TEK 6.4 A, TEK 6.6 A/B - Relationships
TEK 6.4 A Learning Goal: I can compare two rules verbally, numerically, graphically, and symbolically in the form of y=ax or y=x+a in order to differentiate between additive and multiplicative relationships.
TEK 6.6 A Learning Goal: I can identify independent and dependent quantities from tables and graphs.
TEK 6.6 B Learning Goal: I can write an equation that represents the relationship between independent and dependent quantities from a table.
Vocabulary
Relationship: The rule that determines how two quantities are connected with each other.
Variable: A symbol (usually a letter) used to represent a quantity that can change, or an unknown number.
Quantity: The amount or number of things (both material and immaterial). Variables represent a quantity that is unknown and can change.
Independent Quantity: The quantity in a relationship that is subject to choice. You can put any number you want for this quantity.
Dependent Quantity: The quantity in a relationship that changes because the independent quantity was changed.
Linear Relationship: A relationship whose graph is a straight line.
Additive Relationship: Compares the independent and dependent quantities of a relationship using addition.
Multiplicative Relationship: Compares the independent and dependent quantities of a relationship using multiplication.
Input: Another way of saying the independent quantity. It is the number you choose to put into a relationship
Output: Another way of saying the dependent quantity. It is the number that you get out of the relationship after following the rule of that relationship.
Relationships
A relationship is the rule that connects two quantities that are variables, meaning they can change. The independent quantity is the one you change yourself, and the dependent quantity changes based on what you put for the independent quantity and the rule of the relation. The independent quantity is also known as the input value, while the dependent quantity is also known as the output value. This makes sense when you think about it. When you put in (input) a number for the independent quantity, what you get out (output) is the dependent quantity. The set of pairs of independent and dependent quantity forms the relation.
When describing a relationship, we most often use the letters x and y as variables. The x variable represents the independent quantity, and the y variable represents the dependent quantity. Relationships can be written in several different ways:
Table/Chart
You've probably used tables and t-charts as a way to organize information in school. They also come in handy with relationships. Here is an example of a table being used to represent a relationship:
This table shows the relationship between X and Y, which are both quantities. When you look at it for a while, you can already see a pattern. This pattern is the rule of the relationship. You can guess that if the next X value is 4, the Y value will be 8. This is true even though the table doesn't show those numbers. A table just shows enough information to where you can see the pattern and figure out what the rule is. We will be doing this later.
Remember that X is the variable that represents the independent quantity. So, the independent quantity is always on the top row of the table, and the dependent quantity is always on the bottom row. Not all tables use the letters X and Y, but the independent quantity is still always on top. You also read the table top to bottom. You pick an X value and then go down to the Y value directly below to find out what the effect of that X value is. The X value and the Y value below it form an input/output pair.
Here's an example of a t-chart that has the same information:
A t-chart is almost the same as a table, except it is vertical instead of horizontal. The independent quantity is always on the left and the dependent quantity is always on the right. You read the table left to right, and an input/output pair is an X and Y value that are side-by-side.
We read English from left to right and from top to bottom. This is why the independent quantity is always on the top in tables and on the left in charts. You have to read the independent quantity first before moving on to the dependent quantity, because the dependent quantity is based off of the independent quantity.
Tables can also be written vertically like a t-chart. They will just have horizontal lines to separate each pair of X and Y values.
Graph
Graphs are another way to represent relationships. Take a look:
The numbers along the bottom of the graph are independent quantity values. The horizontal line that they are all below is called the x-axis. It doesn't matter if the independent quantity of a relationship is represented by X or not, it will still go on the x-axis. The y-axis is the vertical line on the left. It always has dependent quantity values. When you read the graph, you start with the x-axis value you want to explore, and you draw a vertical line up from that value. Then, you mark where the vertical line crosses the line of the graph (in blue). Draw a horizontal line from that point to the left side of the graph. Where you end up is your matching y value. Let's say you wanted to look at the independent quantity 1. A vertical line is drawn up from 1 in the picture above. A third of the way across the graph, this vertical line crosses the line of the graph. At this crossing point, a horizontal line is drawn. This line ends up at the 2 on the y-axis.
An X value and a Y value that meet at the same place on the line of the graph are called a coordinate pair. This is just another name for an input/output pair. A coordinate pair is written (X, Y). X is first of course, because you start reading the graph on the x-axis. For the example we just did, the pair would be (1, 2).
Like tables and charts, graphs usually don't show all the input/output pairs of a relationship, but they can show an infinite amount of pairs, even if we just look between X=0 and X=3 like the examples above. A graph would have all the input/output pairs that a table or chart would, and then all the decimals in between because the line connects all these points together. However, graphs only have some X and Y values labeled because it would take forever to label an infinite amount of numbers. Computers and advanced calculators can make graphs that tell you what every single coordinate pair is, but humans just don't have time for that.
Equation
A relationship written in equation form will always have two parts that are set equal to each other by an equal sign. A single y is usually written on the left side of the equal sign, while an x is typically on the other side, often accompanied by a number or multiple numbers. This is the equation form of the relationship we have used in the example before this:
y = 2x
As always, X represents the independent quantity, and Y represents the dependent quantity. When using an equation, you plug a number into X, the independent variable, and then figure out what Y is. If you put the number 1 into X for this equation, you would have 2(1) for the right side of the equation. Since there is an equal sign, both sides must be the same. So, Y is also equal to 2(1), which is just 2. Look at the table, chart, and graph we did before. We got 2 as our output when 1 was our input for those cases too.
The equation is basically the rule of the relationship. It tells you how a number is changed when you put it into the equation. In this case, it is multiplied by 2. Later we will work on how to find the equation for a given table.
Words
The last way to write a relationship is using a sentence. Usually, when you describe a relationship with words, you use the quantities in an actual real life scenario instead of the variables X and Y. Let's say that the relationship we have been using so far is a model of how many people have entered a restaurant in the minutes since it was opened. The independent quantity here is the minutes since the restaurant was opened, because that is the quantity you would start with. Unless you have been there the whole day, counting people (which is boring), you have no idea how many people have come in. But, if you have a watch, you always know exactly what time it is. So, you'd plug the time into the X variable because it represents the independent quantity, and the Y variable would be the dependent quantity, the amount of people entered. For this specific relationship, every time X, or the number of minutes passed, increases by one, the number of people that have entered increases by 2. So, every minute, 2 people enter the restaurant. This is the written version of the rule for this relationship.
Different Types of Linear Relationships
All of the relationships you will need to know now are linear relationships. A linear relationship is a relationship that, when graphed, creates a straight line. The graph of a relationship will be straight unless X has an exponent on it in the equation form. Linear relationships can come in two main categories: additive and multiplicative.
Additive Relationships
Additive relationships have the equation form of y = x + a, where Y is dependent quantity, X is the independent quantity, and A is a constant (a number that never changes). Let's use the number 3 for a, which makes the equation y = x + 3, and make a table for that relationship:
To make the table, all we did was pick a few X values, put them into the equation, and then find the Y value. For the X value 0, the equation becomes y = 0 + 3, which is equal to y = 3, so Y is 3 at that X value. For the X value 1, y = 1 + 3, so y = 4, so the Y value is 4. This was done for the other two X values as well.
Notice how X and Y both increase by 1 each time you move one column to the right along the table. They would be the same value all the time, but Y got kind of a head start and began at 3, while X began at 0. Y is always going to be 3 steps ahead of X. These are the kind of signs to look for to see if a relationship is additive. If the different between the X and Y value for every input/output pair is the same, and if the X and Y values both increase by the same amount across the table, then the relationship is additive.
Here is the graph for this relationship:
Looking at the graph can also tell you if the relationship is additive. Additive relationships do not have the coordinate pair (0, 0). (0, 0) is known as the origin of the graph, so we can also say that additive relationships do not go through the origin. This makes sense when you think about it. If you were to put 0 into the equation y = x + a, the equation would simplify to y = a. That means Y has to be equal to whatever the constant number a is for that particular relationship, so it won't be zero.
Multiplicative Relationships
Multiplicative relationships have the equation form of y = ax, where Y is dependent quantity, X is the independent quantity, and A is a constant (a number that never changes). Let's use the number 3 for a, which makes the equation y = 3x, and make a table for that relationship:
Notice how A is 3 for both this relationship and the relationship we used when discussing additive relationships. However, the tables are very different. Multiplying by A instead of adding A changes the relationship a lot.
In our table, we take each independent quantity of our choosing and put it into the equation y = 3x. If we plug in 0 as our independent quantity (x), we will get 0 for the output because zero times any number is just zero. If we put 1 in, the equation becomes y = 3(1), which simplifies to y = 3, so Y is 3. This process is repeated for the other input/output pairs.
You have probably noticed that, as we move across the columns of the table, X increases by 1 each time, but Y increases by 3. X and Y increasing by different amounts is a big clue that the relationship is multiplicative. Another clue is that the difference between the X and Y value of any given input/output pair changes. When X is zero, Y is also zero, so there is no difference between them. But when X is 1, Y is 3. 3 minus 1 is 2, so there is a difference of 2 between them. As you move across the table, the difference gets bigger and bigger instead of staying the same.
Now let's look at the graph:
Remember how an additive relationship wouldn't ever go through the coordinate pair (0, 0)? Well, in a multiplicative relationship, the graph will always go through (0, 0). This makes sense because if X is zero, making y = a(0), then Y also has to be zero because zero multiplied by anything is just zero.
Finding an Equation from a Table
We said before that a table has enough information to determine the rule for a relationship. So, we should be able to look at a table and make an equation that fits that table. When we make this equation, we have to make sure it has X and Y in it, and we have to make sure it works for any value of X that we put in. When you approach a table, use these steps:
1. Figure out whether the table shows an additive or multiplicative relationship. Remember, for an additive relationship, the X and Y values will change by the same amount as you move across the columns of the table. For a multiplicative relationship, they will change at different rates. The difference between the X and Y values of an additive relationship input/output pair will always be the same, while this difference will change for a multiplicative relationship. Move on to either step 2 or step 3 depending on what kind of relationship it is.
2. If the relationship is additive, find where the X value is equal to zero. This may not be shown on the table, so you may need to use the pattern to extend the table until zero for an X value is shown. Now look at the Y value. This is your A in y = x + a. The reason for this is, if X is zero, then y = a. So the Y value when X is equal to zero is equal to A. Now you just plug that number into y = x + a, and you have your equation. You're done and you don't need to move on to step 3.
3. If the relationship is multiplicative, divide the Y value of a single input/output pair (any pair except for the one where X and Y are both equal to zero) by the corresponding X value. This will tell you how many times X goes into Y, or, more importantly, what number X is multiplied by to get to Y. This number is your A in y = ax.
Example 1:
Whenever you see a table, your first instinct might be to only look at the Y and make some kind of observation only based off of that. For this table, you might say that Y keeps increasing by 1 and that is the pattern. However, you shouldn't do this! This is a relationship, which is the connection between X and Y. You need to somehow relate what X is doing to what Y is doing, and it must work for any independent quantity that you put in. Let's do the steps:
1. Look at the table. X and Y both increase by 1 each time you move across a column of the table, and the difference between an X value and its matching Y value is always 5. So, this is an additive relationship and we move on to step 2.
2. Step 2 is finding where X is equal to zero. There is no place on the table where X is equal to zero, and since we don't have the equation yet, we can't just plug it in. What we need to do is make an educated guess about what Y would be when X equals zero based on the pattern that you can see in the table. Every time X changes by a number, Y changes by that number, too. We have already looked at how the numbers increase by 1 as you move across the columns from left to right, but changing X by any number, not just 1, will cause Y to change by that same amount. When X moves from 1 to 3, which is increasing by 2, Y also increases by 2 as it moves from 6 to 8. You can even look at it backwards. When X moves from 3 to 1, it decreases by 2, so Y also decreases by 2 when it moves from 8 to 6. So, let's say X were to move from 1 to 0. That is a decrease by 1. Decrease the Y value of 6 by 1, and you get 5. That is the value of Y when X is zero, and it is also the value of A in the equation. So, the equation is y = x + 5.
Example 2:
1. Look at the table from left to right. From column to column, X increases by 1, but Y increases by 4. Since they change by different values, this is a multiplicative relationship. The difference between the X value and its corresponding Y value also changes, further supporting that this is a multiplicative relationship. Because the relationship is multiplicative, we will move straight to step 3.
3. Now let's divide a Y value by its corresponding X value. Remember, we are not allowed to use the pair where both X and Y are zero. Let's use the pair where X is 2 and Y is 8. 8 divided by 2 is 4. Let's double check that using the pair where X is -1 and Y is -4. -4 divided by -1 is also 4. Your A value, or the value X is being multiplied by to get to Y, is 4. So, the equation is y = 4x.
One Last Thing
All the tables we have used in this lesson show X increasing by 1. However, this doesn't have to be the case. X can increase by any number, and sometimes by a different number each column. This can make it a little bit harder to find a pattern from the table, but everything we have learned still applies to those tables. There will be examples of these with explanations in the practice problems below.
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Use these quizzes to make sure you fully understand what you have just learned. Click "view score" to see explanations for any wrong answers. Click "submit another response" to try again.