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| 15 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
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. |
The y-intercept of the blue line is 32 and the y-intercept of the red line is 24. Note that the y-axis represents the temperature in Celsius degrees. Since the time is zero at those points, the y-intercepts represent the initial temperatures of the computers.
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. |
The blue line passes through the points (0,32) and (4,56), while the red line passes through the points (0,24) and (1,32). Substitute these points into the Slope Formula.
Blue Line | Red Line | |
---|---|---|
Points | (0,32) and (4,56) | (0,24) and (1,32) |
m=x2−x1y2−y1 | 4−056−32 | 1−032−24 |
Evaluate | 424=6 | 18=8 |
In this context, the slope of the functions indicate how quickly the temperature of a computer rises within an hour.
Vincenzo's Computer | Magdalena's Computer | |
---|---|---|
Slope | 6 | 8 |
Its temperature increases by 6∘C per hour. | Its temperature increases by 8∘C per hour. |
Magdalena's computer temperature increases faster than Vincenzo's when both are running the applications. Note also that both functions have continuous domains as the inputs can take any value.
Blue Line | Red Line | |
---|---|---|
y-intercept | 32 | 24 |
Slope | 6 | 8 |
Blue Line | Red Line | |
---|---|---|
y-intercept | 32 | 24 |
Slope | 6 | 8 |
Equation | y=6x+32 | y=8x+24 |
The blue line corresponds to Vincenzo's computer temperature, whereas the red line corresponds to Magdalena's computer temperature.
Consider the graph.
Which of the following is not the equation of the graph?We are asked to write a function in slope-intercept form, which follows a specific format. y= mx+ b In this form, m is the slope and b is the y-intercept. Let's use the given points to calculate the slope of the line. We will start by substituting the points into the Slope Formula.
A slope of - 3 means that for every 1 horizontal step in the positive direction, we take 3 vertical steps in the negative direction. Now that we know the slope, we can write a partial version of the equation. y= - 3 x+ b To complete the equation, we also need to determine the y-intercept, b. Since we know that the given points will satisfy the equation, we can substitute one of them into the partial equation to solve for b. Let's use ( - 1, 10).
A y-intercept of 7 means that the line crosses the y-axis at the point (0, 7). We can now complete the equation. y= - 3x+ 7
Let's write the equation of the line in point-slope form because two of the options are given in that form.
y-y_1= m(x+x_1)
In this form, m is the slope and (x_1,y_1) is a specific point on the line. We need to identify these values using the graph. We can identify three points on the graph of the line.
We see that the line passes through the points (-1.5,- 12), (0.5,0) and (2.5,12). Let's substitute two of the points into the Slope Formula to find m.
The slope of the line is 6. Now we can use each of the points and write the equation.
Equation of The Line | ||
---|---|---|
Point | Equation | Rewrite |
(- 1.5,- 12) | y-(- 12) = 6(x-(- 1.5) ) | y + 12 = 6(x+1.5) |
(0.5,0) | y- 0 = 6(x- 0.5 ) | y = 6x-3 |
(2.5,12) | y-12= 6 (x-2.5) | y -12 = 6(x-2.5) |
All these equations represent the same line. We see these equations in the options, except C. Therefore, the answer is C.
Determine if the equation represents a linear or a nonlinear function.
We can determine whether the given equation represents a linear or nonlinear function without graphing. We can do it by checking if the equation can be rewritten in any of the linear equation forms.
Forms of Linear Equations | ||
---|---|---|
Slope-Intercept Form | Point-Slope Form | Standard Form |
y = mx+b | y−y_1=m(x−x_1) | Ax+By=C |
Let's do it! We will start by distributing - 4.
The equation can be written in slope-intercept form. Therefore, its graph is a straight line. Slope-Intercept Form y = - 4x+(-2) The function has a constant rate of change, - 4. Therefore, the function is linear.
We cannot rewrite the given equation in any of the linear equation forms because x is the denominator of a fraction.
y = 8/x ⇔ y = 8 x^(- 1)
Therefore, this equation represents a nonlinear function.
To confirm it, we can create a table of values and calculate the rates of change between some points. Three points are enough to do so.
x | y = 8/x |
---|---|
1 | 8/1 = 8 |
2 | 8/2 = 4 |
4 | 8/4 = 2 |
Now, we calculate the rate of change between ( 1, 8) and ( 2, 4), as well as between ( 1, 8) and ( 4, 2).
Points | Rate of Change | Are they equal? |
---|---|---|
( 1, 8) and ( 2, 4) | 4- 8/2- 1 = - 4 | No, - 4 ≠ - 2 |
( 1, 8) and ( 4, 2) | 2- 8/4- 1 = - 2 |
The function does not have a constant rate of change, so we are sure that it is a nonlinear function.
Determine if the graph represents a linear or nonlinear function.
The graph of a linear function is portrayed as a single, straight line in a coordinate plane. It appears that the given graph is a composition of two straight lines, each part has a different rate of change.
The rate of change is 1 for the part of the graph to the left of the y-axis, whereas it becomes - 1 for the part on the opposite side. Since the graph does not have a constant rate of change, it is a nonlinear function.
The graph is a non-vertical line, so its a linear function. Let's check if the rate of change between any pair of its points is constant.
When we select any two points on the line, we see that the rate of change between those points is constant and 0.2. We can confirm that it is a linear function.
Recall that discrete variables are those that can only take distinct, separate values, typically integers or whole numbers. Continuous variables, on the other hand, can take any value within a given interval. With this in mind, let's think about the given variables.
In summary, discrete variables are counted in whole numbers, while continuous variables can take on any value within an interval, often including decimals.
What is the graph of the function y=3x−3?
To graph the function y=3x-3, we will first make a function table to obtain ordered pairs. After that, we plot the ordered pairs and connect them with a line. To make a function table, we will select some values of x which will be part of our domain. Since there is no restriction for the domain, the domain is assumed to contain all real numbers. Function: & y = 3x-3 Domain: & All real numbers We will substitute some real number values for x and find the corresponding values of y. These x- and y-values will create a set of ordered pairs.
x | 3x-3 | y | (x,y) |
---|---|---|---|
- 3 | 3( - 3)-3 | - 12 | ( - 3, - 12) |
- 1 | 3( - 1)-3 | - 6 | ( - 1, - 6) |
1 | 3( 1)-3 | 0 | ( 1, 0) |
3 | 3( 3)-3 | 6 | ( 3, 6) |
We found four ordered pairs that lie on the graph of our function. Let's plot the ordered pairs and connect them with a straight line since the function have a continuous domain.
The line is the complete graph of the function. The answer is B.