Point of Intersection
The point of intersection on a graph is the point where two (or more) lines pass through exactly the same point, as shown in graph 1:
For this post we we will focus on the point of intersection for two lines. At the point of intersection the coordinates of each line are equal. This means both lines will have the same value for y and the same value for x.
Therefore to find the coordinates of a point of intersection we make the two lines equal to to each other, because we know that the y values must be equal at the point of intersection.
For example if we have the following equations:
y = 3x + 2 and y = x + 1
We make them equal to each other: 3x + 2 = x +1
We then rearrange this to find the value of x:
So the x coordinate of the point of intersection is -½. We can now substitute this value of x into either of the equations (they will both give us the same answer) to find the y coordinate:
So the point of intersection for these two lines is (-½ , ½).
Two Straight Lines
Two straight that are not parallel with each other will always meet at one specific point, as shown in graph 1. However, if the lines are parallel they will not have a point of intersection because they never meet:
The same applies for any parallel lines. They don’t just have to be horizontal, they could be vertical or at a slant. This means that any lines with the same gradient do not have a point of intersection.
A Straight Line and a Parabola
When we have a straight line and a parabola we have three possible options for the number of points of intersection:
0 points of intersection:
1 point of intersection:
2 Points of intersection:
As parabolas are quadratic functions when we equate the two lines we usually have to solve a quadratic to get the values of the x coordinate. If we get two x values (i.e. two points of intersection) we have to put them both into one of the original equations to get the two y coordinates.
Worked Examples
Example 1
Find the point of intersection of the two lines below and sketch a graph to show this:
y = ½x + 7 and y = 5x – 2
Example 2
How many points of intersection do the two lines below have and what are their coordinates (if there are any)?
y – x2 = 9x and y = 5x + 12
Example 3
The two lines below have one point of intersection, what are the coordinates of this point?
y = x2 + 6x + 3 and x = 3
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