Defining Linear Functions

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

As you’ve seen, a function definition is a way of summarizing a relationship. You’ve seen how a linear relationship can be expressed as a table or graph. But what do these kinds of relationships look like as a definition?

Here we see a function definition written using pyret notation and using function notation.

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The slope-intercept form of the line includes the slope as the coefficient of x and the y-intercept as the numerical term. You will hear people describe this form as y = mx + b$\displaystyle y = mx + b$, where m$\displaystyle m$ stands for slope and b$\displaystyle b$ stands for the y-intercept.

While it is common to write the x-term first and the y-intercept second, they can be written in any order!

When the slope is zero (and the line is horizontal)…​ we may choose whether or not to write the slope term.

When the y-intercept is 0$\displaystyle 0$ (and the line crosses the y-axis at the origin)…​ we may choose whether or not to write the slope term.

To check our work, we can apply the function to the x-value from any coordinate pair on our table or graph, and it should produce the y-value!

As with tables and graphs, a function definition can also reveal whether or not the function is linear. Functions that are not linear will follow other forms, for example they may include exponents or absolute values.

Students can identify slope and y-intercept from functions in Pyret as well, using Exploring Linearity in Definitions Starter File.

Students can practice defining linear functions when given tables and graphs using the Exploring Linearity in Tables Starter File and Exploring Linearity in Graphs Starter File.

It is 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, it’s helpful to be able to switch between representations, and see the connections between them.

But if the y-intercept isn’t zero…​ there is another step to finding the y-value.

The y-intercept always gets added to / subtracted from the product of the slope and the x-value to find the corresponding y-value.

As discussed above, the relationship between the x-values and the y-values can be described using y = mx + b$\displaystyle y = mx + b$, where m$\displaystyle m$ stands for slope and b$\displaystyle b$ stands for the y-intercept.

If we solve that for the y-intercept…​

b = y - mx$\displaystyle b = y - mx$

In other words, the y-intercept can be calculated by subtracting the product of the slope and any x-value from the corrseponding y-value.

To find the y-intercept, subtract 9$\displaystyle 9$ (the y-value of the point) minus 3 × 7$\displaystyle 3 \times 7$ (the product of the slope and the x-value of the point).

y-intercept: -12$\displaystyle -12$

function definition making use of the y-intercept we found: fx = 3x - 12$\displaystyle f(x) = 3x - 12$