Represent linear relations using graphs

A linear relation is represented as a graph of ordered pairs $(x,y)$ , where the horizontal and vertical axes correspond to the independent and dependent quantities, and a consistent scale allows values from a table, equation, or context to be plotted accurately. The resulting straight line reveals the constant rate of change and initial value through its slope and intercepts, while its points represent all pairs that satisfy the relation; this scope is limited to linear relations in two variables, not more advanced functions or nonlinear graphs.

Detailed Explanation: Represent linear relations using graphs

A linear relation can be shown on a coordinate grid by plotting its ordered pairs $(x,y)$ . The $x$ -value goes on the horizontal axis, and the $y$ -value goes on the vertical axis.

Example

Graph the relation

y=2x+1

1. Make a table of values

Choose several $x$ -values and substitute them into the equation.

$x$	$y=2x+1$	Ordered pair
$-2$	$2(-2)+1=-3$	$(-2,-3)$
$-1$	$2(-1)+1=-1$	$(-1,-1)$
$0$	$2(0)+1=1$	$(0,1)$
$1$	$2(1)+1=3$	$(1,3)$
$2$	$2(2)+1=5$	$(2,5)$

2. Choose a consistent scale

Label the horizontal axis with values such as $-2,-1,0,1,2$ and the vertical axis with values from about $-3$ to $5$ . Make sure equal spaces represent equal amounts.

3. Plot the ordered pairs

For each pair, move first along the $x$ -axis, then up or down to the $y$ -value. Plot

(-2,-3),\;(-1,-1),\;(0,1),\;(1,3),\;(2,5).

4. Draw the line

The points line up in a straight pattern, so use a ruler to draw a straight line through them. Extend the line in both directions.

The graph shows every ordered pair that satisfies $y=2x+1$ . The number $2$ is the slope, meaning $y$ increases by $2$ whenever $x$ increases by $1$ . The graph crosses the vertical axis at $(0,1)$ , so the initial value is $1$ .

Learn by doing: Represent linear relations using graphs

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Algebra - Find Equivalent - Function to Graph

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