AustraliaNSW
Stage 5.1-2

# 3.04 Graphs of straight lines

Lesson

## 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=3x5

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=3x5.

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

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 $y$y $1$1 $2$2 $3$3 $4$4 $-2$−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).

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

Plotting points on a number plane

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

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

If we sketch a straight line through the points, we get the graph of $y=3x-5$y=3x5.

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.

Did you know?

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.

## 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.

Intercepts

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

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

### 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

Consider the equation $y=2x-4$y=2x4.

1. Fill in the blanks to complete the table of values.

 $x$x $y$y $0$0 $1$1 $2$2 $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Plot the points that correspond to when $x=0$x=0 and $y=0$y=0:

3. Now, sketch the line that passes through these two points:

##### Question 2

Consider the linear equation $y=2x-2$y=2x2.

1. What are the coordinates of the $y$y-intercept?

Give your answer in the form $\left(a,b\right)$(a,b).

2. What are the coordinates of the $x$x-intercept?

Give your answer in the form $\left(a,b\right)$(a,b).

3. Now, sketch the line $y=2x-2$y=2x2:

##### Question 3

Sketch the line $y=-x-5$y=x5 using the $y$y-intercept and any other point on the line.

##### Question 4

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