A table of values, created using an equation, forms a set of points that can be plotted on a number plane. A line, drawn through the points, becomes the graph of the equation.

We'll begin by creating a table of values for the following equation:

y=3x-5

x	1	2	3	4
y

The first row of the table will contain values for the independent variable (in this case, x). The choice of x-value is often determined by the context, but in many cases they will be given.

To find the corresponding y-value, we substitute each x-value into the equation y=3x-5.

Substituting x=1:

\begin{aligned} y&=3\times1-5\\ &=3-5\\ &=-2 \end{aligned}

Substituting the remaining values of x, allows us to complete the table:

x	1	2	3	4
y	-2	1	4	7

The x and y value in each column of the table can be grouped together to form the coordinates of a single point, (x,y).

Table of x and y values forming ordered pairs. Ask your teacher for more information.

The x and y value in each column of the table can be grouped together to form the coordinates of a single point, (x,y).

Each point can then be plotted on a xy-plane.

To plot a point, (a, b), on a number plane, we first identify where x=a lies along the x-axis, and where y=b lies along the y-axis.

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For example, to plot the point (3, 4), we identify x=3 on the x-axis and construct a vertical line through this point. Then we identify y=4 on the y-axis and construct a horizontal line through this point. The point where the two lines meet has the coordinates (3, 4).

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If we sketch a straight line through the points, we get the graph of y=3x-5.

Notice that when sketching a straight line through a set of points, the line should not start and end at the points, but continue beyond them, across the entire coordinate plane.

To sketch a straight line graph we actually only need to identify two points.

When checking if a set of points forms a linear relationship, we can choose any two of the points and draw a straight line through them. If the points form a linear relationship then any two points will result in a straight line passing through all the points.

Examples

Example 1

Consider the equation y=2x-4.

Complete the table of values.

x	0	1	2	3
y

Worked Solution

Create a strategy

Substitute each x-value into the given equation.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 2\times0-4	Substitute x=0
	\displaystyle =	\displaystyle -4	Evaluate
\displaystyle y	\displaystyle =	\displaystyle 2\times 1 -4	Substitute x=1
	\displaystyle =	\displaystyle -2	Evaluate
\displaystyle y	\displaystyle =	\displaystyle 2\times 2 -4	Substitute x=2
	\displaystyle =	\displaystyle 0	Evaluate
\displaystyle y	\displaystyle =	\displaystyle 2\times 3 -4	Substitute x=3
	\displaystyle =	\displaystyle 2	Evaluate

x	0	1	2	3
y	-4	-2	0	2

Using the table of values, plot the points that correspond to when x=0 and y=0:

Worked Solution

Apply the idea

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Reflect and check

Notice that both points lie on the axes.

Using the points plotted above, sketch the line that passes through the two points.

Worked Solution

Create a strategy

Connect the plotted points.

Apply the idea

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Idea summary

To plot a point, (a, b), on a number plane, we first identify where x=a lies along the x-axis, and where y=b lies along the y-axis.

Sketch a graph using its intercepts

The word intercept in mathematics refers to a point where a line or curve crosses or intersects with the axes.

We can have x-intercepts: where the line or curve crosses the x-axis.
We can have y-intercepts: where the line or curve crosses the y-axis.

Consider what happens as a point moves up or down along the y-axis. It will eventually reach the origin (0,0) where y=0. Now, if the point moves along the x-axis in either direction, the y-value is still 0.

Similarly, consider what happens as a point moves along the x-axis. It will eventually reach the origin where x=0. Now, if the point moves along the y-axis in either direction, the x-value is still 0.

Exploration

This interactive demonstrates the idea behind the coordinates of x and y-intercepts.

Move the points and notice the coordinates of the points of intercepts.

Loading interactive...

The x-intercept occurs at the point where y=0.

The y-intercept occurs at the point where x=0.

x-intercepts occur when the y-value is 0. So let y=0 and then solve for x to find the x-intercept.

y-intercepts occur when the x-value is 0. So let x=0 and then solve for y to find the y-intercept.

Alternatively we can read the y-intercept value from the equation when it is in the form y=mx+c. The value of c is the value of the y-intercept.

Examples

Example 2

Consider the linear equation y=2x-2.

What are the coordinates of the y-intercept?

Worked Solution

Create a strategy

Substitute x=0 into the given equation.

Apply the idea

\displaystyle y	\displaystyle =	\displaystyle 2\times0-2	Substitute x=0
	\displaystyle =	\displaystyle -2	Evaluate

The coordinates are (0,-2).

What are the coordinates of the x-intercept?

Worked Solution

Create a strategy

Substitute y=0 into the given equation.

Apply the idea

\displaystyle 0	\displaystyle =	\displaystyle 2x-2	Substitute y=0
\displaystyle 2x-2	\displaystyle =	\displaystyle 0	Swap the sides
\displaystyle 2x-2+2	\displaystyle =	\displaystyle 0+2	Add 2 to both sides
\displaystyle 2x	\displaystyle =	\displaystyle 2	Evaluate
\displaystyle \dfrac{2x}{2}	\displaystyle =	\displaystyle \dfrac{2}{2}	Divide both sides by 2
\displaystyle x	\displaystyle =	\displaystyle 1	Evaluate

The coordinates are (1,0).

Now, sketch the line y=2x−2.

Worked Solution

Create a strategy

Plot the points found and draw a line through them.

Apply the idea

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Idea summary

The x-intercept occurs at the point where y=0.

The y-intercept occurs at the point where x=0.

We can use the points of x and y-intercepts to sketch a line.

Sketch a graph using its gradient and one point

We can also graph a line by identifying the gradient and the y-intercept from the equation when it is in the form y=mx+c.

We know that the y-intercept occurs at (0,c), and the gradient is equal to m. Using this information we can plot the point at the y-intercept (or any other point by substituting in a value for x and solving for y) and then move right by 1, and up (or down if m is negative) by m.

As as an example, if we have the equation y=2x+3, then we know the y-intercept is at (0,3) and as the gradient is 2, another point will be at \left(1,\,3+2\right)=\left(1,5\right).

Examples

Example 3

Sketch the line that has a gradient of -3 and an x-intercept of -5.

Worked Solution

Create a strategy

Use the gradient to find another point from the x-intercept.

Apply the idea

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The x-intercept ihas coordinates of (-5,0).

The gradient is -3=\dfrac{-3}{1} so the rise is -3 and the run is 1. We can use this to find another point on the line.

By moving right 1 unit and down 3 units we get to the point (-4,-3).

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So now we can draw our line through these two points.

Example 4

Sketch the line y=-x-5 using the y-intercept and any other point on the line.

Worked Solution

Create a strategy

Use the gradient to find another point from the y-intercept.

Apply the idea

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The y-intercept is at (0,-5).

Since m=-1=\dfrac{-1}{1}, we need to move 1 unit right then 1 unit down from the y-intercept leading us to the point (1,-6).

Now we can connect the points to form the line y=-x-5.

Idea summary

To graph a line in the form of y=mx+c:

Plot the point of y-intercept which is (0,c).
From the y-intercept, move 1 unit to the right and move m units up if positive or down if negative to plot another point.
Connect the plotted points.

Outcomes

VCMNA310

Sketch linear graphs using the coordinates of two points and solve linear equations

5.04 Graphs of straight lines

Ideas

Sketch a graph using a table of values

Examples

Example 1

Sketch a graph using its intercepts

Exploration

Examples

Example 2

Sketch a graph using its gradient and one point

Examples

Example 3

Example 4

Outcomes

VCMNA310

What is Mathspace

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