5.04 Graphs of straight lines
Ideas
Sketch 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.
We'll begin by creating a table of values for the following equation:
y=3x-5
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y |
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).
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.
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 |
Using the table of values, plot the points that correspond to when x=0 and y=0:
Using the points plotted above, sketch the line that passes through the two points.
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.
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?
What are the coordinates of the x-intercept?
Now, sketch the line y=2x−2.
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.
Example 4
Sketch the line y=-x-5 using the y-intercept and any other point on the line.
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.