Functions Can Be Linear

Students explore the concept of linearity, as represented in tables and graphs.

We can think of the "x" column as counting the order in which the y-values appear (1st value, 2nd value, etc). When we notice that x-values change at a constant rate and the y-values also change at a constant rate, we know that if we were to plot those values on a graph, all of the points would fall on a straight line.

The line representing the linear relationship would not only include the points represented in the table, but also all of the coordinate pairs that satisfy the same rule, including lots of points whose x and y values are fractions and decimals.

The points in a table are discrete. While ordering the rows in a table can make it easier for us to find the function, they preserve their meaning if the rows are shuffled into a different order.

On a graph, the points on the x-axis cannot be shuffled, because the x-axis must always be ordered. We can stretch the scale of the axes to making the lines look different, but the points will always be in the same order.

We’ve seen that linear relationships can be represented as tables and graphs. Tables only show us some points on a line, whereas a line itself is made up of an infinite number of points. While a table represents a sample of some larger trend, the graph is a way of seeing the trend itself.

Students deepen their understanding of linearity, by seeing counterexamples (non-linear relationships), as well as tables and graphs for which there is no relationship.

Have students turn to Are All Graphs Linear?, where they’ll Notice and Wonder about the six graphs below and consider the question, If all linear relationships can be shown as points on a graph, does that mean all graphs are linear?

Three of the graphs above represent linear relationships, and three show other, non-linear relationships. As we can see, the linear graphs can go in lots of directions and non-linear relationships can follow patterns that aren’t linear!

Have students turn to Are All Tables Linear?, where they’ll Notice and Wonder about the six tables below and consider the question, If all linear relationships can be shown as tables, does that mean all tables are linear?

Three of the tables above show linear relationships, and three show other, non-linear relationships. As we can see, the linear tables can have y-values that change by zero (no change), by a positive number (constant increase), or a negative number (constant decrease) as the x-values increase. The other tables may show patterns, but they aren’t linear!

Sometimes there is no function that will give us a particular table or graph! Take a look at the table and graph below. Can you predict the next two rows? Where will the next point be?

Have students complete Linear, Non-linear, or Bust?. For more (optional) practice, you can have them work with Graphs: Linear, Non-linear, or Bust? and Graphs 2: Linear, Non-linear, or Bust?.

Data has a "shape", and this shape can emerge when we look for patterns in that data. A linear function is one kind of pattern, and we can see it when viewing data as a table or a graph.

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

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 change in the x-value.

2) The y-value when x = 0$\displaystyle x = 0$ is called the y-intercept.

Have students turn to Slope & y-Intercept from Tables (Intro) and facilitate a discussion.

Life isn’t always so simple!

How do we find the slope and y-intercept for these functions, without having to sort or extend the table?

We can exploit the fact that all linear functions 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 two coordinate pairs!

Leave some time for group discussion…​

TO FIND THE 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 described as ChangeInY/ChangeInX$\displaystyle \frac{ChangeInY}{ChangeInX}$ or rise/run$\displaystyle \frac{rise}{run}$.

Taking the first two pairs of values in the the last table on the page, this gives us ( 56 - 5 )/( 20 - 3 )$\displaystyle \frac{56 - 5}{20 - 3}$. We can simplify that to 51/17$\displaystyle \frac{51}{17}$, for a slope of 3$\displaystyle 3$.

We would get the same answer if we subtracted the coordinates in the opposite order…​ ( 5 - 56 )/( 3 - 20 ) = -51/-17 = 3$\displaystyle \frac{5 - 56}{3 - 20} = \frac{-51}{-17} = 3$.

We can use the two points in any order we wish, but we need to use the same order for our x’s and y’s. If we mixed up the order for this example, we’d get ( 56 - 5 )/( 3 - 20 ) = 51/-17 = -3$\displaystyle \frac{56 - 5}{3 - 20} = \frac{51}{-17} = -3$.

We’ll talk more about how to find the y-intercept in the Defining Linear Functions lesson.

Let’s get some practice identifying the slope of a linear function in a table by completing Identifying Slope in Tables

Slope and y-intercept form the essence of linear functions. 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 refine their understanding of linearity, identifying properties like slope and y-intercept from graphs.

On a graph, the y-intercept is the value where the line "intercepts" the y-axis.

On a graph, the slope refers to both the "steepness" and "direction" of the line.

We can compute the slope from a graph the same way we would with a table, by picking two points we know the exact coordinates of.

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Let’s get some practice identifying the slope and y-intercept of a linear function in a graph by completing Identifying Slope and y-intercept in Graphs

We have learned how to find slope and y-intercept from tables and graphs of linear relationships. Check in with yourself and what we’ve learned today.

Linear relationships are everywhere:

What other linear relationships can you think of?