All Four One - Linear Equations IMP 1
All Four, One - Linear Functions
In the last activity, we talked about how situations, rules, x-y tables, and graphs all relate to each other and connect. Now, we’ll look at how situations, rules, x-y tables, and graphs relate and connect to linear functions. A linear function is a function that, if the points from the function were to be put on a graph and connected, it would form a straight line. They are used to show a constant rate of change between two variables.
A very simple example of a linear function would be that the relation between feet and inches is always 12 inches per foot. So, the function for this would be y=12x, where y is the number of inches and x would be the number of feet. This is how you would connect a situation to a linear function. If you use a linear function to make a graph, the connected points from the function would be a straight line, since they would be changing at a constant rate. This can also go the other way. If you have a graph with a straight line, you can find the rate of change, and form that into a linear function. (Desmos example of the y=12x graph: https://www.desmos.com/calculator/k4oqvmjwkf ) Similarly to how you’d use a graph with a linear function, you can also use a table. You can use the function (y=12x, in our case) and plug in points for the x variable to find what the y would be, which you can later graph. If we made a table using our current function, this is what it would look like this:
F(x) Y
1
12
2
24
3
36
4
48 Once again, you can also flip this the other way and originally have the table above and the points in it, and then using that information make the linear function by finding the constant rate of change, or the slope, which in this case would be (+)12. Then, you’d plug that into the the common form for linear functions, f(x) = ax+b. In this form, f(x) is like the y, a is the rate of change, and b is the point at which you would start. In this case,
In the last activity, we talked about how situations, rules, x-y tables, and graphs all relate to each other and connect. Now, we’ll look at how situations, rules, x-y tables, and graphs relate and connect to linear functions. A linear function is a function that, if the points from the function were to be put on a graph and connected, it would form a straight line. They are used to show a constant rate of change between two variables.
A very simple example of a linear function would be that the relation between feet and inches is always 12 inches per foot. So, the function for this would be y=12x, where y is the number of inches and x would be the number of feet. This is how you would connect a situation to a linear function. If you use a linear function to make a graph, the connected points from the function would be a straight line, since they would be changing at a constant rate. This can also go the other way. If you have a graph with a straight line, you can find the rate of change, and form that into a linear function. (Desmos example of the y=12x graph: https://www.desmos.com/calculator/k4oqvmjwkf ) Similarly to how you’d use a graph with a linear function, you can also use a table. You can use the function (y=12x, in our case) and plug in points for the x variable to find what the y would be, which you can later graph. If we made a table using our current function, this is what it would look like this:
F(x) Y
1
12
2
24
3
36
4
48 Once again, you can also flip this the other way and originally have the table above and the points in it, and then using that information make the linear function by finding the constant rate of change, or the slope, which in this case would be (+)12. Then, you’d plug that into the the common form for linear functions, f(x) = ax+b. In this form, f(x) is like the y, a is the rate of change, and b is the point at which you would start. In this case,