Defining Linear Relationships

Students refine their understanding of linearity, identifying properties like slope and y-intercept in both tables and graphs.

Every linear relationship has two properties:

1) The sequence of y-values always changes at a constant rate - called slope - increasing or decreasing by the same amount for each value. On a graph, this value refers to the "steepness" and "direction" of the line.

2) The y-value when x = 0$\displaystyle x = 0$ is called the y-intercept, On a graph, this value is where the line "intercepts" the y-axis.

Take a look at the following Table (shown sideways, with rows going from left-to-right):

Leave some time for group discussion:

We can see that the y-values increase by 2 each time x increases by 1, giving us a slope of 2. Meanwhile, the row for x = 0$\displaystyle x = 0$ tells us that the y-intercept is 1.

But life isn’t always so simple!

The slope and y-intercept in this table are much harder to find, because the x-values are out of order and we can’t see a value for x = 0$\displaystyle x = 0$:

If we had the line for this function, we could generate as many points as we need to fill in the table, or to compute the slope and y-intercept. Can we reconstruct that line?

We can exploit the fact that all linear function form straight lines, and a straight line can be defined with only two points!. That means it is always possible to compute slope and y-intercept, as long as we have just two rows in our table or two points on our graph!

Now that we know it’s possible, how do we do it?

Leave some time for group discussion:

1) SLOPE: find any two pairs of values in the table, and divide the difference in y’s by the difference in x’s.

This is an easy way to see the change in y as a proportion of the change in x, which gives you the slope of the function. This is often called "change in y over change in x", or "rise over run".

Taking the first two pairs of values in the table, this gives us ( 36 - 24 )/( 17 - 11 )$\displaystyle \frac{36 - 24}{17 - 11}$, which simplifies to 12/6$\displaystyle \frac{12}{6}$, for a slope of 2$\displaystyle 2$.

2) Y-INTERCEPT: multiply any x in the table by the slope, and subtract the result from the corresponding y.

You can find the y-intercept by expanding the table and following the pattern to figure out the value of y$\displaystyle y$ when x = 0$\displaystyle x = 0$, but sometimes that’s a lot of work!

Let’s use the last pair of values in the table to demonstrate this shortcut: Starting with the x$\displaystyle x$ value of 9, multiplying 9 × 2$\displaystyle 9 \times 2$ gives us 18$\displaystyle 18$. The corresponding y$\displaystyle y$ is 20, so the y-intercept is 20 - 18 = 2$\displaystyle 20 - 18 = 2$.

But what about graphs? We can compute the slope and y-intercept from a graph the same way, by picking two points and using those as our two sample rows.

Can you identify the slope and y-intercept of a linear function in a table? In a graph?

Slope and y-intercept form the essence of linear function. If we can find them in a sample of data, we can make predictions that go outside that sample. For example: If we know a car is moving at a consistent speed, all we need to know is where it is located at two points in time in order to figure out the speed, and to predict where it will be at any other point in time!

Students explore function definitions as a way of expressing linear relationships, and construct tables and graphs from those definitions.

Writing out an entire table or graph - even if it’s just two rows or two points! - can be time-consuming. It also forces other people to compute the slope and y-intercept by hand!

Fortunately, a function definition can be used to summarize an entire table or graph by putting the slope and y-intercept front-and-center! Let’s see a function definition, written both as regular function notation and as Pyret code. NOTE: the slope and y-intercept can be written in any order!

As with tables and graphs, a function definition can reveal whether or not it is linear.

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If the line is perfectly horizontal the slope will be zero, making the term "invisible"! In the example below, a linear function with a slope of zero is shown with and without this term:

If the line crosses the y-axis at zero, the y-intercept will be 0$\displaystyle 0$. This can make that term "invisible"! In the example below, a linear function with a y-intercept of zero is shown with and without this term:

To check our work, we can apply the function to the x-value in each Row in the table, it produces the y-value! Instead of writing endless rows repeating the rule or drawing an entire graph, we can just declare the rule itself by defining the function.

Let’s get some practice working with Function Definitions

Let’s get some practice connecting Definitions to Graphs

Discuss as a class: What strategies did you use?

Let’s get some practice connecting Definitions to Tables

Discuss as a class: What strategies did you use?

Let’s get some practice connecting Tables, Graphs, and Definitions

It is incredibly common to think of the graph as the "output" of the function, rather than the function itself. Most math textbooks will use language like "matching the graph to the function", suggesting that the graph is somehow not the function! Since this language is pervasive, it’s important to actively push against it.

Function definitions are a way of talking about relationships between quantities: milk costs $0.59/gallon, a stone falls at 9.8m/s^2$\displaystyle 9.8m/s^2$, or there are 30 students for every teacher at a school. If we can figure out the relationship between a small sample of data, we can make predictions about what happens next. We can see these relationships as tables, graphs, or symbols in a definition. We can even think about them as a mapping between Domain and Range!

When we talk about functions, sometimes it’s easiest to look at the graph, the table, or the definition. What’s important is being able to switch between representations, and see the connections between them.