top of page

Math 1 Module 3: Features of Functions

3.1 Ready, Set, Go!

Questions 1 - 2

This video shows you how to graph negative linear equations: https://www.youtube.com/watch?v=X9924cMM-OQ

After you've drawn the graph, pick three points on the line and write out their coordinates. If you need a refresher on how to read coordinates from a graph, this video is helpful: https://www.youtube.com/watch?v=vvv-tATwdTA

Questions 3 - 4

This video shows you how to graph exponential equations: https://www.youtube.com/watch?v=_NVqF3973ew

After you've drawn the graph, pick three points on the line and write out their coordinates. If you need a refresher on how to read coordinates from a graph, this video is helpful: https://www.youtube.com/watch?v=vvv-tATwdTA

Question 5

This video shows you how to graph linear equations: https://www.youtube.com/watch?v=kgD48XXVT1c

After you've drawn the graph, pick three points on the line and write out their coordinates. If you need a refresher on how to read coordinates from a graph, this video is helpful: https://www.youtube.com/watch?v=vvv-tATwdTA

Question 6

This video shows you how to graph exponential equations: https://www.youtube.com/watch?v=_NVqF3973ew

After you've drawn the graph, pick three points on the curve and write out their coordinates. If you need a refresher on how to read coordinates from a graph, this video is helpful: https://www.youtube.com/watch?v=vvv-tATwdTA

Question 7

This graph goes up and down repeatedly.

Question 8

This graph is continuous and increasing over time with some pauses along the way.

Question 9

The graph shows discrete decreasing trend and each period has a different length, representing random times when the action is taking place.

Question 10

The graph shows discrete increasing trend that dips once. Each period is the same length.

Question 11

This graph is continuous and decreasing over time with some pauses along the way.

Question 12

Use the attributes listed in the problem to describe the graphs. This video talks about the difference between linear and exponential functions: https://www.youtube.com/watch?v=TVX9pSPJwYA

Question 13

Use the attributes listed in the problem to describe the graphs. This video talks about the difference between discrete and continuous functions: https://www.youtube.com/watch?v=9HfTW3LEUZI

Question 14

This video explains how to solve exponential equations using log: https://www.youtube.com/watch?v=oVUJMVR1JpA

Questions 15 - 17

Simplify the equation by getting all x-terms on the left side and all numerical terms on the right side. Then follow the strategies in this video to solve the linear equations: https://www.youtube.com/watch?v=GmMX3-nTWbE

Question 18

Simplify the equation by getting all exponential terms on the left side and all numerical terms on the right side. Then follow the log strategies in this video to solve the exponential equations: https://www.youtube.com/watch?v=oVUJMVR1JpA

Question 19

Simplify the equation by getting all x-terms on the left side and all numerical terms on the right side. Then follow the strategies in this video to solve the linear equations: https://www.youtube.com/watch?v=GmMX3-nTWbE

Question 20

Simplify the equation by getting all exponential terms on the left side and all numerical terms on the right side. Then follow the log strategies in this video to solve the exponential equations: https://www.youtube.com/watch?v=oVUJMVR1JpA

Question 21

Simplify the equation by getting all x-terms on the left side and all numerical terms on the right side. Then follow the strategies in this video to solve the linear equations: https://www.youtube.com/watch?v=GmMX3-nTWbE

Question 20

Simplify the equation by getting all exponential terms on the left side and all numerical terms on the right side. Then follow the log strategies in this video to solve the exponential equations: https://www.youtube.com/watch?v=oVUJMVR1JpA

3.2 Ready, Set, Go!

Questions 1 - 6

This video shows you how to graph a system of linear equations and find the point of intersections between the two lines: https://www.youtube.com/watch?v=_EW9AUEUFb8

Questions 7 - 9

This video shows you how to describe the properties of a graph like where it is increasing, decreasing or constant and where the maximum and minimum points are: https://www.youtube.com/watch?v=fVGHrdY8Emo

This video shows you how to find the domain and range of a piecewise functions like this: https://www.youtube.com/watch?v=iIvJAUH-QaE

Questions 10 - 15

First you have to figure out if the table is linear or exponential. This video talks about the difference between linear and exponential functions and how you might recognize them: https://www.youtube.com/watch?v=TVX9pSPJwYA

This video shows you how to write explicit and recursive equations for linear data tables https://www.youtube.com/watch?v=dcSZXlue9IY

This video shows you how to write explicit and recurvise equations for exponential data tables: https://www.youtube.com/watch?v=T4YE1V838ho

3.3 Ready, Set, Go!

Question 1

Fill the tables for f(x) = 3x-5 and g(x) = x+1. Compare each row in both tables, are any of the rows the same? That's the point of intersections, where the two lines share a point!

Question 2

In this table you aren't given the x-values. To figure out what x-values make sense to put in the table, first solve the system of equations f(x) = y = x+2 and g(x) = y = 2x. Use your favorite method (e.g. elimination, substitution) to solve for the point (x, y) that the lines intersect and then fill out the table around that point.

Question 3

Fill the tables for f(x) = 3x-5 and g(x) = x+1. Compare each row in both tables, are any of the rows the same? That's the point of intersections, where the two lines share a point!

Question 4

In this table you aren't given the x-values. To figure out what x-values make sense to put in the table, first solve the system of equations f(x) = y = x+2 and g(x) = y = 2x. Use your favorite method (e.g. elimination, substitution) to solve for the point (x, y) that the lines intersect and then fill out the table around that point.

Question 5

This video talks about increasing, decreasing and constant intervals on graphs: https://www.youtube.com/watch?v=kKsWbhFvoy0

 

Do you think all linear functions are increasing? Can you think of an example of decreasing or constant lines on graphs?

Question 6

Arithmetic sequences are considered "linear" functions however one important property of sequences is that they are discrete - so they appear as points in a line instead of a solid line on a graph. If you want more info, check out this short video: https://www.youtube.com/watch?v=ke0LGvMFPrQ

This is a mathematical definition of arithmetic sequences: https://www.youtube.com/watch?v=_cooC3yG_p0

Question 7

This video discusses the domain, range and asymptote for exponential functions: https://www.youtube.com/watch?v=gVGlA4jdQso

Question 8

Because sequences are discrete functions, their domain is definitely restricted to the integers. However, the domain of a sequence doesn't need to include all integers, just the integers that appear in the sequence!

Question 9

This video discusses the domain, range and asymptote for exponential functions: https://www.youtube.com/watch?v=gVGlA4jdQso

Question 10

Arithmetic sequences are considered "linear" functions however one important property of sequences is that they are discrete - so they appear as points in a line instead of a solid line on a graph. Therefore, they don't have a domain and range containing all real numbers.

Question 11 - 14

This video explains how to find the domain and range of piece wise functions: https://www.youtube.com/watch?v=BxaYyS6lsQ4

3.4 Ready, Set, Go!

Question 1

This video talks about the differences between linear and exponential functions: https://www.youtube.com/watch?v=TVX9pSPJwYA

Questions 2 - 9

This video explains how to find the domain and range of piecewise functions: https://www.youtube.com/watch?v=BxaYyS6lsQ4

This video talks about increasing, decreasing and constant intervals on graphs: https://www.youtube.com/watch?v=kKsWbhFvoy0

Questions 10 - 18

First you have to figure out if the table is linear or exponential. This video talks about the difference between linear and exponential functions and how you might recognize them: https://www.youtube.com/watch?v=TVX9pSPJwYA

This video shows you how to write explicit and recursive equations for linear data tables https://www.youtube.com/watch?v=dcSZXlue9IY

This video shows you how to write explicit and recursive equations for exponential data tables: https://www.youtube.com/watch?v=T4YE1V838ho

3.5 Ready, Set, Go!

Question 1

For part (a) you should evaluate the function h(t)=2t-5 by plugging in t=-4. Similarly, for part (c) you should plug t=13 into h(t).

For part (b) set h(t)=2t-5 equal to 23 and solve for t. Do the same for part (d), setting h(t)=-33.

Question 2

This video shows you how to evaluate a function from a graph, needed for parts (a) and (c), and solve a function for x, needed for part (b): https://www.youtube.com/watch?v=_fO9gx1ncyg

 

This video shows you how to find a exponential equation from a graph: https://www.youtube.com/watch?v=12LDAqnsH_w

Question 3

This video shows you how to evaluate a function from a graph, needed for parts (a) and (c), and solve a function for x, needed for part (b): https://www.youtube.com/watch?v=_fO9gx1ncyg

This video shows you how to find a linear equation from a graph: https://www.youtube.com/watch?v=mmWf_oLTNSQ

Questions 4 - 5

You can do this problem in two ways. The quickest way is to add the lines together graphically like this: https://www.youtube.com/watch?v=GqIan938v-c

You could also find the equations of the two lines separately and then add the line equations together and plot the resulting equation.

Question 6

In part (a) they want you to give them the coordinate where the graphs cross each other because that's where f(x) = g(x).

For part (b), look at the graph to find the y-value of f(x) and g(x) when x = 4 and then add those two numbers together. Part (c) is the same idea.

Lastly, (d) g(x) > f(x) where the line of g(x) is above the line of f(x)

Question 6

For (a), r(x) > h(x) where the line of r(x) is above the line of h(x).

 

For part (b), look at the graph to find the y-value of r(x) and h(x) when x = 1 and then calculate r(1) - h(1). Part (c) is the same idea.


In part (d) you need to write the equations of the functions. h(x) is horizontal line, so it's equation is just a number (where it crosses the y axis). r(x) is an exponential function. This video walks through a similar example: https://www.youtube.com/watch?v=fe1Hsqyetzk

Last but not least, part (e)! Because h(x) is always equal to the same value, you can just translate the r(x) curve down by the value of h(x).

Questions 8 - 15

When you hear "discrete", you can think "in pieces". This video goes through many examples of discrete and continuous functions: https://www.youtube.com/watch?v=ikPkKfRi9dM

3.6 Ready, Set, Go!

Questions 1 - 4

This video shows you how to find the intersection of two linear equations: https://www.youtube.com/watch?v=zkNBSyHh7vw

Questions 5 - 8

This video shows you how to evaluate a function from a graph (needed for parts a. and b.) and solve a function for x (needed for parts c. and d.): https://www.youtube.com/watch?v=_fO9gx1ncyg

Question 9

In this question, the given function is d(t) = 4t where t is time (seconds) and d(t) is the distance (feet) that Fran can walk in time t. For part (a), evaluate the function like this: https://www.youtube.com/watch?v=pJjcqlGUPg4

For part (b) the distance d(t) is equal to 100 ft. You can use the function equation to find the time when the distance is 100 by solving 4t = 100 for t.

For part (c), when t = 16 seconds, the distance walked is d(16). Evaluate the function. For part (d) when 200 feet are walked you set the distance function d(t) equal to 200.

Question 10

In this question, the given function is p(t) = 8(2^t) where t is time (weeks) and p(t) is the population of rodents that grows in time t. For part (a), the population p(t) is equal to 128. You can use the function equation to find the number of weeks when the population is 128 by isolating the exponential term and solving 8(2^t) = 128 for t. If you need some help, check out the example starting at 3:22 in this video: https://youtu.be/5R5mKpLsfYg?t=202

For parts (b) and (c) evaluate the exponential function p(t) 8(2^t) just like this example: https://www.youtube.com/watch?v=GQIcZeKR9RI

For part (d) we want p(t) to be greater than 20,000. Write out the inequality and solve for t.

Finally, for part (e) we're told that t = 16 weeks. Evaluate the exponential function p(t) 8(2^t) just like this example to find out how many rodents there will be after 16 weeks: https://www.youtube.com/watch?v=GQIcZeKR9RI

Question 11

This video shows you how to describe the properties of a graph like where it is increasing, decreasing or constant and where the maximum and minimum points are: https://www.youtube.com/watch?v=fVGHrdY8Emo

This video shows you how to find the domain and range of a piecewise functions: https://www.youtube.com/watch?v=iIvJAUH-QaE

Questions 12  - 13

This video shows you how to describe the properties of a graph like where it is increasing, decreasing or constant and where the maximum and minimum points are: https://www.youtube.com/watch?v=fVGHrdY8Emo

3.7 Ready, Set, Go!

Questions 1 - 3

This video explains the difference between functions and relations and goes over some examples of finding the domain and range for sets of ordered pairs: https://www.youtube.com/watch?v=OxZ0JL4Bjzk

Questions 4 - 5

This video explains how to find the domain and range of piece wise functions like this: https://www.youtube.com/watch?v=BxaYyS6lsQ4

Question 6

This video shows you how to find the domain and range of a similar linear function: https://www.youtube.com/watch?v=9aPuaXUDMEo

Question 7

This video shows you how to find the domain and range of a similar exponential function: https://www.youtube.com/watch?v=-qSvs3brz6M

Question 8

h(x) is a discrete function that contains 4 points. You can specify the domain by listing out the set of x-values and the range by listing out the set of y-values, like this: https://www.youtube.com/watch?v=lJqCN-PDhpI

Question 9

The core definition of a function is that every x-value maps onto just one y-value. The quickest way to determine if a relation is a function is to draw out the relation and do the vertical line test: https://www.youtube.com/watch?v=96VqHrvZdXw

 

In this case, the x-value is time and the y-value is height. If you are on a ferris wheel you go around the circle which means you smoothly move up and down in height over time again and again.

Question 10
The core definition of a function is that every x-value maps onto just one y-value. The quickest way to determine if a relation is a function is to draw out the relation and do the vertical line test: https://www.youtube.com/watch?v=96VqHrvZdXw

The amount of daylight per day is constantly changing depending on where you are located and what time of year is it. This video visualizes that change https://www.youtube.com/watch?v=OoBW9zvpEpE (this isn't going to be tested by feel free to ask me questions about it if you're curious!)

Question 11

The core definition of a function is that every x-value maps onto just one y-value. The quickest way to determine if a relation is a function is to draw out the relation and do the vertical line test: https://www.youtube.com/watch?v=96VqHrvZdXw

When you buy a car, the value is the highest on the day you buy it. Overtime, the value of the car gets lower as you drive it and it starts to wear out.

Question 12

There is no mathematical relationship between a person's name and their phone number.

Question 13

It seems likely that the football player does better in the stadium they are most familiar with but that relationship is probabilistic and can not be written as a function.

Question 14
This video shows you how to describe the properties of a graph like where it is increasing, decreasing or constant and where the maximum and minimum points are: https://www.youtube.com/watch?v=fVGHrdY8Emo

Question 15
The function you are given is d(t)=78t. For part a., d(4) represents the distance traveled at t=4. Similarly, for part c., d(3.5) represents the distance traveled at t=3.5.

For part b., when d(t) is equal to a value is means the distance is that value.

For part d., if the distance traveled is 800 miles then d(t) = 800. To find the time it takes to travel 800 miles, you need to solve d(t) = 78t = 800 for t.

Question 16

(a) f(x) = g(x) where the their graphs cross each other. List out the two points as coordinates for your answer. If you need help with reading coordinates off graphs, check out this video: https://www.youtube.com/watch?v=vvv-tATwdTA

To complete parts b. and d. you need to evaluate f(x) and g(x) at specific values of x and then add them together. Follow this video if you need help evaluating functions when you're just given a graph: https://www.youtube.com/watch?v=_fO9gx1ncyg

For part (c) you need to specify the range of x values where the graph of g(x) is greater than ("on top of") f(x).

bottom of page