In mine last post I wrote around the adhering to standard, and mentioned that I can write a entirety blog post around the an initial comma.

You are watching: What does a straight line on a graph mean

8.F.A.3. Analyze the equation \$y = mx + b\$ as defining a direct function, who graph is a right line; give examples of attributes that are not linear. For example, the role \$A = s^2\$ giving the area of a square together a duty of its side length is no linear because its graph consists of the point out \$(1,1)\$, \$(2,4)\$ and also \$(3,9)\$, which room not on a straight line.

The comma shows that the i “whose graph is a straight line” is nonessential for identifying the noun expression “linear function.” It transforms the clause right into an extra piece of information: “and by the way, did you recognize that the graph that a linear role is a right line?” This truth is frequently presented as obvious; after all, if you draw the graph or produce it making use of a graphing utility, it absolutely looks like a directly line.

When I’ve inquiry prospective teachers why this is so, I’ve obtained answers the look something favor this:

We recognize that a linear duty has a continuous rate that change, \$m\$. If friend go across by 1 on the graph you constantly go increase by \$m\$, choose this:

So the graph is prefer a staircase. It always goes up in steps of the exact same size, for this reason it’s a right line.

This is well as much as the goes. It identifies the specifying property of a linear function—that it has actually a consistent rate the change—and relates that property to a geometric feature of the graph. However it’s a “Here, Look!” proof. In the end it is showing that something is true rather than reflecting why that is true. Which is to say that it’s no a proof.

Still, the move to a geometric building of linear functions is a move in the appropriate direction, due to the fact that it concentrates our psychic on the necessary concept. Us all understand that any kind of two clues lie on a line, yet three points could not. What is it about three point out on the graph that a linear role that suggests they have to lie ~ above a directly line?

Line native \$A\$ to \$B\$ to \$C\$ is dotted since we don’t understand it’s a heat yet

Because a linear duty has a constant rate that change, the steep between any two that the three points \$A\$, \$B\$, and also \$C\$ is the same. Therefore \$|BP|/|AP| = |CQ|/|AQ|\$, which way there is a scale element \$k =|AQ|/|AP| = |CQ|/|BP|\$ so that a dilation with center \$A\$ and also scale element \$k\$ bring away \$P\$ to \$Q\$, and take the vertical line segment \$BP\$ come a vertical line segment based at \$Q\$ with the same size as \$CQ\$. Which way it have to take \$B\$ to \$C\$.

But (drumroll) this way that there is a dilation with facility \$A\$ the takes \$B\$ come \$C\$. Dilations always take clues on a beam from the facility to various other points top top the exact same ray. So \$A\$, \$B\$, and \$C\$ lied on the exact same line.

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I don’t really intend students come get all of this, at least not best away. I’d be happy if they understood that over there is a geometric truth at pat here; that seeing is not constantly believing.