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.