. - when an original function f(x) is multiplied by a number whose absolute value is less than 1; when compared to the original function, the new transformed version will have a smaller output absolute value at every x (or at every input point). It is the squeezing of the graph toward the x-axis. - when an original function f(x) is multiplied by a number whose absolute value is greater than 1; when compared to the original function, the new transformed version will have a larger output absolute at every x (or at every input point). It is the stretching

when the "x" (or input value) of an original function is multiplied by a number whose absolute value is greater than 1, which results in the new transformed version (g(x)) reaching each output value faster than the original function (f(x)). It is squeezing the graph toward the y-axis.

Find (f-g)(x) for the following equations: f(x)= -2x^2 g(x)= -3x^2+5x x^2+5x+5 x^2-5x -9x^2+2x -5x^2+5x

x^2-5x

Find an equation of a line perpendicular to y = -2x − 3 that contains the point (3,4). Write the equation in slope-intercept form. y = -1/2x + 4 y = 1/2x + 5/2 y = 2x + 1/2 y = -2x + 4

y = 1/2x + 5/2