a If we set the functions of a line and a parabola equal to each other, we get a second degree equation. How many solutions can a second degree equation have?
B
b When we set the functions of a line and a parabola equal to each other, we get a second degree equation. How many solutions can a second degree equation have?
C
c When we set the functions of a line and a parabola equal to each other, we get a second degree equation. How many solutions can a second degree equation have?
D
d The second degree equations have to be different.
E
e The second degree equations have to be identical.
F
f Use the fact that a parabola can open upwards or downwards.
G
g Use the fact that a parabola can open upwards or downwards.
A
aExample Diagram:
B
bExample Diagram:
C
c No such graph exists.
D
dExample Diagram:
E
eExample Diagram:
F
fExample Diagram:
G
gExample Diagram:
Practice makes perfect
a Let's start with a line and a parabola that do not intersect.
If we move the line upwards, it will eventually intersect the parabola. If we move it high enough, it will intersect the parabola twice.
b In Part A, we showed that a line and a parabola can have two points of intersections. However, we can also make these functions have just one point of intersection.
c A line and a parabola can at most have two points of intersection. If we equate the functions of a line and a parabola, we will end up with a second degree equation. This type of equation can have zero, one, or two solutions. Therefore, this is not possible.
d When equating the functions of two parabolas, we will get a second degree equation. Like we said in Part C, a second degree equation has either zero, one, or two solutions. Below we see an example of two parabolas with two points of intersection.
e If two parabolas have exactly the same equation, they will show the same graph, and, therefore have an infinite number of intersection points.
f We can draw two parabolas that never intersect by letting them open in different directions. In other words, one parabola opens upward and the other opens downward. If the parabola that opens upward intersects the y-axis above the parabola that opens downward, the parabolas will never intersect.
g From previous parts we know that when we equate the functions of two parabolas, we get a second degree equation which can have zero, one or two solutions as long as its not the same parabola. Below we see an example of two parabolas with one intersection.
