4.04 Linear graphs

Sketching a graph using a table of values

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.

Exploration

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

$y=3x-5$y=3x−5

The first row of the table will contain values for the independent variable (in this case, $x$x). The choice of $x$x-value is often determined by the context, but in many cases they will be given. To find the corresponding $y$y-value, we substitute each $x$x-value into the equation $y=3x-5$y=3x−5.

$x$x	$1$1	$2$2	$3$3	$4$4
$y$y

Substituting $x=1$x=1:

$y$y	$=$=	$3\times1-5$3×1−5
	$=$=	$3-5$3−5
	$=$=	$-2$−2

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

$x$x	$1$1	$2$2	$3$3	$4$4
$y$y	$-2$−2	$1$1	$4$4	$7$7

 

Plotting points from a table of values

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

 

 

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

 

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

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

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

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

 

 

How do I find an $x$x-intercept?

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

 

How do I find a $y$y-intercept?

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

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

 

Sketching a graph using its gradient and one point

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

We know that the $y$y-intercept occurs at $\left(0,c\right)$(0,c), and the gradient is equal to $m$m. Using this information we can plot the point at the $y$y-intercept (or any other point by substituting in a value for $x$x and solving for $y$y) and then move right by $1$1, and up (or down if $m$m is negative) by $m$m. 
As as an example, if we have the equation $y=2x+3$y=2x+3, then we know the $y$y-intercept is at $\left(0,3\right)$(0,3) and as the gradient is $2$2, another point will be at $\left(1,3+2\right)=\left(1,5\right)$(1,3+2)=(1,5).

 

Practice questions

Question 1

Question 2

Question 3

Question 4

Lines parallel to x and y-axis

 

The line with equation given by $x=a$x=a, with $a$a as any real number, is drawn parallel to the $y$y axis passing through the point $\left(a,0\right)$(a,0) on the $x$x axis.  Three examples, specifically $x=-1$x=−1, $x=1$x=1, and $x=2\sqrt{2}$x=2√2 are shown in the diagram below:

A nice way to think about the line given by the equation $x=a$x=a is to realise that every point on it has the $x$x part of the coordinate address equal to $a$a, and the $y$y part can be any number. Thus points like $\left(a,-3\right),\left(a,0\right),\left(a,5\right),\left(a,23\right)$(a,−3),(a,0),(a,5),(a,23) are all on the line, one directly above the other.  Hence, the line is perpendicular to the $x$x-axis.

In a similar way, the line with equation $y=b$y=b, where $b$b is a constant, is parallel to the $x$x-axis and passes through the point $\left(0,b\right)$(0,b) on the $y$y-axis. The lines $y=-3,y=2$y=−3,y=2 and $y=5$y=5 are shown in the following diagram:

 

Practice questions

QUESTION 5

QUESTION 6