Graph lines by hand using slope or intercepts

Ontario Grade 10 Mathematics

Modelling Linear Relations

A Grade 10 guide to plotting straight lines on a coordinate grid

A straight-line graph can be drawn by plotting points that belong to the line. Two useful ways to find those points are to use the line’s slope or to find its intercepts. You will practise both methods. You only need a coordinate grid, a ruler, and the ability to plot ordered pairs.

What you will learn

Read and plot ordered pairs on a coordinate grid.
Use slope to find points on a line and draw the line.
Find and plot the $x$ - and $y$ -intercepts of a line.
Choose a clear hand-graphing method and check whether a graph matches its equation.

1. Bridge: coordinates and plotting points

A coordinate grid has a horizontal axis, called the

x

-axis, and a vertical axis, called the

y

-axis. The axes meet at the origin, written

(0,0)

. An ordered pair tells you where to plot a point. In

(x,y)

, move along the

x

-axis first, then move up or down to the

y

-value.

For example, to plot

(2,3)

, move

2

units right and

3

units up from the origin. A negative coordinate means move left or down. A line graph is drawn by plotting points that follow the same relationship and joining them with a straight edge.

Plot the first coordinate horizontally and the second vertically.
A line needs at least two distinct points to be drawn.

2. Use slope to graph a line

Slope describes how much a line rises or falls as you move from left to right. It is often written as

m

. The rise is the vertical change between two points. The run is the horizontal change. Keep the order consistent: compare the vertical change with the horizontal change.

A line written as

y=mx+b

gives slope

m

and the

y

-intercept

b

. The

y

-intercept is the point where the line crosses the

y

-axis. On that axis,

x=0

, so the point is

(0,b)

. This form is useful because it gives you a starting point and a direction.

For example, if the slope is

2

, then for every

1

unit you move right, the line rises

2

units. From a point on the line, repeat that movement to find another point. A negative slope means the line falls as you move right. If the slope is a fraction, use its numerator as the rise and its denominator as the run. You can also multiply both by the same number to make a larger, equivalent movement.

Plot the intercept, use the slope to find a second point, and draw a straight line through both points. Extend the line in both directions. If you have room, check your graph by using the slope again to find a third point.

m=\frac{\text{rise}}{\text{run}}

Slope is vertical change divided by horizontal change.
A positive slope rises from left to right; a negative slope falls from left to right.
Use the intercept as a known point, then apply the slope.

3. Use intercepts to graph a line

An intercept is where a line crosses one of the axes. The

x

-intercept is the crossing on the

x

-axis. Every point on the

x

-axis has

y=0

. The

y

-intercept is the crossing on the

y

-axis, where

x=0

.

To find an intercept from an equation, set the other coordinate to zero and solve. For the

x

-intercept, substitute

0

for

y

. For the

y

-intercept, substitute

0

for

x

. Plot both intercepts, then draw a straight line through them. This method is especially direct when the equation makes those values easy to find.

The small table below shows what to set to zero. Once you calculate each intercept, write it as an ordered pair so you can plot it correctly.

At the $x$ -intercept, $y=0$ .
At the $y$ -intercept, $x=0$ .
Two different intercepts give two points for drawing the line.

4. Guided example and independent practice

The worked example compares the slope method and the intercept method on one line. Notice that each method produces points that fit the same line. The intercept method is convenient when both intercepts are easy to calculate. The slope method is convenient when slope and one point are already clear.

After the example, try this independently: graph

y=-\frac{1}{2}x+3

using slope and the

y

-intercept. Then check whether the point

(4,1)

lies on your line. Substitute its coordinates into the equation to check. Your graph should have a negative slope, and the point should fit the equation.

Label axes and use a consistent scale before plotting.
Plot points carefully and use a ruler to draw the line.
A point lies on a line if its coordinates satisfy the line’s equation.

What to set to zero when finding intercepts

Intercept	Coordinate set to zero	Point format
$x$ -intercept	$y=0$	$(x,0)$
$y$ -intercept	$x=0$	$(0,y)$

Worked example

Graph a line using slope or intercepts

Graph the line

y=2x-1

by hand. Find two points using slope, then check the graph using intercepts.

Read the slope and starting point
The equation is in the form $y=mx+b$ . The slope is $2$ , and the $y$ -intercept is $-1$ . Plot the intercept at $(0,-1)$ .
$m=2,\quad (0,b)=(0,-1)$
Use slope to find another point
Write the slope as a rise of $2$ and a run of $1$ . From $(0,-1)$ , move $1$ unit right and $2$ units up. This reaches $(1,1)$ . Plot that point.
$\frac{2}{1}=2,\quad (0,-1)\rightarrow(1,1)$
Draw and check the line
Draw a straight line through $(0,-1)$ and $(1,1)$ , extending it in both directions. To check with intercepts, set $y=0$ to find the $x$ -intercept, then set $x=0$ to confirm the $y$ -intercept.
$0=2x-1\Rightarrow x=\frac{1}{2};\quad x=0\Rightarrow y=-1$

Answer: The line passes through

(0,-1)

and

(1,1)

. Its intercepts are

(\frac{1}{2},0)

and

(0,-1)

. Plot the points and draw the straight line through them.

Check: Substituting

(1,1)

into the equation gives

1=2(1)-1

, which is true. The calculated intercepts also satisfy the equation.

Common mistakes and how to avoid them

Treating slope as run divided by rise.

Correction: Use vertical change divided by horizontal change. For example, a rise of

3

and a run of

2

gives slope

\frac{3}{2}

.

Plotting the

y

-intercept on the horizontal axis.

Correction: The

y

-intercept is on the vertical axis, so its point has an

x

-coordinate of

0

.

Setting

x=0

when finding the

x

-intercept.

Correction: At the

x

-intercept, the point is on the horizontal axis, so set

y=0

.

Drawing a line through points without checking their coordinates.

Correction: Check that each plotted point matches the equation. A misplaced point can change the entire graph.

Lesson summary

Plot ordered pairs by moving horizontally first and vertically second.
To use slope, plot a known point and follow the rise and run.
To use intercepts, set $y=0$ for the $x$ -intercept and set $x=0$ for the $y$ -intercept.
Use two points to draw a straight line, then check that the points fit the equation.

Check your understanding

Question 1

A line has slope

-\frac{2}{3}

. Starting at a point on the line, which move follows this slope?

Move $3$ units right and $2$ units up.
Move $3$ units right and $2$ units down.
Move $2$ units right and $3$ units down.
Move $2$ units left and $3$ units down.

Show answer and explanation

Move

3

units right and

2

units down.

A slope of

-\frac{2}{3}

means a vertical change of

-2

for a horizontal change of

3

. So move

3

units right and

2

units down.

Question 2

Which coordinate must be zero at the

y

-intercept?

The $x$ -coordinate
The $y$ -coordinate
Both coordinates
Neither coordinate

Show answer and explanation

The

x

-coordinate

The

y

-intercept is on the vertical axis. Points on that axis have an

x

-coordinate of

0

.

Question 3

For

y=x+4

, what is the

x

-intercept?

$(0,4)$
$(4,0)$
$(-4,0)$
$(0,-4)$

Show answer and explanation

(-4,0)

At the

x

-intercept, set

y=0

. Then

0=x+4

, so

x=-4

. The intercept is

(-4,0)

.

Key terms

Ordered pair: A pair of coordinates written as $(x,y)$ that names a point on a grid.
Slope: A measure of how much a line rises or falls for a horizontal movement.
Intercept: A point where a graph crosses an axis.
$x$ -intercept: The point where a line crosses the $x$ -axis.
$y$ -intercept: The point where a line crosses the $y$ -axis.

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