A function's graph

A function’s graph

A function’s graph is all the points $\displaystyle (x,y)$ in the coordinate system where $\displaystyle y$ is the value the function gives $\displaystyle x$ .
The function $\displaystyle f$ is the rule “multiply by 2 and subtract 1”:

$\displaystyle x$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$
$\displaystyle y$	$\displaystyle -3$	$\displaystyle -1$	$\displaystyle 1$	$\displaystyle 3$

Each column of the table is a point of the graph.

\displaystyle f(x)

\displaystyle (-1,-3)

\displaystyle (0,-1)

\displaystyle (1,1)

\displaystyle (2,3)

\displaystyle (1.5,2)

Between the points of the table there are points of the graph too. For example $\displaystyle {x}={1.5}$ : $\displaystyle {1.5\cdot2-1}={2}$ , so the point $\displaystyle (1.5,2)$ is on the graph as well. As more and more such points are worked out, they join into a line.

Reading values off the graph

From the graph you can read the $\displaystyle y$ of any $\displaystyle x$ , and also find the values of $\displaystyle x$ that have a given $\displaystyle y$ .

The $\displaystyle y$ of $\displaystyle {x}={-1}$ : from $\displaystyle -1$ on the $\displaystyle x$ -axis go up to the graph, and from there straight across to the $\displaystyle y$ -axis: $\displaystyle {y}={1}$ .
The values of $\displaystyle x$ where $\displaystyle {y}={-1}$ : from $\displaystyle -1$ on the $\displaystyle y$ -axis go across, and find all the points of the graph at that height.

\displaystyle f(x)

\displaystyle (-1,1)

\displaystyle (-2,-1)

\displaystyle (1,-1)

\displaystyle (3,-1)

At the height $\displaystyle {y}={-1}$ the graph has three points: $\displaystyle (-2,-1)$ , $\displaystyle (1,-1)$ and $\displaystyle (3,-1)$ . Each $\displaystyle x$ has one $\displaystyle y$ , but the same $\displaystyle y$ may belong to several values of $\displaystyle x$ .

A story graph: a bike ride

Noa left home for a bike ride, and came back home.
The graph shows her distance from home at every moment of the ride.
The $\displaystyle x$ -axis: the time in hours since leaving. The $\displaystyle y$ -axis: the distance from home in kilometres.

\displaystyle d(x)

In the first hour the graph goes up: Noa moves away from home, to 12 km.
Between one hour and an hour and a half the graph is flat: the distance does not change – Noa stopped to rest.
Two and a half hours after leaving she is farthest from home: 18 km.
Then the graph goes down: Noa rides back, and after 4 hours she is home.

The graph shows the distance from home, not the shape of the road: it goes up while Noa moves away from home, even if the road is flat.

The vertical line test

Each vertical line meets a function’s graph at one point at most.

All the points on a vertical line have the same $\displaystyle x$ .
A function gives each $\displaystyle x$ one $\displaystyle y$ , so its graph has one point at most on each vertical line.

\displaystyle ?

\displaystyle x=3

\displaystyle (3,4)

\displaystyle (3,-4)

The circle is not a function’s graph: the vertical line $\displaystyle {x}={3}$ meets it at two points, $\displaystyle (3,4)$ and $\displaystyle (3,-4)$ . That is, $\displaystyle {x}={3}$ gets two values of $\displaystyle y$ , 4 and $\displaystyle -4$ .

Exercise 1

Mark the points where the graph of $\displaystyle g$ meets the $\displaystyle x$ -axis and the $\displaystyle y$ -axis.
Write:

$\displaystyle y$ when $\displaystyle {x}={-2}$ , and when $\displaystyle {x}={4}$ ;
all the values of $\displaystyle x$ where $\displaystyle {y}={2}$ .

\displaystyle g(x)

\displaystyle {x}={-2}

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