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Here are a few recommended readings before getting started with this lesson.
Vincenzo and Magdalena are two friends eager to explore the world of gaming and programming. They are developing their own mobile game by coding and testing together. As part of their development process, they are monitoring the temperature of their computers. The diagram shows how the temperature of each changes over time.
A linear function is a function with a constant rate of change. A linear function can be represented by a linear equation in two variables. Graphically, a linear function is a nonvertical line.
Using that line, the rate of change can be determined by finding the horizontal change Δx and the vertical change Δy between any two points on the line. Any function whose graph is not a straight line cannot be linear.What is the condition for being a function? What happens if the x-coordinates of M and N are the same.
Analyze the given graph and determine whether it represents a function. If it is a function then determine if it is a linear function or a nonlinear function.
Numerical data, also called quantitative data, is measurable data expressed using numbers. Some examples of numerical data are age, height, and weight of a person. Consider the following personal information.
Person | Age | Weight | Shoe Size | Number of Siblings | Eye Color |
---|---|---|---|---|---|
Dejen | 15 | 43.4 | 6 | 0 | Blue |
Madge | 16 | 56.7 | 7 | 3 | Green |
Yulia | 20 | 64.8 | 7.5 | 1 | Brown |
A person's age, weight, shoe size, and number of siblings are examples of numerical data. In contrast, a person's eye color cannot be expressed in numbers and is an example of categorical data. Additionally, numerical data can be either discrete data that can only take specific values or continuous data that can take any value within an interval.
Data | Numerical? | Type |
---|---|---|
Age | Yes ✓ | Continous |
Weight | Yes ✓ | Continous |
Shoe Size | Yes ✓ | Discrete |
Number of Siblings | Yes ✓ | Discrete |
Eye Color | No × | Non-applicable |
A continuous quantity can take any value in a certain interval. A discrete quantity can only take distinct values, so there is a gap between possible values.
To determine if a variable is continuous or discrete, check if there is a gap between possible values. Analyze each given variable separately to reach an answer.
The number of levels is typically counted in whole numbers. It cannot take a real number value, so it is discrete.
The time it takes for a player to complete a level can be any real number. Time can be measured with high precision, including fractions of seconds. Therefore, T is continuous.
A player's score can typically take whole numbers. Scores are limited to whole numbers, so it is discrete.
The speed at which the character moves can be measured on a continuous scale. That is, it can take any real value, including fractions. Therefore, V is continuous.
In conclusion, discrete variables take on distinct values, often whole numbers. Continuous variables, on the other hand, can take any value within a given interval.
Variable | Can it take all values in a certain interval? | Type |
---|---|---|
L | No | Discrete |
T | Yes | Continuous |
S | No | Discrete |
V | Yes | Continuous |
A function is said to have a continuous domain when the independent variable of the function is a continuous quantity. The graph of such a function is typically a curve or a line. Consider the graph of the function, which shows the relationship between a person's age and weight.
The domain of this function consist of all real numbers from 0 to 10, indicating that the function has a continuous domain.A function is said to have a discrete domain when the independent variable of the function is a discrete quantity. A function of this type can be identified by its graph, which is composed of any number of unconnected points. Consider the function that models the costs of concert tickets, specifically for 0, 1, 2, 3, and 4 tickets.
The domain of this function is the set of integers from 0 to 4, indicating that the function has a discrete domain.Vincenzo and Magdalena are continuing the development of their own mobile game. In their game, players start with 100 virtual coins and spend 5 coins per day on in-game activities.
Days, x | Number of coins, y |
---|---|
2 | 90 |
4 | 80 |
6 | 70 |
8 | 60 |
10 | 50 |
The function have a discrete domain.
Days x | -5x+100 | Number of Coins y |
---|---|---|
2 | ||
4 | ||
6 | ||
8 | ||
10 |
How about finding the remaining number of coins for a player after 2, 4, 6, 8, and 10 days from starting the game? Try substituting the values of days into the formula and evaluate.
Days x | -5x+100 | Number of Coins y |
---|---|---|
2 | -5(2)+100 | 90 |
4 | -5(4)+100 | 80 |
6 | -5(6)+100 | 70 |
8 | -5(8)+100 | 60 |
10 | -5(10)+100 | 50 |
Great work so far!
Days x | Number of Coins y |
---|---|
2 | 90 |
4 | 80 |
6 | 70 |
8 | 60 |
10 | 50 |
Plot the ordered pairs (x,y) on a coordinate plane where the x-axis represents the number of days and the y-axis represents the remaining number of coins.
Since the function has a discrete domain, the points are not connected with a straight line. There are missing points in this graph. When other points are included, the graph of the function will be as follows.
As part of their development process, Vincenzo and Magdalena monitor the battery power consumption of a test laptop.
The table shows the percentage, y in decimal form, of battery power remaining x hours after the laptop is turned on.
Hours, x | 0 | 2 | 3 |
---|---|---|---|
Power Remaining, y | 1.0 | 0.4 | 0.1 |
Graph:
Hours, x | 0 | 2 | 3 |
---|---|---|---|
Power Remaining, y | 1.0 | 0.4 | 0.1 |
y=0
Subtract term
LHS/(-0.3)=RHS/(-0.3)
ba=b⋅(-10)a⋅(-10)
Add fractions
Rewrite 3+31 as 331
Rearrange equation
Rewrite 331 h as 3 h+31 h
1 h=60 min
b1⋅a=ba
Calculate quotient
Rewrite 3 h+20 min as 3 h and 20 min
y=0.25
Subtract term
LHS/(-0.3)=RHS/(-0.3)
Calculate quotient
LHS+3=RHS+3
Rearrange equation
Finally, the challenge presented at the beginning of the lesson will be solved. In the challenge, Vincenzo and Magdalena are monitoring the temperatures of their computers which tend to increase while running applications.
It is helpful to answer the questions by recalling the concepts and rules presented throughout the lesson.
Vincenzo's Computer | Magdalena's Computer | |
---|---|---|
y-intercept | 32 | 24 |
The initial temperature of the computer is 32∘C. | The initial temperature of the computer is 24∘C. | |
Slope | 6 | 8 |
Its temperature increases by 6∘C per hour. | Its temperature increases by 8∘C per hour. |