Modelling Linear Relationships
We can show it in a table, plot it on the xy-axis, and express it using a mapping diagram. Since we have repetitions or duplicates of x-values with different y- values, Example 1: Is the relation expressed in the mapping diagram a function ?. Example: "Multiply by 2" is a very simple function. Here are the three parts: and we see something like: y = x2. But there is still: an input (x); a relationship ( squaring); and an output (y) x → x2. function. Could also be written as a table. in the table if x is the number of school years since. b. Draw a graph showing the relationship between the year x is the number of pictures sent, and y is the cost in dollars. a. . Sample answer: You can determine whether each.
These are two ways of saying the same thing. Now the relation can also say, hey, maybe if I have 2, maybe that is associated with 2 as well. So 2 is also associated with the number 2. And so notice, I'm just building a bunch of associations. I've visually drawn them over here.
Here I'm just doing them as ordered pairs. We could say that we have the number 3. Our relation is defined for number 3, and 3 is associated with, let's say, negative 7.
Write and graph an exponential function by examining a table | LearnZillion
So this is 3 and negative 7. Now this type of relation right over here, where if you give me any member of the domain, and I'm able to tell you exactly which member of the range is associated with it, this is also referred to as a function. And in a few seconds, I'll show you a relation that is not a function.
Because over here, you pick any member of the domain, and the function really is just a relation. It's really just an association, sometimes called a mapping between members of the domain and particular members of the range.
So you give me any member of the domain, I'll tell you exactly which member of the range it maps to. You give me 1, I say, hey, it definitely maps it to 2. You give me 2, it definitely maps to 2 as well. You give me 3, it's definitely associated with negative 7 as well.
So this relation is both a-- it's obviously a relation-- but it is also a function. Now to show you a relation that is not a function, imagine something like this.
So once again, I'll draw a domain over here, and I do this big, fuzzy cloud-looking thing to show you that I'm not showing you all of the things in the domain. I'm just picking specific examples.
And let's say that this big, fuzzy cloud-looking thing is the range. And let's say in this relation-- and I'll build it the same way that we built it over here-- let's say in this relation, 1 is associated with 2. So let's build the set of ordered pairs. So 1 is associated with 2. Let's say that 2 is associated with, let's say that 2 is associated with negative 3. So you'd have 2, negative 3 over there.
What is a Function
And let's say on top of that, we also associate, we also associate 1 with the number 4. So we also created an association with 1 with the number 4. So we have the ordered pair 1 comma 4. Now this is a relationship. We have, it's defined for a certain-- if this was a whole relationship, then the entire domain is just the numbers 1, actually just the numbers 1 and 2. It's definitely a relation, but this is no longer a function.
And the reason why it's no longer a function is, if you tell me, OK I'm giving you 1 in the domain, what member of the range is 1 associated with?
Over here, you say, well I don't know, is 1 associated with 2, or is it associated with 4? It could be either one. So you don't have a clear association. If I give you 1 here, you're like, I don't know, do I hand you a 2 or 4?
That's not what a function does. A function says, oh, if you give me a 1, I know I'm giving you a 2. If you give me 2, I know I'm giving you 2.
Now with that out of the way, let's actually try to tackle the problem right over here. So let's think about its domain, and let's think about its range. So the domain here, the possible, you can view them as x values or inputs, into this thing that could be a function, that's definitely a relation, you could have a negative 3.
What is a Function?
For this column, I can choose any number I wish. Try to choose numbers that can be graphed on your graph. For example, if your x axis only extends to 10, don't choose 12 as an x coordinate. The second column is for substituting x into the equation in order to solve for y. So, whatever value I chose for x, I will substitute back into the equation and solve to find the y value.
The third column is for the y value. After substituting your x value into the equation, your answer is the y coordinate. The last column is for your ordered pair. Your ordered pair is the x value and the y value.
This is the point on your graph. Let's look at a few examples, and it will all make more sense!
Using a Table of Values to Graph Linear Equations
After substituting those values into the equation: Therefore, the ordered pairs that I found on my graph were: I plotted those points on my graph. I then used my ruler and drew a straight line through those points. If you had done this problem on your own, you may have found three different points using the table of values.
That's ok, because even if your three points are different, your line will still look exactly the same! We can also find other solutions for the equation just by reading the graph. I see that 3,7 is a point on the graph. If I substitute 3 for x into the equation, I will get 7 as my y coordinate. This line goes on forever, so there are infinite solutions to the equation. Let's look at another example. This becomes a little trickier in choosing x coordinates because we could end up with a fraction for the y coordinate.
This then becomes a little more difficult to graph.