The Number of Intersection Points


Reviewing Points of Intersection

Recall

A point of intersection of two relations is a point \((x, y)\) that satisfies both relations. On a graph, this is a point where the two relations cross.

Two linear relations can have:

  • No points of intersection if the lines are parallel and distinct

    A graph of two linear relations where one passes through the points (negative 2, negative 1) and (0, 3) and the other passes through the points (0, negative 1) and (2, 3).

  • One point of intersection (in this case, \((0, 3)\)) if the slope of the lines are different

    A graph of two linear relations where one functions passes through the points (negative 2, negative 1) and (0, 3) and another function that passes through the points (0, 3) and (2, 1).

  • Infinite points of intersection if both relations are the same

    The graph of two linear relations but both pass through the point (0, negative 1) and (1, 1)

It is possible to discuss the point(s) of intersection of any two relations, not just lines.


Explore This 1


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Explore This 1 Summary

While exploring the two graphs, you may have noticed that a quadratic relation and a linear relation can have:

  • Exactly two points of intersection.
  • Exactly one point of intersection.
  • No points of intersection.

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Two Points of Intersection

A quadratic relation and a linear relation can have exactly two points of intersection.

 

No Points of Intersection

A quadratic relation and a linear relation can have no points of intersection.

 

 

 

 

One Point of Intersection

A quadratic relation and a linear relation can have exactly one point of intersection.

 

 

 

 

Cases With One Point of Intersection — Case 1

 

Cases With One Point of Intersection — Case 2

A tangent line indicates how steep a curve is at a particular point (or point of tangency). 

In the case of a quadratic relation, a tangent line intersects the curve exactly once. 

 

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Check Your Understanding 1


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Example 1

Are the points \((-2, 5)\) and \((-5, 11)\) the points of intersection of \(y=-2x+1\) and ​​​​​​\(y=4x^2+24x+37\)? 

 

 

Example 1 Continued

Are \((-2, 5)\) and \((-5, 11)\) the points of intersection of the linear relation \(y=-2x+1\) and the quadratic relation \(y=4x^2+24x+37\)? 

Correct!

Correct!

 

 

Example 1 Continued

Are \((-2, 5)\) and \((-5, 11)\) the points of intersection of the linear relation \(y=-2x+1\) and the quadratic relation \(y=4x^2+24x+37\)? 

Correct!

Incorrect

 

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Example 2

Two cylindrical water tanks have exactly the same dimensions – a radius of \(11\) m and a height of \(25\) m.

One of the tanks starts out empty, and then water is added to it at a constant rate. The height of the water in this tank (\(h\), in metres) after \(t\) seconds is \(h=0.03t\).

The other tank starts out full, but water is leaking out of a large hole in the bottom of the tank. The height of the water in this tank is represented by \(h=0.0004t^2-0.2t+25\).

The image of a water tank being drained through a hole in the bottom.

The image of a tank being filled with water from a pipe.

After how many seconds do the two tanks contain the same amount of water?  

Solution

Since the dimensions of the tanks are exactly the same, they contain the same amount of water when the height of the water is the same in both tanks.

These times correspond to the point(s) of intersection of the two relations. One is linear and the other is quadratic.

We can find any points of intersection by graphing. Use a graphing calculator to graph both relations on the same axes.

The quadratic and linear function intersect twice, once between x equals 100 and 150, and once between x equals 400 and 450.

There are two points of intersection, or two points where the line and the parabola cross. However, the tank that is leaking won't start magically filling up again.  We can see from the graph that this tank is totally empty after \(250\) seconds. We only need to consider the point of intersection that occurs before \(250\) seconds.

Using a graphing calculator, we can identify the coordinates of the points of intersection to a high degree of accuracy. The relevant point of intersection is approximately \((145.527, 4.366)\). This means that the amount of water in both tanks is the same when \(t \approx 145.527\) and \(h \approx 4.366\).  

Therefore, the two tanks contain the same amount of water after approximately \(146\) seconds (\(2\) minutes and \(26\) seconds). The height of the water in both tanks at this time is approximately \(4.4\) metres. 

Math in Action

The height of water in a tank that is leaking does not decrease at a constant rate. When there is more water remaining in the tank, the pressure is higher and the water leaks more quickly. When there is less water in the tank, the pressure is lower and the water does not leak as quickly. Thus, the relationship between the height of the water in a leaking tank and time is non-linear (and is, in fact, quadratic).