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Math 1 Module 2: Linear & Exponential Functions

Access the PDF of this module here

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2.1 Ready, Set, Go!

Questions 1 - 8

In arithmetic sequences, each term is a constant amount (called the common difference) greater than the term before it. For geometric sequences, each term is multiplied by a constant number to get the next term (called a common ratio). https://www.youtube.com/watch?v=s2IEyZLrAUQ

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Question 9

They want you to make up any sequence (a, b, c, d) that doesn't follow arithmetic or geometric rules. To make sure it's not arithmetic, choose values of a, b, c, d so that b-a ≠ c-b. To make sure it's also not geometric, choose values of a, b, c, d so that b/a ≠ c/b.

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Question 10

Discrete data is countable and doesn't have decimals or fractions. A good example of a discrete number is the number of people in a room - it can't be 1.25! Continuous data is measured with as much precision (and as many decimal places) as you want - for example, an object weighs 0.32421 lbs.

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Question 11 - 15

In this problem, you are given an expression and three "ordered pairs" or "coordinates" which look like: (x,y). However, you are only given the x value for each. For the first ordered pair, take the value x = -4 and find the value of y in the expression y = 2x − 1. Your answer is the y value for the first ordered pair. Do the same for the other two!

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This video talks about the difference between discrete and continuous: https://www.youtube.com/watch?v=ikPkKfRi9dM

Questions 16 - 18

You need to find the price of the specified item based on the information given, like this: https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-rates/v/finding-unit-prices

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Question 19

You need to calculate how many dozens were purchased. I found this helpful: https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-rates/v/multiple-rates-word-problem

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Question 20

This video helped me find average price for one pair of shoes https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-rates/v/usain-bolt-s-average-speed

 

Question 21

The average price for one pair of shoes (from question 20) is the price you have to pay!

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Questions 22 - 30

This video shows you how to solve for x: https://www.youtube.com/watch?v=HwOZEXDwD2o

2.2 Ready, Set, Go!

Question 1

To compare the rates, you need to get both situations into the same units. I found the number of inches stretch per pound of weight for both. E.g. The bungee stretched 6 inches per 3 pounds which is the same as 2 inches per pound. For the slinky, you need to convert feet to inches (1 ft = 12 in).

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This problem is about "unit rates". This video shows you how to find the unit rate in a similar word problem: https://www.youtube.com/watch?v=Zm0KaIw-35k

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Questions 2 - 4

To compare the rates, you need to get both situations into the same units.

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Questions 5 - 11

This video talks about the difference between discrete and continuous functions: https://www.youtube.com/watch?v=ikPkKfRi9dM

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They also want you to figure out if the situation described is an arithmetic or geometric relationship. You can review the difference between arithmetic and geometric sequences here: https://www.youtube.com/watch?v=s2IEyZLrAUQ.

Questions 11 - 16

You need to solve the linear equation to find x. This video shows you how to solve similar linear equations: https://www.youtube.com/watch?v=TxqBn43hijY

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Question 17 - 19

You can multiply by the reciprocal of the fraction to solve for x. This video will help answer your question: https://www.youtube.com/watch?v=6EJHam1nv9A

2.3 Ready, Set, Go!

Question 1

This video shows you how to find rate of change from a table: https://www.youtube.com/watch?v=6cqQzI3RrqQ

 

This video shows you how to find the rate of change (more commonly called slope) on a linear graph: https://www.youtube.com/watch?v=7t2SLKMDKRI

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Question 2

In this problem, the two graphs have the same scale and as you can see, they both cross the y-axis at y = 1. Look at another x-value on both graphs (x=1 maybe) and compare the values of y on each graph for that x. The graph that has a higher y with the same x will have the greater rate of change.

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Question 3

In situation (a), the rate is $25/week. This is a fixed rate and represents a linear function. In situation (b), the rate is "half of what is owed". That means the first week he pays back $25, but the second week he pays only $12.50 and the week after that $6.25. Which situation is paying the debt back at a faster rate?

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Question 4

The y-values in the table for part (a) increase by a fied amount each time. In part (b) I recommend you make a table of figure number (x) and number of tiles (y). With two tables you can compare the rate easily.

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Question 5

I compared the rate of change by making a table of x and y values for (a) and another one for (b). I just put x - 1, 2, 3, 4, 5 for both tables and then plotted the curves to see which one was steeper.

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This video shows you how to actually calculate the rate of change: https://www.youtube.com/watch?v=VhGgZja2ghQ

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Question 6

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

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Question 7
If Joan just earned $30,000 a year, it would be a linear function because each year she gets the same amount BUT she also gets a commission which is a percentage of the sales she makes. We have no idea how many sales she makes and it could go up or down from one year to the next.

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Question 8

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

Question 9
Let's walk through a few steps together. On day 1, you get 1 gift (1 partridge). On day 2, you get 3 gifts (2 turtle doves and a partridge!) Day 3: 6 gifts... I put this data into a table and used this strategy to figure out whether it was linear or exponential: https://www.youtube.com/watch?v=TVX9pSPJwYA

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Question 10
This video talks about the difference between linear and exponential functions: https://www.youtube.com/watch?v=TVX9pSPJwYA

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Question 11

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

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Did you notice that the right column values are the squares of the left column values? E.g. (3 in)² = 9 in², (4 in)² = 16 in²

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Questions 12 - 16

You can fill the missing data in this table by finding the geometric mean of the values in the table, like this: https://www.youtube.com/watch?v=K1iOmYRA9ng

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Questions 17 - 19

With at least two (x,y) points we can find the slope of the the linear function that runs between them.This video shows you how: https://www.youtube.com/watch?v=wasEKQkA7MA

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Question 20

With at least two (x,y) points we can find the slope of the the linear function that runs between them.This video shows you how when you have two points: https://www.youtube.com/watch?v=wvzBH46D6ho

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Question 21

With at least two (x,y) points we can find the slope of the the linear function that runs between them.This video shows you how when you have a graph: https://www.youtube.com/watch?v=c-iK1SCCINc

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Question 22

With at least two (x,y) points we can find the slope of the the linear function that runs between them.This video shows you how when you have two points: https://www.youtube.com/watch?v=wvzBH46D6ho

2.4 Ready, Set, Go!

Questions 1 - 3

You can fill the missing data in the "Arithmetic" row of the table like this: https://www.youtube.com/watch?v=g7Tl3clSDYw

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You can fill the missing data in the "Geometric" row of the table by finding the geometric mean, like this: https://www.youtube.com/watch?v=K1iOmYRA9ng

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Question 4

Real numbers are continuous but integers and natural numbers can only be whole numbers.

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Question 5

This video discusses "Natural numbers" (which have the symbol N) https://www.youtube.com/watch?v=VRe7LLgkL3o

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Question 6

I started by eliminating any functions that weren't exponential and there were two options left. Recursive functions aren't continuous because they only take in x=1, 2, 3.. (integers), which left me with one function which is continuous and exponential.

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Question 7

For this one, I got rid of the recursive functions (because they aren't continuous) and then decided which of the two remaining functions was linear.

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Question 8

This video describes R, Q, Z and N (incase you forgot!) https://www.youtube.com/watch?v=vbPUS-0Wbv4

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Question 9

Sequences don't have to be integers (whole numbers) but they're always a list of individual data points.

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Question 10

This video is all about linear functions: https://www.youtube.com/watch?v=MXV65i9g1Xg

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Linear functions are related to arithmetic sequences: https://youtu.be/C5titprQAc4

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Question 11

This video is all about exponential functions: https://www.youtube.com/watch?v=6WMZ7J0wwMI

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Exponential functions are related to geometric sequences: https://youtu.be/yZ-GufE_uyA

Questions 12 - 15

The data in this table is either linear or exponential. This video shows you what linear tables look like and how to write equations for them: https://www.youtube.com/watch?v=dcSZXlue9IY

 

This video shows you what exponential tables look like and how to write their equations: https://www.youtube.com/watch?v=T4YE1V838ho

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Question 16

Based on the shape of the graph, we can tell it's a linear function. This video shows you how to find the equation for a similar graph: https://www.youtube.com/watch?v=u9YZxBh1AxQ

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Question 17

Based on the shape of the graph, we can tell it's an exponential function. This video shows you how to find the equation for a similar graph: https://www.youtube.com/watch?v=ueCcMc1FUsw

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Question 18

Based on the shape of the graph, we can tell it's a linear function. This video shows you how to find the the equation for a similar graph: https://www.youtube.com/watch?v=u9YZxBh1AxQ

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Question 19

Based on the shape of the graph, we can tell it's an exponential function. This video shows you how to find the equation for a similar graph: https://www.youtube.com/watch?v=ueCcMc1FUsw

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Question 20

Based on the shape of the graph, we can tell it's a linear function. This video shows you how to find the equation for a similar graph: https://www.youtube.com/watch?v=u9YZxBh1AxQ

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Question 21

Based on the shape of the graph, we can tell it's an exponential function. This video shows you how to find the equation of a similar graph: https://www.youtube.com/watch?v=12LDAqnsH_w

2.5 Ready, Set, Go!

Questions 1 - 3

Just just need to put the values of m and b that they give you into the slope intercept form of the linear equation: y = mx + b

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Questions 4 - 9

This video shows you how to find the slope-point form of a linear equation for a similar example: https://www.youtube.com/watch?v=0bqr9Fo3Qi8

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Question 10

In this problem, the y-intercept is c = 5. When they say "3 units smaller at each stage", it means the function is linear. Put the intercept and slope into the form y = mx + c and then sketch it like this: https://www.youtube.com/watch?v=kgD48XXVT1c

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Question 11

In this problem, the y-intercept is 16. When they say "1/4 smaller at each stage", it means the function decays exponentially. This video goes over exponential decay functions: https://www.youtube.com/watch?v=AXAMVxaxjDg

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Question 12

In this problem, the y-intercept is 1. When they say "10 times as big at each stage", it means the function grows exponentially. This video goes over exponential functions: https://www.youtube.com/watch?v=6WMZ7J0wwMI

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Question 13

In this problem, the y-intercept is c = -8. When they say "2 units bigger at each stage", it means the function is linear and has a slope of m = 2. Put the intercept and slope into the form y = mx + c and then sketch it like this: https://www.youtube.com/watch?v=kgD48XXVT1c

Questions 14 - 17

Simplify the right side of the equation so it matches the left side. This video shows you how to simplify linear equations: https://www.youtube.com/watch?v=n3FbQGGKVfc

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Question 18

You should distribute the parenthesis and simplify the right side of the equation so it matches the left side. This video shows you how to simplify linear equations: https://www.youtube.com/watch?v=n3FbQGGKVfc

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Question 20

(x-13) is an expression that can't be simplifed anymore but you can "unsimplify" it by adding/subtracting/multiplying/dividing different terms (just make sure that the final expression simplifies back to (x-13) https://www.youtube.com/watch?v=n3FbQGGKVfc

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As an example, if we want to make an equivalent expression with two sets of parentheses and a minus sign for 10, I could do (5*4) - (5 + 5) because 20 - 10 = 10!

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2.6 Ready, Set, Go!

Question 1

They want you to identify which function is linear and which is exponential. This video talks about the difference between linear and exponential functions: https://www.youtube.com/watch?v=TVX9pSPJwYA

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Question 2

We've learned about types of sequences - arithmetic https://youtu.be/C5titprQAc4 and geometric https://youtu.be/yZ-GufE_uyA

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One of the functions represents an arithmetic sequence and the other represents a geometric sequence.

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Question 3

Make a table with x = -3, -2, -1, 0, 1, 2, 3 and then make a row for each of the equations. Plug in all the x-values for each equation to fill the table.

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This video shows you a table of linear and exponential data if you want a similar example to look at: https://www.youtube.com/watch?v=721RrH6auoU

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Questions 4 - 5

This video about linear and exponential functions discusses the rate of change for both: https://www.youtube.com/watch?v=TVX9pSPJwYA

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Question 6

This example shows you how to find the y-intercept for an exponential function https://www.youtube.com/watch?v=e2kIJv7sXnc and to find the y-intercept of a linear function, you just need to set x = 0 and evaluate for y.

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Questions 7 - 8

"Intercepts" are the points where a graph crosses the x- and y- axes. To find the x- intercept, set y = 0 and solve the equation for x. To find the y-intercept, set x = 0 and evaluate the function.

 

This video shows you how to find intercepts for exponential equations (which is the same as geometric functions): https://www.youtube.com/watch?v=yYaT18oz8vg

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Notice that sometimes there ISN'T an x-intercept for these functions

Question 9

Let x be the amount of money Jasmine deposits and let y be the amount of money in Jasmine's bank account. Then follow the process in this video to write the equation: https://www.youtube.com/watch?v=W7H-VcaSSu8

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Question 10
This is a table of exponential data. To find out how many rectangles you get with 5 folds and 14 folds, I recommend you start by writing the explicit equation for the data like this: https://www.youtube.com/watch?v=T4YE1V838ho. When you have the equation, you can plug in x = 5 and x = 14.

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Questions 11 - 12
This situation can be modeled with a linear equation where y is the amount of weight lost and x is the number of weeks. This video shows you a similar example where they write an equation for linear word problem: https://www.youtube.com/watch?v=W7H-VcaSSu8

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Questions 13 - 14

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

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Questions 15 - 20

This video shows you how to solve algebra equations like this one: https://www.youtube.com/watch?v=l3XzepN03KQ

2.7H Ready, Set, Go!

Question 1

To find the scale of the y-axis, I noticed that the y2 equation is linear and when x = 0, the y-intercept is 4. You can see on the graph that the line crosses the y-axis at the 8th tick mark, so the y-axis must have a scale of 2. That is, every two tick marks represents 1 unit.

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Set Y2 = 0 and solve for x to find the x-intercept of the linear equation. With that intercept, you can figure out the scale of the horizontal axis.

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If you have a graphing calculator, you can set y1 = y2 and solve for x. The value of x you get will be the points where the lines intersect each other. You can use those intersections to figure out the scale of the graph

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Question 2

To find the scale of the y-axis, I noticed that the y2 equation is linear and when x = 0, the y-intercept is 1. You can see on the graph that the line crosses the y-axis at the first tick mark, so the y-axis must have a scale of 1.

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Set Y2 = 0 and solve for x to find the x-intercept of the linear equation. With that intercept, you can figure out the scale of the horizontal axis.

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Question 3

If you're feeling stressed about this problem, you aren't alone - it's really confusing!!!! In Y1, when x = 0, Y1 = 150(10⁰) = 150. That means the y-axis starts from 150 and goes up. Therefore Y min is 150.

 

I found that the two curves cross each other at x = 0.92 and again at x = 1.81.The first dot is halfway across the graph and is just after the first intersection, so it must be x = 1. The second dot is just after the second intersection and must be x = 2.

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To find X min, X max and Y max: we have the y-intercept on the screen, and that happens when x = 0, which means the left side of the box must start at x = 0, which is X min, and end at x = 2, which is X max. You can plug x = 2 into Y1 to find the y-value of the curve in the top right corner of the graph, which will tell you Y max.

Questions 4 - 6

This video shows you how to find rate of change from a table: https://www.youtube.com/watch?v=6cqQzI3RrqQ

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Question 7

Sketch the two curves on the same graph. This video talks about the difference between linear and exponential functions: https://www.youtube.com/watch?v=TVX9pSPJwYA

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Question 8

You can write anything you want here to describe how you thought about problem 7!

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Question 9

This video helps to introduce the secant line (don't worry too much about the details of the problem): https://www.youtube.com/watch?v=wLtBPvMv6kw.

 

If you have a positive parabola like in this video and you draw a secant line that goes horizontally between two points, those points would have the same y-value and the y-value would decrease between those points.

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Questions 10 - 13

This video shows you how to find the slope between two points: https://www.youtube.com/watch?v=wvzBH46D6ho

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